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1.3120863.pdf	Spin-transfer torque driven magnetic antivortex dynamics by sudden excitation of a spin-polarized dc Xiang-Jun Xing and Shu-Wei Li Citation: Journal of Applied Physics 105, 093902 (2009); doi: 10.1063/1.3120863 View online: http://dx.doi.org/10.1063/1.3120863 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/105/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Micromagnetic simulations of spin-wave normal modes and the spin-transfer-torque driven magnetization dynamics of a ferromagnetic cross J. Appl. Phys. 115, 17D123 (2014); 10.1063/1.4863384 Steady-state domain wall motion driven by adiabatic spin-transfer torque with assistance of microwave field Appl. Phys. Lett. 103, 262408 (2013); 10.1063/1.4860455 Bloch-point-mediated magnetic antivortex core reversal triggered by sudden excitation of a suprathreshold spin- polarized current Appl. Phys. Lett. 93, 202507 (2008); 10.1063/1.3033400 Understanding eigenfrequency shifts observed in vortex gyrotropic motions in a magnetic nanodot driven by spin- polarized out-of-plane dc current Appl. Phys. Lett. 93, 182508 (2008); 10.1063/1.3012380 Domain wall motion by spin-polarized current: a micromagnetic study J. Appl. Phys. 95, 7049 (2004); 10.1063/1.1667804 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.254.87.149 On: Fri, 19 Dec 2014 19:03:25Spin-transfer torque driven magnetic antivortex dynamics by sudden excitation of a spin-polarized dc Xiang-Jun Xinga/H20850and Shu-Wei Lib/H20850 State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, China /H20849Received 25 December 2008; accepted 21 March 2009; published online 1 May 2009 /H20850 Spin dynamics of antivortices excited by sudden action of a spin-polarized dc is reported. Two main excitation modes are found with increased current density, involving a translational /H20849gyrotropic /H20850 mode and a core reversal mode. The former mode can be described by Thiele’s equation, whichaccounts for the orbital distortion in view of the modiﬁed restoring force by nontrivial structuresnucleated at sample edges. The ﬁnal states of the system in the translational mode are obtained,being either a domain wall state or a vortex state, depending on the current density. The frequencyof gyromotion is dependent on dot sizes. Within a threshold radius, the off-centered antivortex canfreely relax back to the dot center. © 2009 American Institute of Physics . /H20851DOI: 10.1063/1.3120863 /H20852 I. INTRODUCTION Over past several years, magnetic vortex dynamics1–10 has been the focus of research in magnetism due to its po- tential applications in data storage, sensing, oscillators, andmagnetologics. Antivortices, almost the same frequently ob-served as vortices, are fundamental magnetic structuresformed on submicron-scale ferromagnetic elements. 11–15De- spite numerous researches on vortices,1–9few studies have been made on dynamics of magnetic antivortices,11–15espe- cially for the case wherein the antivortices are excited by aspin-polarized current, 11although the current-induced mag- netic vortex dynamics has been well studied.1–5Wang and Campbell12and Gliga et al.13,14reported spin dynamics of antivortices triggered by magnetic ﬁelds. Depending on ex-citation parameters, they observed three kinds of excitationmodes, i.e., antivortex translational motion, spin-wave exci-tation, and antivortex core /H20849AC/H20850reversal. What is the situa- tion for spin-polarized current-excited antivortices? The an-swer is unknown. This fact stimulates the study. In this work, we performed micromagnetic simulation study on the spin dynamics of single magnetic antivorticesexcited by sudden application of a spin-polarized dc. Weidentify the dynamic excitation spectra of the current-drivenantivortices which, dependent on the magnitude of the cur-rent density, involve two characteristic modes: an AC trans-lational mode and an AC reversal mode. At the initial stageof the translational motion, spin wave, accompanying the ACmotion, is also excited. The translational motion can be de-scribed by Thiele’s equation. 1,6,16The spiral orbital shape evolution characterized by /H9011jis attributed to the enhanced tangential acceleration determined by the increased spin-transfer-torque /H20849STT /H20850-induced force. 1The ﬁnal equilibrium states for the translational mode are obtained, which are ei-ther a domain wall /H20849DW /H20850state or a centered vortex /H20849V ↑/H20850state/H20851Figs. 1/H20849c/H20850and1/H20849d/H20850/H20852. The gyrotropic frequency is found to rely on the sample sizes.6There is a threshold radius for the antivortex relaxation, within which the antivortex can relaxback to the dot center and outside which it is expelled fromthe boundary. II. MICROMAGNETIC SIMULATIONS The single antivortices were stabilized on astroid-shaped Permalloy nanodots12–15with lateral size Lequal to 200 and 300 nm and thickness dranging from 5 to 30 nm. The four short stretching strips had widths of w=/H208494%/H20850Land /H2084910%/H20850L. The antivortex conﬁguration /H20851Fig.1/H20849b/H20850/H20852with core polarity of p=−1 was initially obtained by relaxing the nanodots from a quasiantivortex state /H20849supplemental Fig. 1 of Ref. 11/H20850.T o simulate the current-driven spin dynamics, the modiﬁedLandau-Lifshitz-Gilbert equation 1–3with a STT term17,18was numerically integrated over the nanodot, which was dis-cretized into cells with the size of 2 /H110032/H110032.5 nm 3to per- a/H20850Electronic mail: xxjun@mail2.sysu.edu.cn. b/H20850Author to whom correspondence should be addressed. Electronic mail: stslsw@mail.sysu.edu.cn.DW StateCore Translational Motion (involving Spin-Wave Excitation) V/CID1StateCore Reversal Unknown jc jscj jm (b) (c) (d)(a) FIG. 1. /H20849Color online /H20850/H20849a/H20850Excitation spectra of the spin-polarized current- driven antivortex /H20849on the 200-nm-wide and 10-nm-thick dot /H20850. Top: the ex- citation modes. Bottom: the ﬁnal states formed on the dot for correspondingmodes. j m=0.5/H110031011A/m2is the simulated minimum current density, jsc =1.75/H110031011A/m2is the secondary critical current density, dividing the types of the ﬁnal states in the core translational mode, and jc=3.6 /H110031011A/m2is the critical current density, dividing the types of the exci- tation modes. /H20849b/H20850The initial antivortex /H20849with negative polarity /H20850used in simulations. /H20851/H20849c/H20850and /H20849d/H20850/H20852Final states of the system in the translation mode: /H20849c/H20850DW state and /H20849d/H20850vortex state /H20849with positive polarity, opposite to that of the initial antivortex /H20850.JOURNAL OF APPLIED PHYSICS 105, 093902 /H208492009 /H20850 0021-8979/2009/105 /H208499/H20850/093902/4/$25.00 © 2009 American Institute of Physics 105 , 093902-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.254.87.149 On: Fri, 19 Dec 2014 19:03:25form the fully three-dimensional simulations.19Typical Per- malloy material parameters were used, with saturationmagnetization M s=8.6/H11003105A/m, exchange stiffness A =1.3/H1100310−11J/m, and damping parameter /H9251=0.01. The cur- rent with the spin polarization of /H9257=0.7 /H20849Refs. 4and5/H20850was injected into the nanodot perpendicular to plane, and it wasalso polarized in the same direction. III. RESULTS AND DISCUSSION The antivortex dynamics were calculated systematically with the current density ranging from 0.5 /H110031011to 1.5 /H110031012A/m2with the smallest increments down to 0.5 /H110031010A/m2. The excitation spectra of the current-driven antivortices are shown in Fig. 1/H20849a/H20850. With the increase in the current density, the antivortices experience successively twoexcitation modes. Below the threshold current density j c, the antivortex performs a translational motion, in which the ACcirculates in a spiral orbital around the dot center and thenturns into a curved orbital when it approaches the dot bound-ary. In fact, this mode is accompanied by spin-wave modes atthe initial stage of AC movement. Above j c, the antivortex translational mode is suppressed, and instead the AC reversaloccurs on time scales of /H11011200 ps via two Bloch point nucle- ation, propagation, and annihilation /H20849for details, see Ref. 11/H20850. The ﬁnal equilibrium states /H20851Figs. 1/H20849c/H20850and 1/H20849d/H20850/H20852for the translational mode are existent, which are a DW state for j lower than the secondary threshold valve j scand a vortex state with positive polarity /H20849V↑/H20850centered in the dot for j higher than jsc. As an example, the vortex with clockwise chirality is shown in Fig. 1/H20849d/H20850; actually the anticlockwise chiral direction is also observed in simulations. However,regarding the AC reversal mode, the ﬁnal states are notreached yet /H20849for details, see Ref. 11/H20850. Figures 2/H20849a/H20850–2/H20849h/H20850represent AC trajectories in the trans- lational mode for the antivortex situated in the 200-nm-wideand 10-nm-thick dot under different current densities. All ofthe trajectories include a spiral orbital and a subsequentcurved orbital. To describe the antivortex translational mo-tion, well-known Thiele’s equation 16is introduced. Here, Thiele’s equation has the form1,6 FS+FG+FR+F/H9251=0 , /H208491/H20850 where FSis the STT-induced force, FGis the gyroforce, FRis the restoring force, and F/H9251is the dissipative force. For the antivortex centered in the astroid-shaped dot, because theproﬁle of magnetization m=M/M shas not been established thus far, the precise expressions of these forces cannot bededuced. Nevertheless, an approximation can be made basedon comparison to the vortex case. 1,6,20In this scenario, the STT-induced force drives the AC to quit the dot center if thecurrent exceeds a critical value /H20849so that F S/H11022F/H9251/H20850.2,10The dis- placed AC is then subjected to a gyroforce and a restoringforce, which are perpendicular to the AC velocity and thusresult in the gyrotropic nature of the translationalmotion. 2,5,10With higher excitation current, the spiral orbital becomes increasingly ﬂat, owing to enhancement of the tan-gential acceleration provided by the STT-induced force. Theﬂatness of a spiral orbital can be characterized by a quantity/H9011 j=/H208491//H9270/H20850/H20885 0/H9270 /H20849R//H9021/H20850dt, /H208492/H20850 where R=/H20841R/H20849t/H20850/H20841, with Ra vector pointing from the dot center to the displaced AC position, /H9021=/H9021/H20849t/H20850is the angle swept by R, and/H9270=min /H20853/H9270j;jm/H11349j/H11021jc/H20854with/H9270jas the time of the AC gyromotion under the current j. Smaller /H9011jvalues describe more circulated orbitals, while larger values give ﬂatter ones.As the antivortex moves outward, there exhibit nontrivialstructures at the edges, especially for larger applied currents.The creation of these edge structures /H20849a vortex for higher j/H20850 should be attributed to the metastability of antivortices. 12–14 A similar situation was found by Gliga et al.13,14who ob- served vortex nucleation at the edges when stronger mag-netic ﬁeld pulses were employed. The presence of thesestructures modiﬁes the restoring force. This modiﬁcation fur-05 1 0 1 5-0.40.00.40.8 012345 <mx> <my><mx>&< my> Time (ns)(i) f (GHz)Amplitude0.465 GHz FFT of <mx> 0 5 10 15-0.10.00.10.20.3 51 0 1 5 2 0<mz> Time (ns)(j) 20.4 15.7Amplitude f( G H z )6.8 GHz 12.4FaFGFSFR(a) (b) (c) (d) (e) (f) (g) (h) FIG. 2. /H20849Color online /H20850/H20851 /H20849a/H20850–/H20849h/H20850/H20852Orbital trajectories /H20849Ref. 21/H20850of an antivor- tex /H20849on the 200-nm-wide and 10-nm-thick dot /H20850under different current den- sities: /H20849a/H20850j=0.5/H110031011,/H20849b/H208500.8/H110031011,/H20849c/H208501.0/H110031011,/H20849d/H208501.75/H110031011,/H20849e/H20850 2.5/H110031011,/H20849f/H208503.25/H110031011,/H20849g/H208503.5/H110031011, and /H20849h/H208503.55/H110031011A/m2. The red rhombus and wine circle in /H20849a/H20850denote the positions from which the antivor- tex is relaxed. The arrows in /H20849b/H20850represent the forces exerted on the antivor- tex core. /H20851/H20849i/H20850and /H20849j/H20850/H20852Temporal evolution of the averaged magnetization for the in-plane /H20851/H20849i/H20850j=0.5/H110031011A/m2/H20852and out-of-plane /H20851/H20849j/H20850j=1.0 /H110031011A/m2/H20852components. The insets show the corresponding FFT spectra of the magnetization evolution on the left side of the dotted lines.093902-2 X.-J. Xing and S.-W. Li J. Appl. Phys. 105 , 093902 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.254.87.149 On: Fri, 19 Dec 2014 19:03:25ther breaks the previous force balance between FS,FG,FR, and F/H9251and ﬁnally results in the AC entering into the curved orbital. Steady-state AC motions5,6,10around a circular limit cycle is not excited3throughout the studied current densities. The temporal evolution of the in-plane magnetization components /H20849/H20855mx/H20856and /H20855my/H20856/H20850for the antivortex under j=0.5 /H110031011A/m2is shown in Fig. 2/H20849i/H20850. The sinusoidal-like os- cillations characterize the spiral orbital. The enhanced ampli-tude results from the continually injected energy by thecurrent. 1The frequency /H20849f/H20850of the gyromotion is identiﬁed through fast Fourier transform /H20849FFT /H20850, the /H20855mx/H20856oscillating curve, being /H110110.465 GHz irrelevant to the current density. The later sharp jumps of /H20855mx/H20856and /H20855my/H20856correspond to the AC moving along the curved orbital. Correspondingly, the out- of-plane magnetization /H20849/H20855mz/H20856/H20850evolution is shown in Fig. 2/H20849j/H20850. Using FFT technique, several additional modes with flo- cated around 6.8, 12.4, 15.7, and 20.4 GHz are found, asshown in the inset of Fig. 2/H20849j/H20850, indicating spin-wave excita- tion. At the end of the movement, the antivortex is sub-merged, and subsequently, the ﬁnal state of the system isattained. Below j sc, it is ejected out of the dot, resulting in a DW conﬁguration. Above jsc, it is annihilated with a vortex with opposite polarity originating from the dot edge. Afterthe annihilation, another vortex with positive polarity nucle-ates at the edge and then moves toward the dot center, form-ing the ﬁnal state of the system. The production of a vortexhaving opposite polarity from an original antivortex throughsuch a process is a presently unexplored micromagnetic pro-cess. In vortex translational mode, the gyrofrequency was ap- proximately proportional to the dot aspect ratio /H9252=dv/Lv,6 where dvand Lvare the thickness and radius of a submicron cylindrical dot, respectively. For our astroid-shaped dot, thegeometric parameters are the lateral size L, the thickness d, and the strip width w, and so far the dependence of its gy- rofrequency on these parameters is not fully realized. Toclarify this, we simulated different sized nanodots that aresubjected to the identical current density /H20849j=0.8 /H1100310 11A/m2/H20850. Table Isummarizes the simulated results, ex- hibiting that the frequency is closely related to the sample sizes. When the dot thickness /H20849d/H20850is increased or its lateral sizes /H20849Land w/H20850are decreased, the gyrofrequency rises.12 Thus, the size dependence of the gyrofrequency of antivorti- ces on astroid-shaped dots is in qualitative agreement withthat for vortices on cylindrical dots. 1,4–6,8,12 Finally, the relaxation properties of the off-centered an- tivortex were investigated. Various positions /H20851marked by red rhombus and wine circle in Fig. 2/H20849a/H20850/H20852were examined as the starting point of the antivortex relaxation. When setting outfrom the circle-denoted position, the antivortex rotates in awell-deﬁned spiral orbital /H20851Fig.3/H20849a/H20850/H20852, returning slowly to thedot center. In this process, the AC motion is governed by the restoring force F R, the gyroforce FG, and the dissipative force F/H9251. The temporal evolution of /H20855mx/H20856and /H20855my/H20856during the antivortex relaxation is shown in Fig. 3/H20849b/H20850. The slow decay arises from the weak dissipative force determined by thedamping coefﬁcient /H20849 /H9251=0.01, characteristic of Permalloy material /H20850. The inset in Fig. 3/H20849b/H20850shows the FFT spectra of /H20855mx/H20856. Compared to its counterpart in Fig. 2/H20849i/H20850, one ﬁnds that the gyrofrquency of the AC free motion is approximately equal to that of the AC forced motion by the currents butwith a much narrower linewidth. When leaving from therhombus-denoted position, the antivortex moves outwardalong a curved orbital /H20851Fig. 3/H20849a/H20850/H20852and ﬁnally is ejected from the boundary, leading to a DW state. This is very similar tothe AC late-staged motion under the currents of j/H11021j sc, at- testing again that the AC motion in the vicinity of the bound-ary is dominated by the dipolar force, which is now no morea restoring force toward the dot center. Conclusively, there isa threshold radius R cin the astroid-shaped dot, and only the AC situated inside the periphery of 2 /H9266Rccan relax back to the dot center. IV. CONCLUSION In conclusion, we studied spin dynamics of the magnetic antivortices triggered by sudden excitation of a perpendicularspin-polarized dc. The excitation spectra of the current-excited antivortices are derived, which involve several char-TABLE I. The frequency of the antivortex gyromotion in various sized dots under a current with j=0.8/H110031011A/m2. L,d,w /H20849nm/H20850 200, 10, /H208494%/H20850L300, 10, /H208494%/H20850L200, 5, /H208494%/H20850L200, 10, /H2084910% /H20850L f/H20849GHz /H20850 0.465 0.255 0.322 0.456 0 2 04 06 08 0-0.3-0.2-0.10.00.10.20.3 012345<mx>&< my> Time (ns)<mx> <my>(b) Amplitude f (GHz)FFT of <mx>0.475 GHz(a) FIG. 3. /H20849Color online /H20850/H20849a/H20850Trajectories /H20849Ref. 21/H20850of an AC /H20849on the 200-nm- wide and 10-nm-thick dot /H20850relaxing from two different starting points. The symbols marking the starting positions are the same as in Fig. 2/H20849a/H20850.T h e green circle deﬁnes the periphery with the threshold radius Rc/H1101530 nm /H20849Ref. 22/H20850./H20849b/H20850Time evolution of /H20855mx/H20856and /H20855my/H20856magnetization components. The inset is the FFT spectra of /H20855mx/H20856.093902-3 X.-J. Xing and S.-W. Li J. Appl. Phys. 105 , 093902 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.254.87.149 On: Fri, 19 Dec 2014 19:03:25acteristic excitation modes. Under low current densities, the antivortex is in a translational mode, in which the AC ini-tially revolves around the dot center in a spiral orbital, laterenters a curved orbital, and ﬁnally is quenched near theboundary region, resulting in the equilibrium state of thesystem. Under high current densities, the antivortex is in acore reversal mode, in which the AC polarity is switched at atypical time scale of 200 ps. /H20849This mode was detailedly re- ported in Ref. 11./H20850The antivortex gyrofrequency, like in the vortex case, is revealed to be dependent on the dot sizes. Theantivortex situated within a threshold radius can freely relaxback to the dot center. ACKNOWLEDGMENTS The National Natural Science Foundation of China /H20849Grant Nos. 50572124 and U0734004 /H20850funded this work. 1Y . Liu, H. He, and Z. Zhang, Appl. Phys. Lett. 91, 242501 /H208492007 /H20850. 2D. D. Sheka, Y . Gaididei, and F. G. Mertens, Appl. Phys. Lett. 91, 082509 /H208492007 /H20850. 3Y .-S. Choi, S.-K. Kim, K.-S. Lee, and Y .-S. Yu, Appl. Phys. Lett. 93, 182508 /H208492008 /H20850. 4Y . Liu, S. Gliga, R. Hertel, and C. M. Schneider, Appl. Phys. Lett. 91, 112501 /H208492007 /H20850. 5S. Kasai, Y . Nakatani, K. Kobayashi, H. Kohno, and T. Ono, Phys. Rev. Lett. 97, 107204 /H208492006 /H20850. 6K. Y . Guslienko, B. A. Ivanov, V . Novosad, Y . Otani, H. Shima, and K. Fukamichi, J. Appl. Phys. 91, 8037 /H208492002 /H20850.7Y . Gaididei, D. D. Sheka, and F. G. Mertens, Appl. Phys. Lett. 92, 012503 /H208492008 /H20850. 8R. Hertel, S. Gliga, M. Fähnle, and C. M. Schneider, Phys. Rev. Lett. 98, 117201 /H208492007 /H20850. 9M. Buess, R. Höllinger, T. Haug, K. Perzlmaier, U. Krey, D. Pescia, M. R. Scheinfein, D. Weiss, and C. H. Back, Phys. Rev. Lett. 93, 077207 /H208492004 /H20850. 10D. D. Sheka, Y . Gaididei, and F. G. Mertens, in Electromagnetic, Magne- tostatic, and Exchange-Interaction Vortices in Conﬁned Magnetic Struc-tures , edited by E. O. Kamenetskii /H20849Research Signpost, Kerala, 2008 /H20850,p p . 59–75. 11X. J. Xing, Y . P. Yu, S. X. Wu, L. M. Xu, and S. W. Li, Appl. Phys. Lett. 93, 202507 /H208492008 /H20850. 12H. Wang and C. E. Campbell, Phys. Rev. B 76, 220407 /H20849R/H20850/H208492007 /H20850. 13S. Gliga, M. Yan, R. Hertel, and C. M. Schneider, Phys. Rev. B 77, 060404 /H20849R/H20850/H208492008 /H20850. 14S. Gliga, R. Hertel, and C. M. Schneider, J. Appl. Phys. 103, 07B115 /H208492008 /H20850. 15A. Drews, B. Krüger, M. Bolte, and G. Meier, Phys. Rev. B 77, 094413 /H208492008 /H20850. 16A. A. Thiele, Phys. Rev. Lett. 30, 230 /H208491973 /H20850. 17J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 18L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 19M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, NIS- TIR 6376, National Institute of Standards and Technology, Gaithersburg,MD, September 1999 /H20849http://math.nist.gov/oommf/ /H20850. 20N. A. Usov and S. E. Peschany, J. Magn. Magn. Mater. 118, L290 /H208491993 /H20850. 21The position of the AC is determined by the point in which the zcompo- nent of the magnetization has the smallest value /H20849/H20855mz/H20856/H11011−1/H20850over the dot area. 22The existence of the threshold radius has been elucidated in the text, and its value of /H1101130 nm is deduced from simulation results. The intrinsic relation between the quantitative values of 30 nm and 200 nm/10 nm /H20849dot sizes /H20850cannot be clariﬁed at present.093902-4 X.-J. Xing and S.-W. Li J. Appl. Phys. 105 , 093902 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.254.87.149 On: Fri, 19 Dec 2014 19:03:25
1.4813763.pdf	Interfacial effect on the ferromagnetic damping of CoFeB thin films with different under-layers Shaohai Chen, Minghong Tang, Zongzhi Zhang, B. Ma, S. T. Lou et al. Citation: Appl. Phys. Lett. 103, 032402 (2013); doi: 10.1063/1.4813763 View online: http://dx.doi.org/10.1063/1.4813763 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v103/i3 Published by the AIP Publishing LLC. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 22 Jul 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsInterfacial effect on the ferromagnetic damping of CoFeB thin films with different under-layers Shaohai Chen,1Minghong Tang,1Zongzhi Zhang,1,a)B. Ma,1S. T. Lou,2and Q. Y . Jin2 1Key Lab of Micro and Nano Photonic Structures (Ministry of Education), Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China 2State Key Laboratory of Precision Spectroscopy and Department of Physics, East China Normal University, Shanghai 200062, China (Received 3 June 2013; accepted 27 June 2013; published online 15 July 2013) Interfacial effects on magnetic properties are investigated for the as-deposited and annealed Co64Fe16B20ﬁlms with different under-layers (Cu, Ru , or Pd). The intrinsic Gilbert damping factor is inferred to be slightly lower than the obtained value of 0.007. We found that both the in- plane coercivity Hcand ferromagnetic resonance linewidth DHpprely on the interfacial morphology. The Cu under-layer provides a rough surface, which offers an extra contribution totheDH pp. The surface roughness was greatly enhanced by post-annealing for Cu, while little affected for Ru and Pd. Resultingly, the DHppandHcof Cu/CoFeB increase s igniﬁcantly after annealing. However, for the annealed Ru/CoFeB sample, the DHppeven decreases implying Ru is a proper under-layer material for CoFeB-based spintronic devices. VC2013 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4813763 ] In recent years, soft CoFeB thin ﬁlms have been exten- sively studied owing to the extremely large tunneling magne-toresistance (TMR) ratio and the spin transfer torque (STT) phenomenon in CoFeB-MgO-CoFeB magnetic tunnel junc- tions (MTJs). 1–5Spin polarized current ﬂowing through a spin valve or MTJ element will exert a spin torque on the free layer magnetization, which can drive the magnetization switching if the current exceeds a threshold value. The spintorque induced magnetization reversal provides a new data writing method which can be used in magnetic random access memories (MRAM). 5,6One of the key challenges for STT-MRAM is to reduce the critical writing current density while maintaining enough thermal stability for achieving better performance such as low power consumption, gooddata retention, and high information density. 6–8The magni- tude of threshold current depends on the free layer magnet- ization, ﬁlm thickness, spin polarization, anisotropy ﬁeld,and especially the effective magnetic damping factor a. Understanding and manipulation of the dynamic magnetic properties of the CoFeB ﬁlms is of vital importance to effec-tively reduce the critical switching current for memory application. The magnetic damping makes the magnetization relax to the local equilibrium state by dissipating the magnetic energy via the damped magnetization oscillations. It has been found that there are different contributions to theenergy dissipation processes. In addition to the intrinsic Gilbert damping resulting from the spin-orbit coupling of the ferromagnetic materials, the local ﬂuctuation of magnetiza-tion, magnetic anisotropy, as well as the magnetostatic ﬁelds at different sample locations would also give rise to some extrinsic contributions to the magnetic damping. These ex-trinsic damping caused by the inhomogeneous magnetic properties is usually termed as two-magnon scattering, whichstrongly relies on the surface/interface roughness and other ﬁlm defects. 9–11Moreover, the coupling between a ferro- magnetic layer and an adjacent normal non-magnetic (NM) metal layer may also enhance the effective damping for the precessing magnetization via spin-pumping effect.12,13 For the MTJs containing a CoFeB layer, post annealing treatment2–4and proper under-layer14,15are required to achieve perpendicular magnetic anisotropy and high TMRsignal essential for developing high density STT-MRAM. However, the introduction of non-magnetic surrounding layer and thermal heating will vary the ﬁlm microstructure,which would inevitably affect the avalue of the free CoFeB layer in the spin torque devices. Therefore, understanding of the interfacial and annealing effects on the damping factor isessential for the successful adaptation of the spin-transfer writing scheme for advanced MRAM. In this work, we have performed a comprehensive study on the CoFeB ﬁlms. The ﬁlms were deposited at room temperature (RT) in a magnetron sputtering system with a base pressure better than 1/C210 /C08Torr.16The sample structure is glass substrate/Ta (3 nm)/under-layer (5 or 10 nm)/Co 64Fe16B20(5 or 15 nm)/ Ta (3 nm), where Cu, Pd, or Ru was chosen as the under- layer. The respective Ar working pressure is 3 mTorr for Cuand CoFeB, 5 mTorr for Ru, and 6 mTorr for Pd and Ta. After deposition, each sample was cut into two pieces, one was subjected to heat treatment at 350 /C14C for an hour in a vacuum chamber without applying any magnetic ﬁeld. Meanwhile, samples of glass/Ta (3 nm)/under-layer (5 nm) capped with a 1 nm thin Ta layer were also prepared for sur-face morphology analyses. Magnetic hysteresis loops were measured by a vibrating sample magnetometer (VSM) with the external ﬁeld parallel to the ﬁlm plane. The ferromag-netic resonance (FMR) measurements were carried out by a JEOL JESFA-300 electron spin-resonance spectrometer, with a ﬁxed microwave frequency of 9.0 GHz and swept dcﬁeld. The surface roughness and crystallographic texture were examined by Atomic Force Microscope (AFM) and a)Author to whom correspondence should be addressed. Electronic mail: zzzhang@fudan.edu.cn 0003-6951/2013/103(3)/032402/5/$30.00 VC2013 AIP Publishing LLC 103, 032402-1APPLIED PHYSICS LETTERS 103, 032402 (2013) Downloaded 22 Jul 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsX-ray diffraction (XRD) with Cu-K aradiation ( k¼1.54 A ˚), respectively. All the measurements were conducted at RT. All the CoFeB samples with different under-layers have very similar XRD results, so only the XRD patterns of Cu/CoFeB are presented in Fig. 1. In the as-deposited state, there is only one peak corresponding to the Cu (111) texture, demonstrating that the as-deposited CoFeB layer is in anamorphous state or very ﬁne nanocrystalline state. After 1 h heating at 350 /C14C, although no additional peak is observable for the sample with a thin CoFeB (5 nm thick) layer, we could identify a clear CoFe (110) peak for the 15 nm thick CoFeB, implying the amorphous CoFeB layer crystallizes af-ter annealing treatment. The ﬁeld-swept FMR absorption derivative spectra were recorded at various ﬁeld angles h H(0/C14–90/C14)r e l a t i v et ot h e ﬁlm normal direction. The angular dependence of resonance ﬁeld Hresis shown in Fig. 2(a), for the as-deposited samples with a structure of Ta 3 nm/under-layer 5 nm/CoFeB 5 nm/Ta3 nm. A typical in-plane FMR spectrum with the external magnetic ﬁeld applied parallel to the ﬁlm plane (i.e., h H¼90/C14) is given in the inset, from which the resonance Hres and peak-to-peak linewidth DHppcan be determined. The res- onance ﬁeld Hresdecreases monotonically with increasing hH from 0/C14to 90/C14due to the co-effect of large demagnetization ﬁeld and the in-plane magnetic anisotropy of our samples. The rapid decrease of Hresat low angle reveals the misalign- ment between the static dc ﬁeld and the magnetization. Aboveh H¼30/C14, the curve gradually turns to ﬂat showing the mag- netization is already in-plane.9Considering the amorphous na- ture of the CoFeB layer, the in-plane anisotropy ﬁeld Hkcan be neglected. Therefore, from the in-plane FMR ﬁeld and the well-known Kittel formula17 x¼cﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðHresþHkÞðHresþHkþ4pMsÞp ; (1) where x¼2pfis the microwave circular frequency and cis the gyromagnetic ratio, we get the saturation magnetizationMsfor CoFeB is about 1180 66 emu/cm3, which is in good agreement with our VSM result (1176 630 emu/cm3). Figure 2(b) shows the FMR linewidth DHppas a func- tion of angle hHbetween the external ﬁeld and the ﬁlm nor- mal, for the as-deposited samples in the structure of Ta 3 nm/ Cu, Pd, or Ru 5 nm/CoFeB 5 nm/Ta 3 nm. The DHppshows a very signiﬁcant angular variation. It increases very rapidly asthe external FMR ﬁeld is rotated away from the perpendicu- lar orientation at h H¼0/C14, and then decreases after reaching a maximum at hH¼6/C14. For a homogeneous ﬁlm with magnet- ization aligned parallel to the applied ﬁeld, the intrinsic damping factor acan be simply determined from the FMR linewidth according to the relation of DHpp¼2ax=ﬃﬃﬃ 3p c.18 However, for the practical magnetic thin ﬁlms generally with magnetic inhomogeneities, the DHppwould be broadened by two-magnon scattering which can transfer magnetic energy from the uniform precession to the degenerate spin wave FIG. 1. Typical XRD patterns for the as-deposited and post-annealed CoFeB samples with a Cu under-layer.FIG. 2. (a) The FMR resonance ﬁeld as a function of ﬁeld angle hHwith respect to the ﬁlm normal direction measured in the as-deposited state. The inset shows a representative FMR absorption derivative spectrum. (b) The peak-to-peak linewidth DHppversus ﬁeld angle hH. (c) The in-plane (hH¼90/C14) and perpendicular ( hH¼0/C14)DHppfor the samples of Ta 3 nm/ under-layer 5 nm/CoFeB 5 nm/Ta 3 nm.032402-2 Chen et al. Appl. Phys. Lett. 103, 032402 (2013) Downloaded 22 Jul 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsstates. The two-magnon broadening is ineffective when the magnetization is nearly perpendicular to the ﬁlm plane,9,19 hence the in-plane linewidth would be greater than the per- pendicular linewidth at the presence of extrinsic two-magnon scattering. Figure 2(c)shows the in-plane linewidth ath H¼90/C14and the perpendicular at hH¼0/C14. Obviously, the perpendicular linewidth is smaller than the in-plane value,conﬁrming the contribution of two-magnon scattering. Importantly, when compared to the other ﬁlms, the DH pp reduction is more signiﬁcant for the sample with a Cu under- layer, indicating severe magnetic inhomogeneity occurs at the Cu/CoFeB interface. It is also noticed that, at the perpen-dicular conﬁguration where the two-magnon broadening is suppressed, the DH ppis not identical for the CoFeB samples with different under-layers. The sample with Ru or Pdunder-layer exhibits a higher perpendicular DH pp, which is attributed to the spin-pumping effect. It is known that in order to observe a signiﬁcant spin pumping contribution tothe magnetic damping, the spin diffusion length k SDin the normal metal layer should be smaller than or comparable to its thickness LNM.20,21The Ru or Pd has a short spin diffu- sion length, the measured kSDis only 14 nm for Ru (Ref. 22) and 25 nm for Pd (Ref. 23) at 4.2 K, which is usually longer than that at RT owing to the weak electron-phonon scatteringat low temperatures. The k SDcould decrease by at least a factor of two between 4.2 K and RT. In our work, since the under-layer thickness LNMis 5 or 10 nm, the criterion of kSD/C20Lfor spin pumping broadening is basically satisﬁed. In contrast, for the sample with a 5 nm Cu under-layer, the thickness is far smaller than its long spin diffusion length(/C24350 nm) at room temperature, 24thus, the non-local spin pumping contribution to the linewidth is negligible, leading to the observed smaller perpendicular DHpp. Based on the above analyses, the total measured reso- nance linewidth of our samples is typically composed of three contributions: the intrinsic Gilbert damping, two-magnon scat-tering, and spin pumping effect. The perpendicular DH pp measured for the sample with Cu under-layer is very close to the intrinsic FMR linewidth since the two-magnon scatteringand spin pumping effect has been excluded. So, according to the formula of DH pp¼2ax=ﬃﬃﬃ 3p c, the magnetic damping fac- torais calculated to be 0.007. We point out that the actual intrinsic avalue should be slightly smaller than 0.007, as the thin Ta buffer and capping layers (with a short spin diffusion length of about 10 nm at RT) may also contribute to theincrease of damping. 13,27The obtained damping factor value is a little different from the results reported for CoFeB system by others.13,25,26This difference can be attributed to the differ- ent elemental compositions. In Refs. 13,25,a n d 26,t h em e a s - ured damping factors are for the alloys of Co 56Fe24B20, Co40Fe40B20, and Co 31.5Fe58.5B10, respectively, while in this work the obtained value of 0.007 is for the Co 64Fe16B20. The FMR linewidth broadening due to two-magnon scattering is known to originate from the microstructuralimperfections, which will affect the static magnetic proper- ties such as magnetic coercivity as well. Figure 3shows the in-plane magnetic hysteresis loops of the as-deposited andpost-annealed samples. Similar to the DH pp, the coercivity Hcvalue measured in as-deposited state is also very small, only 3–10 Oe, seen from the loops in Figs. 3(a)–3(c). Thelow coercivity and saturation ﬁeld reveal the CoFeB layer is magnetically soft and the magnetic easy axis lies in the ﬁlmplane. Thermal annealing will change the CoFeB microstruc- ture and hence the magnetic properties, as shown in Figs. 3(d)–3(f). After post-annealing, the coercivity value H c increases for all the samples, especially for the Cu/CoFeB sample. The increase of Hcprobably arises from the CoFe crystallization, interface roughness increase, and/or atomicinterdiffusion. In order to clarify the relationship between the coerciv- ity and the FMR linewidth, Fig. 4displays the in-plane DH pp andHcvalues for the samples with both 5 and 10 nm thick under-layers. Apparently, the Hcfollows a very similar vari- ation trend to the DHpp, implying the increase of FMR line- width and coercivity is more likely related to the same origin. Moreover, although the post-annealing treatment in our experiment was performed under the identical condition,the increase of H candDHppinduced by thermal heating is distinctly different for the different under-layer samples. For the as-deposited Cu 5 nm/CoFeB sample, the in-plane Hc andDHppare only 9.4 and 56 Oe, respectively, which dra- matically rises up to 322 and 623 Oe after annealing. As a comparison, the increase of HcandDHppis intermediate for the case of Pd. Nevertheless, for the Ru 5 nm/CoFeB sample, thermal annealing did not produce an increase of DHpp.O n the contrary, it even decreases from the as-deposited 47 Oedown to 35 Oe. We consider that such different varying tend- ency probably originates from the different interfacial micro- structure between the CoFeB and the various under-layers. To explain the experimental results that the two-magnon broadening of DH ppis more pronounced for the sample with a Cu under-layer, and gain further insight on the annealingFIG. 3. Magnetic hysteresis loops for samples of Ta 3 nm/under-layer 5 nm/ CoFeB 5 nm/Ta 3 nm measured in the as-deposited state ((a)–(c)) and post- annealed state ((d)–(f)).032402-3 Chen et al. Appl. Phys. Lett. 103, 032402 (2013) Downloaded 22 Jul 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionseffect, AFM scanning was conducted in an atmosphere envi- ronment on the sample surface with a structure of Ta 3.0 nm/Cu, Pd, or Ru 5 nm/Ta 1 nm. The root-mean-square (RMS) roughness was determined from the AFM images shown in Fig.5. In the as-deposited state, the RMS value is low, which is 0.31, 0.25, and 0.41 for Ru, Pd, and Cu, respectively. Roughness will bring ﬂuctuations of the static stray ﬁeld, which correspond to the local shape of the interface and thusresult in DH ppbroadening. The relatively rough surface of the as-deposited Cu ﬁlm is responsible for the observed slightly higher in-plane DHpp. For the post-annealed Cu sam- ple, because of the formation of much more large grains seen from the island style image, the RMS increases considerably which introduces great inhomogeneities at the Cu/CoFeBinterface. Consequently, both the DH ppandHcshow a dra- matic enhancement. The surface roughness of the Pd or Ru ﬁlm is not so sensitive to the annealing treatment as the Cu.The RMS basically does not vary much for the annealed Pd sample, so there must be other factors related to the increase ofDH ppandHc. Considering that thermal heating can cause the interfacial diffusion and atomic alloying,28we ascribe the corresponding moderate DHppincrease to the formation of CoPd or FePd alloy which has strong spin-orbit coupling.For the annealed Ru/CoFeB sample, the observed decrease ofDH ppis considered as a result of the slightly reduced sur- face roughness. In conclusion, we have studied the magnetic properties and microstructure for CoFeB thin ﬁlms with various under- layers. The in-plane magnetic CoFeB layers, which areamorphous in the as-deposited state, become crystallized af- ter annealing at 350 /C14C. The intrinsic damping factor ais determined to be close to 0.007. Before annealing treatment,the extrinsic contribution to magnetic damping mainly results from the spin-pumping effect for the sample with a Pd or Ru under-layer due to the short spin diffusion length,whereas for Cu/CoFeB ﬁlm the two-magnon scattering plays a dominant role because of the slightly rough Cu surface. In the post-annealed state, the sample with a Cu under-layershows a pronounced increase of both in-plane DH ppandHc due to the greatly increased surface roughness. Although no obvious change can be detected on the Pd surface, the DHpp and Hcrise twice for the annealed Pd/CoFeB, which is ascribed to the formation of CoPd or FePd alloy. Interestingly, the DHppof the Ru/CoFeB sample drops in the annealed state as a consequence of the slight reduction of the interfacial roughness. These results indicate that the Ru is an appropriate adjacent layer material for CoFeB ﬁlms for pos-sible applications in the STT-MRAM. This work was supported by the National Natural Science Foundation of China (Grant Nos. 51222103, 11274113,11074046, 51171047, and 61078030), and the National Basic Research Program of China (2009CB929201). Z. Zhang thanks for the support from the Program for New CenturyExcellent Talents in University (NCET-12-0132). B. Ma thanks for the support from the NSFC (Grant Nos. 51071046 and 11174056). 1D. Djayaprawira, K. Tsunekawa, M. Nagai, H. Maehara, S. Yamagata, N. Watanabe, S. Yuasa, Y. Suzuki, and K. Ando, Appl. Phys. Lett. 86, 092502 (2005). 2S. Ikeda, J. Hayakawa, Y. Ashizawa, Y. M. Lee, K. Miura, H. Hasegawa, M. Tsunoda, F. Matsukura, and H. Ohno, Appl. Phys. Lett. 93, 082508 (2008).FIG. 4. The coercivity Hcand in-plane DHppfor samples with various under-layers measured in the as-deposited state ((a), (b)) and post-annealedstate ((c), (d)). FIG. 5. 3D AFM images measured on the surface of ﬁlms with a structure of glass/Ta 3 nm/Ru, Pd, and Cu 5 nm/Ta 1 nm. The images correspond to a 1lm/C21lm sample area, and the black to white color contrast scales z-axis from 0 to 1.5 nm.032402-4 Chen et al. Appl. Phys. Lett. 103, 032402 (2013) Downloaded 22 Jul 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions3P. K. Amiri, Z. M. Zeng, J. Langer, H. Zhao, G. Rowlands, Y. J. Chen, I. N. Krivorotov, J. P. Wang, H. W. Jiang, J. A. Katine, Y. Huai, K. Galatsis, and K. L. Wang, Appl. Phys. Lett. 98, 112507 (2011). 4W. G. Wang, S. Hageman, M. Li, S. Huang, X. Kou, X. Fan, J. Q. Xiao, and C. L. Chien, Appl. Phys. Lett. 99, 102502 (2011). 5S. Ikeda, K. Miura, H. 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1.4867298.pdf	Control of the magnetic in-plane anisotropy in off-stoichiometric NiMnSb F. Gerhard, C. Schumacher, C. Gould, and L. W. Molenkamp Citation: Journal of Applied Physics 115, 094505 (2014); doi: 10.1063/1.4867298 View online: http://dx.doi.org/10.1063/1.4867298 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Stress dependence of ferromagnetic resonance and magnetic anisotropy in a thin NiMnSb film on InP(001) Appl. Phys. Lett. 89, 242505 (2006); 10.1063/1.2405885 Structural and magnetic properties of NiMnSb/InGaAs/InP(001) J. Appl. Phys. 97, 073906 (2005); 10.1063/1.1873036 Magnetic properties of NiMnSb(001) films grown on InGaAs/InP(001) J. Appl. Phys. 95, 7462 (2004); 10.1063/1.1687274 Epitaxial NiMnSb films on GaAs(001) Appl. Phys. Lett. 77, 4190 (2000); 10.1063/1.1334356 Optical spectroscopy investigations of half metallic ferromagnetic Heusler alloy thin films: PtMnSb, NiMnSb, and CuMnSb J. Appl. Phys. 81, 4164 (1997); 10.1063/1.365167 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 209.183.185.254 On: Thu, 27 Nov 2014 19:38:20Control of the magnetic in-plane anisotropy in off-stoichiometric NiMnSb F . Gerhard, C. Schumacher, C. Gould, and L. W. Molenkamp Physikalisches Institut (EP3), Universit €at W €urzburg, Am Hubland, D-97074 W €urzburg, Germany (Received 30 January 2014; accepted 19 February 2014; published online 4 March 2014) NiMnSb is a ferromagnetic half-metal which, because of its rich anisotropy and very low Gilbert damping, is a promising candidate for applications in information technologies. We have investigated the in-plane anisotropy properties of thin, molecular beam epitaxy-grown NiMnSb ﬁlms as a function of their Mn concentration. Using ferromagnetic resonance to determine theuniaxial and four-fold anisotropy ﬁelds, 2KU Msand2K1 Ms, we ﬁnd that a variation in composition can change the strength of the four-fold anisotropy by more than an order of magnitude and cause a complete 90/C14rotation of the uniaxial anisotropy. This provides valuable ﬂexibility in designing new device geometries. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4867298 ] INTRODUCTION NiMnSb is a half-metallic ferromagnetic material offer- ing 100% spin polarization in its bulk,1and was therefore long considered a very promising material for spintronicapplications such as spin injection. Experience has shown, however, that preserving sufﬁciently high translation symme- try to maintain this perfect polarization at surfaces and interfa-ces is a major practical challenge, reducing its attractiveness for spin injection. The material nevertheless continues to be very promising for use in other spintronic applications; in par-ticular, in spin torque devices such as spin-transfer-torque (STT) controlled spin valves and spin torque oscillators (STO). This promise is based on its very low Gilbert damping,of order 10 /C03or lower2which should enhance device efﬁ- ciency, as well as on its rich and strong magnetic anisotropy which allows for great ﬂexibility in device engineering. For example, it has been shown that STO oscillators formed from two layers of orthogonal anisotropy can yield signiﬁcantly higher signal than those with co-linear magneticeasy axis. 3–6Being able to tune the magnetic anisotropy of individual layers is clearly useful for the production of such devices. Previous results have shown a dependence of the anisot- ropy of NiMnSb on ﬁlm thickness,7which offers some con- trol possibilities when device geometries allow forappropriate layer thicknesses, but that is not always possible due to other design or lithography limitations. Here, we show how the anisotropy of layers of a given range of thick-ness can effectively be tuned by slight changes in layer com- position, achieved by adjusting the Mn ﬂux. EXPERIMENTAL The NiMnSb layers are grown epitaxial by molecular beam epitaxy (MBE) on top of a 200 nm thick (In,Ga)As buffer on InP (001) substrates. All samples have a protectivenon-magnetic metal cap (Ru or Cu) deposited by magnetron sputtering before the sample is taken out of the UHV envi- ronment, in order to avoid oxidation and/or relaxation of theNiMnSb. 8The ﬂux ratio Mn/Ni, and thus the composition, is varied between samples by adjusting the Mn cell temperaturewhile the ﬂux ratio Ni/Sb is kept constant. The thickness of most of the studied NiMnSb layers is 38 62 nm. Two sam- ples have a slightly larger ﬁlm thickness (45 nm, markedwith () in Fig. 3(a)), caused by the change in growth rate due to the change in Mn ﬂux. We veriﬁed that there is no correla- tion between anisotropy and sample thickness in this range. High Resolution X-Ray Diffr action (HRXRD) measure- ments are used to determine the ve rtical lattice constant of each sample. Fig. 1shows standard x-2h-scans of the (002) Bragg reﬂection on layers with the lowest and highest Mn concentra- tions used in the study, as well as a scan for a sample with me- dium Mn concentration. The sample with the lowest Mncontent has a vertical lattice constant of 5.939 A ˚(sample A) and that with the highest Mn content (sample C) has a vertical lattice constant of 6.092 A ˚. To get an estimate of the vertical lattice constant of stoichiomet ric NiMnSb in our layer stacks, we used an XRD measurement of a stoichiometric, relaxed sample. 9We determine a relaxed lattice constant of are- l¼(5.92660.007) A ˚. Together with the lattice constant of our InP/(In,Ga)As substrate, 5.8688 A ˚, and an estimated Poisson ra- tio of 0.3 60.03, we get the minimal and maximal values for the vertical lattice constant of stoichiometric NiMnSb: a?;max¼5:999 ˚A;a?;min¼5:957 ˚A. The vertical lattice con- stant of the sample with medium Mn concentration (sample B,5.968 A ˚) lies in this range. We conclude that the composition of sample B is approximately stoichiometric. In Refs. 10and 11, the effects of off-stoichiometric defects in NiMnSb are discussed. Among the possible defects related to Mn, Mn Ni(Mn substituting Ni) is most likely (it has lowest formation energy) and the predicteddecrease of the saturation magnetization is consistent with our observation (see Fig. 3(b)). Furthermore, an increase of the lattice constant with increasing concentration of this kindof defect is predicted theoretically and observed experimen- tally. Thus, we can use the (vertical) lattice constant as a measure for the Mn concentration in our samples. The crystal quality is also assessed by the HRXRD measurements. The inset in Fig. 1shows the x-scans of the same three NiMnSb layers. The x-scans of both the low and medium Mn concentration sample are extremely narrow with a full width half-maximum (FWHM) of 15 and 0021-8979/2014/115(9)/094505/4/$30.00 VC2014 AIP Publishing LLC 115, 094505-1JOURNAL OF APPLIED PHYSICS 115, 094505 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 209.183.185.254 On: Thu, 27 Nov 2014 19:38:2014 arcsec, respectively. A broadening for the sample with highest Mn concentration can be seen (FWHM of 35 arcsec). Reasons for the broadening can be partial relaxation of the layer due to the increased lattice mismatch with the (In,Ga)As buffer, and/or defects related to the surplus of Mn. Using the experimental data of the lattice constant in Ref. 11, we can estimate a difference in Mn concentration between sample A and C (extreme samples) of about 40%. For sample C (extreme high Mn concentration), we deter-mine a saturation magnetization of 3 :4l Bohr(see Fig. 3(b)). According to Ref. 11, this corresponds to a crystal where about 20% of Ni is replaced by Mn. It should be noted thatwe investigated the effect of extreme surplus/deﬁcit of Mn within the limits of acceptable crystal quality. As can be seen in Fig. 3(a), already a much smaller change in composi- tion can change the strength and orientation of the magnetic anisotropy signiﬁcantly. To map out the in-plane anisotropy of our samples, we use frequency-domain ferromagnetic resonance (FMR) measurements at a frequency of 12.5 GHz. The resonance ﬁelds are determined as a function of an external magneticﬁeld applied at ﬁxed angles ranging from 0 /C14(deﬁned as the [100] crystal direction) to 180/C14. Fig. 2shows results of these measurements for four different samples with four distincttypes of anisotropy: Sample A and D both exhibit large uni- axial anisotropies with an additional four-fold component, however of opposite sign. The hard axis of sample A is alongthe [1 /C2210] crystal direction, where for sample D the hard axis is along the [110] crystal direction. Sample B and C both show mainly uniaxial anisotropies, again with oppositesigns. The FMR data can be simulated with a simple phenome- nological magnetostatic model to extract the anisotropy com-ponents (derivation taken from Ref. 12). The free energy equation for thin ﬁlms of cubic materials is given by/C15 c¼/C0Kk 1 2ða4 xþa4 yÞ/C0K? 1 2a4 z/C0Kua2 z; (1) where ax;ay,a n d azdescribe the magnetization with respect to the crystal directions [100], [010], and [001]. Kk 1is the four- fold in-plane anisotropy constant, KuandK? 1represent the perpendicular uniaxial anisotropy (second and fourth order,respectively). In our in-plane FMR geometry, the fourth order perpendicular anisotropy term K ? 1can be neglected. Instead, an additional uniaxial in-plane anisotropy term is added /C15u¼/C0Kk uð^n/C1^MÞ2 M2 s with the unit vector ^nalong the uniaxial anisotropy and the saturation magnetization Ms;^M. The Zeeman term coupling to the external ﬁeld H0and a demagnetization term originat- ing from the thinness of the sample are deﬁned as /C15Z¼/C0M/C1H0;/C15 demag ¼/C04pDM2 ? 2(2) and added as well to the free energy. The effective magnetic ﬁeld Hef f¼/C0@/C15total @M(3) with /C15total¼/C15cþ/C15uþ/C15Zþ/C15demag (4) is used to solve the Landau-Lifshitz-Gilbert-Equation (LLG) /C01 c@M @t¼½M/C2Hef f/C138/C0G c2M2 s/C20 M/C2@M @t/C21 (5)FIG. 1. HRXRD x/C02h-scans of 3 NiMnSb samples with various Mn con- centrations. The curves are vertically offset for clarity. Inset: x-scans show- ing high crystal quality.FIG. 2. FMR measurements and simulation for four different samples. The symbols are measurements of the resonance frequency for magnetic ﬁelds along speciﬁc crystal directions, where 0/C14lies along [100]. The lines are simulations (see below) and also serve as a guide to the eye. Samples A, B,and C correspond to the samples with lowest, medium, and highest Mn con- centration shown in Fig. 1. Sample D completes the various kinds of anisot- ropy observed in NiMnSb.094505-2 Gerhard et al. J. Appl. Phys. 115, 094505 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 209.183.185.254 On: Thu, 27 Nov 2014 19:38:20with the gyromagnetic ratio c¼glB /C22hand the Gilbert damping constant G. The resonance condition can be found by calcu- lating the susceptibility,13v¼@M @H /C18x c/C192 ¼Bef fH/C3 ef f: (6) In the following, we neglect the damping contribution since G cMsin our samples is of the order of 10/C03or lower. Thus, Beff andH/C3 ef fin our case can be found to be H/C3 ef f¼H0cos½/M/C0/H/C138þ2Kk 1 Mscos½4ð/M/C0/FÞ/C138 þ2Kk U Mscos½2ð/M/C0/UÞ/C138; (7) Bef f¼H0cos½/M/C0/H/C138þKk 1 2Msð3þcos½4ð/M/C0/FÞ/C138Þ þ4pDM s/C02K? U MsþKk U Msð1þcos½2ð/M/C0/UÞ/C138Þ:(8) Here, /M;/H,a n d /Udeﬁne the angles of the magnetization, external magnetic ﬁeld, and in-plane easy axis of the uniaxial anisotropy, respectively, with respect to the crystal direction[100]. / Faccounts for the angle of the four-fold anisotropy. At the magnetic ﬁelds used in these studies, it is safe to assume /M¼/H.14In Eq. (8),4pDM s/C02K? U MScan be deﬁned as an effective magnetization 4 pMef f, containing the out-of- plane anisotropy. It is used as a constant in our simulation. For each sample, we extract2K1 Msand2KU Ms, the four-fold and uniaxial in-plane anisotropy ﬁeld, from the simulationand plot them versus the vertical lattice constant (Fig. 3(a)). The vertical, dotted lines mark the range where stoichiomet- ric NiMnSb is expected. For vertical lattice constants in therange from 5.96 to 6.00 A ˚, both anisotropy ﬁelds are rela- tively small. The four-fold contribution increases for samples with decreasing vertical lattice constant (lower Mn concen-tration) but remains small for larger vertical lattice constant (increasing Mn concentration). The uniaxial anisotropy gets more strongly negative with increasing vertical lattice con-stant, whereas in samples with lower vertical lattice con- stants, the uniaxial ﬁeld can be either positive or negative while its absolute value grows signiﬁcantly with decreasingvertical lattice constant. The change in sign of the uniaxial anisotropy ﬁeld at a vertical lattice constant of about 5.99 A ˚ corresponds to a rotation of the easy axis from the [110] direction (positive anisotropy ﬁelds) to the [1 /C2210] direction. One can see that already a small change of the vertical lattice constant (small change in composition) is sufﬁcient to rotatethe uniaxial anisotropy as well as to induce a signiﬁcant four-fold anisotropy. The ﬁtting accuracy of the extracted anisotropy ﬁelds is /C245%, giving error bars smaller than the symbols in Fig. 3(a). It should be noted that in order to exactly extract the anisot- ropy constants K 1andKUfrom the anisotropy ﬁelds, the sat- uration magnetization Msof each sample is needed. This can be determined by SQUID measurements. We haveperformed such measurements on a representative fraction of the samples (Fig. 3(b)). Samples with medium Mn concen- tration show saturation magnetizations which, to experimen- tal accuracy of about 8% are consistent with the theoretically expected 4 :0lBohr per unit formula for stoichiometric NiMnSb.15The estimated measurement accuracy of 8% accounts for uncertainty in the sample thickness extracted from the HRXRD data of about 5%, as well as errors indetermining the exact sample area, SQUID calibration and SQUID response due to ﬁnite sample size. Our samples with highest and lowest magnetization show a slight decrease insaturation magnetization, of order 12%. This change is sufﬁ- ciently small to be neglected in the overall assessment of the anisotropy vs. vertical lattice constant of Fig. 3(a). In an attempt to understand the effect of higher or lower Mn concentration on the crystal structure in our samples, we consider the possible non-stoichiometric defects which canexist in NiMnSb, as discussed in Ref. 10. Formation energies, magnetic moment change, and effect on the half-metallic character are presented there for each type of defect. Mn-related defects are (a) Mn substituting Ni or Sb ðMn Ni;Mn Sb), (b) Mn on a vacancy position ðMn IÞ, (c) Ni or Sb substituting MnðNiMn;SbMnÞ, or (d) a vacancy position at the Mn site ðvac MnÞ. With a surplus of Mn, both Mn substituting Ni or Sb and Mn incorporated on the vacancy position seem plausible. However, the formation energy of Mn Sbis more than three times larger than for the other defects, suggesting it should be very rare. On the other hand, in the case of a Mn deﬁciency, either Ni or Sb could substitute Mn or vacancies can be builtinto the crystal. Those three defects have similar formation energies, making them equally possible.FIG. 3. (a) Uniaxial anisotropy ﬁeld2KU Msand four-fold anisotropy ﬁeld2K1 Msfor NiMnSb layers with various Mn concentrations. The vertical lattice con- stant is used as a gauge of the Mn content. Samples with a rotated RHEED pattern (see last section) are indicated by open symbols. The dotted lines mark the range where stoichiometric NiMnSb is expected. The samples oflowest, medium, and highest Mn concentration (A, B, and C) together with sample D are marked. The two samples marked with () exhibit slightly higher ﬁlm thickness than the other samples. (b) Saturation magnetization M sdepending on the vertical lattice constant.094505-3 Gerhard et al. J. Appl. Phys. 115, 094505 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 209.183.185.254 On: Thu, 27 Nov 2014 19:38:20Except for Mn Iand Mn Sb, all of these possible defects reduce the magnetic moment per formula unit. Our observa- tions of a lower magnetic moment for samples with eitherhigh or low Mn ﬂux, are thus consistent with the defects Mn Ni;NiMn;SbMn, and vac Mn. The positive contribution of Mn Ito the magnetic moment is, however, some 5 times smaller than the decrease induced by the other defects, so some fraction of defects of the Mn Ivariety could also be present in the samples. A detailed discussion on the transi-tion from stoichiometric NiMnSb towards off-stoichiometric Ni 1/C0xMn 1þxSb is given in Ref. 11. It is shown that the lattice constant of off-stoichiometric NiMnSb increases for increas-ing substitution of Ni by Mn. This behavior is clearly seen in our samples for increasing Mn concentration and we con- clude that this kind of defect is most prominent in our sam-ples. An explanation for a decreasing lattice constant for decreasing Mn concentration is yet to be found. A further observation which may provide insight into the observed anisotropy behavior comes from Reﬂective High Energy Electron Diffraction (RHEED), which is used to monitor the surface of the sample in-situ during the growth. RHEED provides information about the surface reconstruction, which turns out to be sensitive to the Mn con- tent. In all samples, at the beginning of the growth (afterapproximately 1 min), the surface reconstruction exhibits a clear 2 /C21 pattern, meaning a d/2 reconstruction in the [110] crystal direction and a d/1 reconstruction along [1 /C2210] direc- tion (see Fig. 4). How this pattern then evolves during growth depends on the Mn ﬂux. For ideal Mn ﬂux, the pat- tern is stable throughout the entire 2 h growth time corre-sponding to a 40 nm layer. A reduced Mn ﬂux results in a more blurry RHEED pattern, but does not lead to any change in the surface reconstruction. A higher Mn ﬂux, on the otherhand, causes a change of the reconstruction such that the d/2 pattern also becomes visible along the [1 /C2210] direction and fades over time in the [110] direction until a 90 /C14rotation of the original pattern has been completed. The length of time (and thus the thickness) required for this rotation depends strongly on the Mn ﬂux. A slightly enhanced Mn ﬂux causesa very slow rotation of the reconstruction that can last the entire growth time, whereas a signiﬁcant increase of the Mn ﬂux (sample with vertical lattice constants above 6.05 A ˚) will cause a rotation of the reconstruction within a few minutes of growth start, corresponding to a thickness of only very few monolayers. Based on these observations, our sam-ples can be split into two categories: samples with a stable 2/C21 reconstruction and those with a 2 /C21 reconstruction that rotates during growth. In Fig. 3(a), samples with a stable RHEED pattern are indicated with ﬁlled symbols while empty symbols show samples with a rotated RHEEDreconstruction. It is interesting to note that all samples with a rotated reconstruction exhibit a very low four-fold anisotropy ﬁeld. In addition, the sooner the rotation of the RHEED pat-tern occurs, the stronger the uniaxial anisotropy is. SUMMARY We have shown that the anisotropy of NiMnSb strongly depends on the composition of the material. A variation ofthe Mn ﬂux results in different (vertical) lattice constants (measured by HRXRD) that can be used for a measure of the Mn concentration. RHEED observations ( in-situ ) during the growth already give an indication of high or low Mn concen- tration. The anisotropy shows a clear trend for increasing Mn content. Using this together with the RHEED observations,NiMnSb layers with high crystal quality and anisotropies as- requested can be grown. The microscopic origin of this behavior remains to be understood, and it is hoped that thispaper will stimulate further efforts in this direction. The phe- nomenology itself is nevertheless of practical signiﬁcance in that it provides interesting design opportunities for devicessuch as spin-valves that could be made of two NiMnSb layers with mutually parallel or orthogonal magnetic easy axes as desired. ACKNOWLEDGMENTS We thank T. Naydenova for assistance with the SQUID measurements. This work was supported by the European Commission FP7 Contract ICT-257159 “MACALO”. 1R. A. de Groot, F. M. Mueller, P. G. v. Engen, and K. H. J. Buschow, Phys. Rev. Lett. 50, 2024 (1983). 2A. Riegler, “Ferromagnetic resonance study of the Half-Heusler alloy NiMnSb: The beneﬁt of using NiMnSb as a ferromagnetic layer in pseudo spin-valve based spin-torque oscillators,” Ph.D. thesis (Universitaet Wuerzburg, 2011). 3T. Devolder, A. Meftah, K. Ito, J. A. Katine, P. Crozat, and C. Chappert,J. Appl. Phys. 101, 063916 (2007). 4D. Houssameddine, U. Ebels, B. Dela €et, B. Rodmacq, I. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda, M.-C.Cyrille, O. Redon, and B. Dieny, Nature Mater. 6, 447 (2007). 5G. Consolo, L. Lopez-Diaz, L. Torres, G. Finocchio, A. Romeo, and B. Azzerboni, Appl. Phys. Lett. 91, 162506 (2007). 6S. M. Mohseni, S. R. Sani, J. Persson, T. N. Anh Nguyen, S. Chung, Y. Pogoryelov, and J. A ˚kerman, Phys. Status Solidi RRL 5, 432 (2011). 7A. Koveshnikov, G. Woltersdorf, J. Q. Liu, B. Kardasz, O. Mosendz, B. Heinrich, K. L. Kavanagh, P. Bach, A. S. Bader, C. Schumacher, C.R€uster, C. Gould, G. Schmidt, L. W. Molenkamp, and C. Kumpf, J. Appl. Phys. 97, 073906 (2005). 8C. Kumpf, A. Stahl, I. Gierz, C. Schumacher, S. Mahapatra, F. Lochner, K. Brunner, G. Schmidt, L. W. Molenkamp, and E. Umbach, Phys. Status Solidi C 4, 3150 (2007). 9W. Van Roy and M. W /C19ojcik, “Half-metallic alloys,” in Lecture Notes in Physics , edited by I. Galanakis and P. Dederichs (Springer, Berlin, Heidelberg, 2005), Vol. 676, pp. 153–185. 10B. Alling, S. Shallcross, and I. A. Abrikosov, Phys. Rev. B 73, 064418 (2006). 11M. Ekholm, P. Larsson, B. Alling, U. Helmersson, and I. A. Abrikosov,J. Appl. Phys. 108, 093712 (2010). 12Ultrathin Magnetic Structures II , edited by B. Heinrich and J. A. C. Bland (Springer, Berlin, Heidelberg, 1994). 13C. Kittel, Phys. Rev. 73, 155 (1948). 14We have conﬁrmed from frequency dependent measurements that this assumption leads to errors smaller than the size of the symbols in Fig. 3(a). 15T. Graf, C. Felser, and S. S. Parkin, Prog. Solid State Chem. 39, 1 (2011). FIG. 4. Typical RHEED reconstruction of the NiMnSb surface illustrating the two reconstructions discussed in the text.094505-4 Gerhard et al. J. Appl. Phys. 115, 094505 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 209.183.185.254 On: Thu, 27 Nov 2014 19:38:20
5.0030016.pdf	AIP Advances 11, 015045 (2021); https://doi.org/10.1063/5.0030016 11, 015045 © 2021 Author(s).Implementation of complete Boolean logic functions in single spin–orbit torque device Cite as: AIP Advances 11, 015045 (2021); https://doi.org/10.1063/5.0030016 Submitted: 18 September 2020 . Accepted: 28 December 2020 . Published Online: 27 January 2021 Yunchi Zhao , Guang Yang , Jianxin Shen , Shuang Gao , Jingyan Zhang , Jie Qi , Haochang Lyu , Guoqiang Yu , Kui Jin , and Shouguo Wang COLLECTIONS Paper published as part of the special topic on Chemical Physics , Energy , Fluids and Plasmas , Materials Science and Mathematical Physics ARTICLES YOU MAY BE INTERESTED IN Spin-orbit torques: Materials, physics, and devices Applied Physics Letters 118, 120502 (2021); https://doi.org/10.1063/5.0039147 Large damping-like spin–orbit torque and perpendicular magnetization switching in sputtered WTe x films Applied Physics Letters 118, 042401 (2021); https://doi.org/10.1063/5.0035681 Enhancement of spin–orbit torque in WTe 2/perpendicular magnetic anisotropy heterostructures Applied Physics Letters 118, 052406 (2021); https://doi.org/10.1063/5.0039069AIP Advances ARTICLE scitation.org/journal/adv Implementation of complete Boolean logic functions in single spin–orbit torque device Cite as: AIP Advances 11, 015045 (2021); doi: 10.1063/5.0030016 Submitted: 18 September 2020 •Accepted: 28 December 2020 • Published Online: 27 January 2021 Yunchi Zhao,1,2 Guang Yang,3,a) Jianxin Shen,4Shuang Gao,5 Jingyan Zhang,4 Jie Qi,4 Haochang Lyu,4Guoqiang Yu,1,2 Kui Jin,1,2 and Shouguo Wang4,a) AFFILIATIONS 1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom 4Beijing Advanced Innovation Center for Materials Genome Engineering, School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China 5CAS Key Laboratory of Magnetic Materials and Devices, Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China a)Authors to whom correspondence should be addressed: gy251@cam.ac.uk and sgwang@ustb.edu.cn ABSTRACT All 16 Boolean logic functions in a single Ta/CoFeB/MgO device with perpendicular magnetic anisotropy were experimentally demonstrated based on the spin–orbit torque (SOT) effect. Furthermore, by combining with the voltage-controlled magnetic anisotropy (VCMA) effect, a novel SOT-MTJ (magnetic tunnel junction) prototype device with the assistance of the VCMA effect was further designed to perform mag- netic field-independent logic operations. The numerical simulations were carried out, demonstrating the feasibility to realize all 16 Boolean logic functions in a single three-terminal device by applying the bias voltage and current injection as input variables. This approach pro- vides a potential way toward the application of energy efficient spin-based logic, which is beyond the current von Neumann computing architecture. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0030016 The continued scaling of complementary metal-oxide- semiconductor (CMOS) technology according to Moore’s law has brought to us upgraded electronic devices with a smaller volume, higher speed, and lower price for decades. However, the challenge to such rapid development still exists.1,2One critical limitation is the explosive growth of static power dissipation arising from the leakage current as the CMOS feature size scales down to a few nanometers.3On the other hand, the operating speed of electronic devices is seriously limited by the ineluctable data exchange between separate processor and memory units, encountering the so-called von Neumann bottleneck that further brings massive dynamic power dissipation.4To overcome the above-mentioned obstacles rooted in the existing Si-based electronic devices, new problem-solving strategies beyond CMOS or even beyond vonNeumann are urgently needed and have aroused extensive research interests.5,6 One promising strategy is to introduce the emerging non- volatile memories such as magnetoresistive memory,7–12resistive memory,13–16and phase-change memory17into logic circuits. For- tunately, due to their nonvolatile feature, these memories can eradi- cate the static power dissipation. More importantly, the exploration of their logic functions will lead to the unity of logic and mem- ory units, hence thoroughly breaking the von Neumann bottleneck. In particular, the magnetic tunnel junction based on spin trans- fer torque (called the STT-MTJ) combines the advantages of non- volatility, CMOS compatibility, and unlimited endurance,18showing great potential to construct a “stateful” logic circuit where intrinsic logic-in-memory cells both perform logic operations and store logic AIP Advances 11, 015045 (2021); doi: 10.1063/5.0030016 11, 015045-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv values.6Recently, a reprogrammable logic gate consisting of three input STT-MTJs and one output STT-MTJ was designed to realize the basic Boolean logic functions AND, OR, NAND, NOR, and the Majority operation.9,10,19 However, despite the reprogrammable gates and implication gates being good examples for the realization of stateful logic, there are still some shortcomings that need to be addressed. In the aspect of device performance, the STT-MTJ suffers from serious failure and reliability issues due to the high writing current densities as well as erroneous writing by the read current.20Furthermore, both the sin- gle implication gate and the single reprogrammable gate can only implement just one or a limited number of logic operations, indi- cating more steps and complex combinations are needed to com- plement the other basic Boolean logic functions. Therefore, it is extremely attractive to explore a better alternative to the STT-MTJ and to finally realize as many Boolean logic functions as possible in such a single device. Spin–orbit torque (SOT) originating from the spin Hall effect and Rashba–Edelstein effect provides another method to realize magnetization switching21–23and high-speed domain wall motion24,25by an in-plane current injection. Moreover, the critical current of the SOT-induced switching can be modulated with a bias voltage due to the voltage-controlled magnetic anisotropy (VCMA) effect,26,27exhibiting great potential in the application of MTJ-based logic devices. In this work, a design based on the SOT mechanism was experimentally demonstrated to realize all 16 Boolean logic func- tions in a simple Ta/CoFeB/MgO trilayer with perpendicular mag- netic anisotropy (PMA). The key idea is to combine different logic inputs to tune the magnetization state and to measure the anoma- lous Hall resistance as the logic output. Furthermore, by utiliz- ing the tunnel magnetoresistance (TMR) value as the logic out- put, this method is applicable to the emerging perpendicular SOT- MTJ, a novel three-terminal device with high reliability, symmet- ric switching, and scalable energy consumption compared to the conventional STT-MTJ. More importantly, we also conceive that by combing the voltage-controlled magnetic anisotropy (VCMA) effect, the modified method (using current injection and bias volt- age as logic inputs) can realize all 16 Boolean logic functions in the SOT-MTJ. The multilayers with a core structure of Ta (3)/Co 40Fe40B20 (1.1)/MgO (2)/Ta (3) (in nm) were deposited on thermallyoxidized Si (001) substrates by a magnetron sputtering system at room temperature. The films were patterned into 15- μm-wide Hall bars for transport measurements using photolithography and Ar-ion etching. The devices exhibit PMA after an annealing pro- cess at above 300○C, which can be proved by the anomalous Hall signal with an out-of-plane magnetic field (Fig. S1, supple- mentary material). Figure 1(a) shows SOT-induced magnetization switching, with current injection ( Ix) and magnetic field ( Hx) both along the x-axis. Opposite switching can be clearly observed when Hxis reversed, suggesting similar features to the previous stud- ies on perpendicularly magnetized heavy metal/ferromagnet het- erostructures.21,28Figure 1(b) presents current-induced switching loops under different Hx, indicating that the critical switching current is positively correlated with Hx, which can be explained by the Marco model.29Moreover, programmable logic devices based on the SOT mechanism can be designed according to this relationship. Figure 2(a) shows two schematic current-induced switching loops under different external fields in consideration of a single domain switching paradigm. The critical switching currents can be distinguished in the high external field ( HH) and low external field (HL) configurations. Hence, a median value of two critical switching currents is defined as ∣IM∣. Similar to the schematic curves, distinctly different switching loops corresponding to different external fields can be observed in the experimental measurements, as shown in Fig. 2(b). Here, HHandHLare 200 Oe and 30 Oe, respectively, and the value of ∣IM∣is 8.5 mA. The external field Hxand injected cur- rent Ixare used as two input variables to perform logic operations. For example, Hxis 30 Oe or 200 Oe for input 0 or 1, respectively, andIxis set to−8.5 mA or+8.5 mA for input 0 or 1, as listed in the table in Fig. 2(c). The measured Hall resistance functions as the logic output, which can be identified as the spin-up state (logic “1”) and the spin-down state (logic “0”), as shown in Fig. 2(c). Figure 2(d) suggests the logic output transformation corresponding to different logic inputs (external field/current, HxIx). It can be seen from the diagram that for the HxIx=“11” configuration ( Hx=200 Oe and Ix=+8.5 mA), the logic output is 1 (spin-up state), regardless of the initial magnetization state. Similarly, the HxIx=“10” configura- tion leads to the logic output 0, and SOT-induced switching will not take place with the logic input HxIx=“00” or “01” configuration. Based on these working principles, all 16 Boolean logic operations can be performed, and the detailed operations are listed in Fig. 2(e). FIG. 1. (a) Illustration of a Ta/CoFeB/MgO Hall bar device. Hxand Ixrefer to in- plane field and current injection along the x axis, respectively. (b) Current switch- ing loops under different Hx(the arrows shown in the figure indicate the switching polarities of the SOT-induced switching). AIP Advances 11, 015045 (2021); doi: 10.1063/5.0030016 11, 015045-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 2. (a) Schematic switching loops under high external field ( HH) and low external field ( HL); the median value of two critical switching currents is defined as ∣IM∣. (b) Corresponding experimental switching loops under 200 Oe ( HH) and 30 Oe ( HL), where ∣IM∣is 8.5 mA. (c) The correspondence between input logic values and Hxas well as Ix. The illustra- tion suggests the output logic values cor- responding to magnetization states. (d) Transition of the output logic states by HxIxcombined operations. (e) Detailed operation methods of all 16 Boolean logic functions (W1, W2, and W3 stand for a write sequence). Therefore, “ p“ and “ q“ are the two input logic variables, “-” means no operation, and “ ¬” represents the negation operation, i.e., ¬0=1 and ¬1=0. Figure 3 shows the experimental results of Boolean logic func- tions based on the SOT-induced switching. Combining with a truth table of all 16 Boolean logic gates (shown in supplementary material), the feasibility of complementary logic operation in a sin- gle device with programmability and non-volatility can be demon- strated. For example, a TRUE gate can be performed in one step with the logic operation “11” ( Hx=200 Oe and Ix=+8.5 mA), regard- less of the value of inputs pandq. For this configuration, the final magnetization is in the spin-up state, outputting a logic value 1, as shown in Fig. 3(a1). For an AND gate, if the input p=q=0, a logic operation HxIx=“10” ( Hx=200 Oe and Ix=−8.5 mA) is needed as an initialization step. Then, the writing operation “00” is performed byHx=30 Oe and Ix=−8.5 mA. In this case, the final magneti- zation state is “spin-down,” outputting a logic value 0, as shown in Fig. 3(b1). For the configurations with other “ p q” values, “01,” “10,” and “11,” the logic function can be realized by writing “ p q” after an initialization step with “10,” outputting the corresponding results finally, as shown in Figs. 3(b2)–3(b4). For an OR gate, the logic inputIxHx=“11” ( Hx=200 Oe, Ix=+8.5 mA) is initially operated if the input p=1 and q=0. Then, the write operation “ ¬p q” (“00,” Hx=30 Oe, and Ix=−8.5 mA) is performed as the second step, resulting in the spin-up state (output 0), as shown in Fig. 3(b5). For configurations with other “ p q” values, “11,” “00,” and “01,” the logic operations can be completed by writing “ ¬p q” after an initializa- tion step with “11,” and the corresponding experimental results are shown in Figs. 3(b6)–3(b8). Moreover, an XOR gate can be realized in a process with three steps. If the logic input p=0 and q=1, the write operation HxIx =“10” ( Hx=200 Oe and Ix=−8.5 mA) is performed at first. The second step is to input “ ¬p q” (“11,” Hx=200 Oe, and Ix =+8.5 mA). Write operation “ p¬q” (“00,” Hx=30 Oe, and Ix =−8.5 mA) is performed at last. After the three-step operations, the final magnetization state is spin-up, outputting a logic value 1, as shown in Fig. 3(c4). As for the other “ p q” values of “10,” “11,” and “00,” the initial operation is to input “10” and then input “¬p q” as follows. The third step is to perform a logic operation “p¬q” and read the corresponding output values finally. The rele- vant experimental results are shown in Figs. 3(c1)–3(c3). In addi- tion, the operations of other logic gates not mentioned here are AIP Advances 11, 015045 (2021); doi: 10.1063/5.0030016 11, 015045-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 3. Experimental results of all 16 Boolean logic functions based on the SOT switching mechanism. (a)–(c) represent the realization of logic functions by using one, two, and three write cycles, respectively. summarized in the truth table (Fig. S2, supplementary material). It clearly indicates that these experimental results demonstrate the fea- sibility to successfully realize all 16 Boolean logic operations in the Ta/CoFeB/MgO device. It means that the SOT-based logic opera- tions can be introduced into the device design, which is expected to realize the combination of non-volatile memory and computing unit in an integrated circuit, breaking the von Neumann bottleneck in the future. Furthermore, the external field can be replaced by bias volt- age as a logic input based on the VCMA effect.30,31As shown in Fig. 4(a), charge accumulation will take place at the ferro- magnetic metal/oxide interface in the MTJ structure with a bias voltage (V b) applied. The energy barrier of magnetization switch- ing is thus modulated due to the change in the relative occu- pancy of the 3d-orbitals.32Figure 4(b) shows the specific influ- ence of the bias voltage on the energy barrier of magnetization switching in the free layer, suggesting a lower barrier correspond- ing to a positive bias voltage and a higher barrier correspond- ing to a negative bias voltage. The voltage required to com- pletely eliminate the barrier is defined as V c. The voltage-driven switching induced by SOT can be realized by precisely control- ling the duration of the bias voltage when 0 <Vb<Vc.32,33 Based on this feature, a VCMA-assisted SOT MTJ device was designed, as shown in Fig. 4(c). A low energy barrier is achieved witha bias voltage applied between terminals T1 and T3, and the SOT originated from the heavy metal layer is induced by an injected cur- rent between T2 and T3 that results in a current-induced switching. This procession can be described by the Landau–Lifshitz–Gilbert (LLG) equation,34 d⇀ M dt=−γ⇀ M×⇀ Heff+α⇀ M×d⇀ M dt+⇀ Γ, (1) where⇀ Mis the magnetization vector, γis the gyromagnetic ratio, andαis the Gilbert damping coefficient. The effective magnetic field Heff=Hext+Hd+Haniis the sum of the external field ( Hext), the demagnetizing field ( Hd), and the anisotropy field ( Hani). Tak- ing into account the VCMA effect, Hanican be expressed in the form33 ⇀ Hani(Vb)=(2Ki(0)tox−2ξVb Mstftox)⇀ M, (2) where Ki(0)is the interfacial anisotropy energy without bias volt- age applied, toxandtfare the thickness of the barrier and the free layer, respectively, ξis the VCMA coefficient, and Msis the satu- ration magnetization of the free layer. The additional SOT item in AIP Advances 11, 015045 (2021); doi: 10.1063/5.0030016 11, 015045-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 4. Illustrations of (a) MTJ structure, (b) impact of different voltages on the energy barrier of an MTJ, and (c) the VCMA-assisted SOT MTJ device. the LLG equation can be described as ⇀ Γ=−γ̵hθSHEJ 2etfMs⇀ M×(⇀ M×⇀σ), (3) where θSHEis the spin Hall angle and⇀σis the spin polarization vec- tor. Hence, the dynamic evolution of the magnetization in the free layer can be characterized according to Eqs. (1) and (3). Based on this, Verilog-A language was applied to build electrical models for a VCMA-assisted SOT MTJ device to perform numerical simulationswith a 40-nm CMOS kit.33The detailed parameters for the model can be found in Table S3 listed in the supplementary material. The simulation results suggest that all 16 Boolean logic functions can be realized based on the VCMA and SOT effects. Figure 5 shows the transient waveforms of the proposed VCMA-assisted SOT MTJ device corresponding to AND, OR, and XOR logic gates. Bias voltage V band injected current Ixare used as two input variables to perform logic operations. V bis set as 0 mV or 600 mV for input 0 or 1, respectively, and Ixis set to−65μAor +65μAfor input 0 or 1, respectively. The application of bias FIG. 5. Transient waveforms of the proposed VCMA-assisted SOT MTJ device corresponding to (a) AND, (b) OR, and (c) XOR logic gates. AIP Advances 11, 015045 (2021); doi: 10.1063/5.0030016 11, 015045-5 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv voltage can lead to a lower energy barrier, and the SOT is gener- ated by the injected current, resulting in a switching of the mag- netization state of the free layer in the device. The logic output is defined based on the tunneling magnetoresistance (TMR) cor- responding to the anti-parallel state (high resistance, +1 V, logic 1) or parallel state (low resistance, −1 V, logic 0) of the two fer- romagnetic layers. The transient waveforms of each logic opera- tion shown in Fig. 5 are divided into four color areas (areas I- IV), representing the operation process when the initial logic input “p q”=“00,” “10,” “01,” and “11,” respectively. For an AND gate shown in Fig. 5(a), if the inputs p=q=0 (area I), the initial- ization step is to perform a logic operation “10,” setting the bias voltage to 600 mV (1 ns) with a pulsed injected current of −65μA (2 ns). Then, the logic operation “00” is performed by Ix=−65μA without bias voltage applied. The device is in the low-resistance state finally in this case, outputting a logic value 0. For the configura- tions with other “ p q” values, “10” (area II), “01” (area III), and “11” (area IV), the logic operation can be completed by writing “ p q” after an initialization step with “10” and outputting the corre- sponding results. For an OR gate shown in Fig. 5(b), if the inputs p=1 and q=0 (area II), a write operation “11” (V b=600 mV and Ix=+65μA) is performed at first. Then, the operation is completed by a write operation “ ¬p q” (“00,” V b=0 mV, and Ix=−65μA) that results in a high-resistance state (1 V, output 1). For configura- tions with other “ p q” values, “00” (area I), “01” (area III), and “11” (Area IV), the logic operation can be completed by writing “ ¬p q” after an initialization step with “11,” similar to the discussion above. Figure 5(c) shows the realization of an XOR gate with three steps. If the logic input p=0 and q=1 (area III), the write operation “10” (Vb=600 mV and Ix=−65μA) is performed at first, leading to the low-resistance state. The second step is to input “ ¬p q” (“11,” V b =600 mV, and Ix=+65μA), and write operation “ p¬q” (“00,” V b=0 mV, and Ix=−65μA) is performed at last. Finally, the device is in the high-resistance state outputting a logic value 1 after the three-step operations. As for the other input values of “ p q,” “00” (area I), “10” (area II), and “11” (Area IV), the logic function can be performed with three operations of input “10,” “ ¬p q,” and “ p¬q,” reading the corresponding logic output value finally. Based on this, a VCMA- SOT assisted MTJ device with three terminals is designed, and the feasibility of realizing all 16 Boolean logic functions in a single unit is demonstrated. The simulation results suggest that the operation with a single step only needs 2.26 ns, featuring an ultra-fast writing speed. In summary, all 16 Boolean logic functions in a single Ta/CoFeB/MgO device with PMA were experimentally demon- strated based on the SOT effect by applying the external field and current injection as input variables. Furthermore, the VCMA effect was introduced to design a three-terminal MTJ device, which can implement magnetic field-independent logic operations. The approach can be improved through optimization of structure design and device fabrication, paving the way for the application of energy efficient spin-based logic, which is beyond the current von Neumann computing architecture. SUPPLEMENTARY MATERIAL See the supplementary material for the anomalous Hall signal of the device, the truth table of all 16 Boolean logic functions, and thedetailed parameters for the electrical model to perform numerical simulations. ACKNOWLEDGMENTS This work was supported by the National Key Research and Development Program of China (Grant No. 2019YFB2005800), by the Natural Science Foundation of China (Grant Nos. 51625101, 11874082, 51971026, 61704178, and 61974179), by the NSFC-ISF Joint Research Program (Grant No. 51961145305), by the State Key Laboratory for Advanced Metals and Materials (Grant No. 2019Z- 10), and by the Beijing Natural Science Foundation Key Program (Grant Nos. Z190007 and Z190008). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon request. REFERENCES 1K. L. Wang, X. Kou, P. Upadhyaya, Y. Fan, Q. Shao, G. Yu, and P. K. Amiri, Proc. IEEE 104, 1974 (2016). 2D. S. Jeong, K. M. Kim, S. Kim, B. J. Choi, and C. S. Hwang, Adv. Electron. Mater. 2, 1600090 (2016). 3N. S. Kim, T. Austin, D. Baauw, T. Mudge, K. Flautner, J. S. Hu, M. J. Irwin, M. Kandemir, and V. Narayanan, Computer 36, 68 (2003). 4J. Backus, Commun. 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1.4987007.pdf	A switchable spin-wave signal splitter for magnonic networks F. Heussner , A. A. Serga , T. Brächer , B. Hillebrands , and P. Pirro Citation: Appl. Phys. Lett. 111, 122401 (2017); doi: 10.1063/1.4987007 View online: http://dx.doi.org/10.1063/1.4987007 View Table of Contents: http://aip.scitation.org/toc/apl/111/12 Published by the American Institute of PhysicsA switchable spin-wave signal splitter for magnonic networks F.Heussner,1A. A. Serga,1T.Br€acher,1,2,3,4B.Hillebrands,1and P . Pirro1 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit €at Kaiserslautern, D-67663 Kaiserslautern, Germany 2University Grenoble Alpes, INAC-SPINTEC, F-38000 Grenoble, France 3CNRS, SPINTEC, F-38000 Grenoble, France 4CEA, INAC-SPINTEC, F-38000 Grenoble, France (Received 8 June 2017; accepted 28 July 2017; published online 18 September 2017) The inﬂuence of an inhomogeneous magnetization distribution on the propagation of caustic-like spin-wave beams in unpatterned magnetic ﬁlms has been investigated by utilizing micromagnetic simulations. Our study reveals a locally controllable and reconﬁgurable tractability of the beamdirections. This feature is used to design a device combining split and switch functionalities for spin-wave signals on the micrometer scale. A coherent transmission of spin-wave signals through the device is veriﬁed. This attests the applicability in magnonic networks where the information isencoded in the phase of the spin waves. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4987007 ] Currently, spin wave (SW) based logic networks are widely discussed due to their potential as a CMOS complemen- tary or even subsequent technology with extended functiona lity and improved performance. 1–9In the emerging research ﬁeld of magnonics, magnons, the quanta of spin waves, are used totransport information and to perform logic operations. Advantages arise from the possibility to use the SW phase as an additional degree of freedom, to utilize interference effects, and to reduce the size of devices. 10In combination with charge- less information transport and the consequent absence of ohmiclosses, it allows for the realization of innovative and energy efﬁcient information processing. 11Furthermore, recent works demonstrate the ﬂexibility and reconﬁgurability of SW propa- gation in two-dimensional magnetic textures12–14promising an additional efﬁciency boost. The design and realization of differ-ent magnonic devices 15–26show the way of this emerging ﬁeld of data processing, and architectures for magnon-based logic networks are already proposed.27However, their constructions demand different basic elements for SW excitation, manipula- tion, and detection, which still have to be developed. In this Letter, we explore two-dimensional magnonic devices employing the phenomenon of caustic-like focused SW beams28–33in microstructures. These speciﬁc beams, which concentrate and direct energy of a large number ofplane spin-wave modes with different wave vectors, can occur in in-plane magnetized magnetic ﬁlms due to the anisotropic nature of the dipole-dipole interaction. Moreover, it has been shown recently that a strong focusing of large wave-vector spin waves can occur even without dipolar interactions, 34pav- ing the way for utilizing these SW beams at dimensions where the exchange interaction dominates the propagation character- istics of the spin waves. The excitation of the caustic-like SW beams by opening of a one-dimensional waveguide into a two-dimensional medium has been shown in different mag- netic materials, just as the tuneability of their propagation characteristics due to the strong dependency on the frequencyand on the direction of a uniform external magnetic ﬁeld. 29–32 In view of the utilization of these SW beams in magnonic devices, their local manipulation is investigated in this Letter.In the ﬁrst part, the tractability of the SW beams by current controlled inhomogeneities of the external ﬁeld and conse- quent inhomogeneous magnetization distributions are studied by micromagnetic simulations. Subsequently, we exploit the results to design a switchable SW signal splitter on the micro- meter scale, a pivotal element for magnon-based networks. The basic concept of the studied structure to create focused SW beams and to control them by an inhomoge- neous external ﬁeld is depicted in Fig. 1.A1 lm wide and 30 nm thick SW waveguide is connected to an unpatterned magnetic ﬁlm of the same thickness. The structure is magne- tized in the ﬁlm plane by an external bias ﬁeld Bext¼50 mT applied perpendicular to the long axis of the input wave-guide. A 2 lm wide and 150 nm thick current carrying micro- strip is placed above the ﬁlm, separated by a 300 nm thick insulation layer. Passing a DC-current I DCthought this microstrip leads to the creation of a localized Oersted mag- netic ﬁeld BDC. Figure 1(b) shows the in-plane component BDC,xin the unpatterned magnetic ﬁlm, calculated according to Biot- Savart’s law for IDC¼–100 mA. As shown in the same graph, this ﬁeld component with a maximum absolute strength of around 24 mT causes a change of the in-plane angle bof the total local bias ﬁeld Blocwhich is measured relative to the x-axis. In the micromagnetic simulation, only the magnetic layer as shown in Fig. 1(c) is implemented together with the homogeneous external ﬁeld Bext, along with both compo- nents BDC,xandBDC,zof the DC-Oersted ﬁeld. The material parameters of the magnetic layer are chosen as for Permalloy35since this is a widely used material which is less complex as it relates to deposition and microstructuring. The dimensions of the design and the parameters of the applied ﬁelds and currents as described earlier are chosen in such away as to ensure the realizability of the ﬁnally presented device by conventional methods. The micromagnetic simula- tions were carried out by utilizing the open-source simula- tion program MuMax3. 36The 19 lm long, 10 lm wide, and 30 nm high magnetic structure was discretized into cells of a 0003-6951/2017/111(12)/122401/5/$30.00 Published by AIP Publishing. 111, 122401-1APPLIED PHYSICS LETTERS 111, 122401 (2017) size of 10 nm /C210 nm /C230 nm. This cell size ensures that in-plane wave vectors of the spin waves up to approximately 0.1prad/nm can be resolved. To ensure that reﬂections of SW energy along the x-axis are suppressed, the SW damping at the vertical boundaries in Fig. 1(c) is incrementally increased in 25 steps over a distance of 0.5 lm to a value of a¼0.5. After calculating the ground state of the magnetiza- tion distribution inside the structure, a microwave magnetic ﬁeld bexc¼ðb0;0;0Þsinð2pftþ/0Þwith an extent of 0.5 lm in the x-direction in the middle of the input waveguide [seeFig.1(c)] is used to excite spin waves. The amplitude of this ﬁeld has been set to b0¼10 mT and the frequency to f¼7 GHz. The spin waves excited in this way represent the coherent signal of a magnonic network. To evaluate the simu- lation, the magnetization distribution is saved every 25 ps for a duration of 10 ns right after applying the excitation ﬁeld. For every cell, this data is independently Fourier transformed in time to access the time averaged, frequency dependent spatial distribution of the SW intensity. In the following, the SW intensity distributions are shown integrated over the frequency interval from f¼6.75 GHz to f¼7.25 GHz. Figure 2(a) shows the result of the simulation without any DC-current applied to the microstrip. It can clearly be seen that the SW energy splits in two focused SW beams at the opening of the 1D input waveguide into the 2D magnetic ﬁlm area (see Refs. 29and 30). The beam creation is explained by the broad angular spectrum of SW wave vec- tors originating from the waveguide opening in combination with the anisotropic SW dispersion relation. Due to this dipo- lar induced anisotropy, the SW group velocity features twospeciﬁc beam angles h Bwith respect to the local magnetiza- tion direction. Since the local ﬁeld and, accordingly, the magnetization distribution inside the ﬁlm are homogeneous, the beam angles do not change and the beams propagate in straight lines at angles of hB1¼75/C14andhB2¼105/C14in rela- tion to the y-axis. To estimate the occurring losses, the SW intensity shown in Fig. 2(a) is integrated along the y-direc- tion at the input ( x1¼0lm) and at a reference point (x2¼5lm). A comparison reveals an attenuation of 11 dB. However, effects like reﬂections of the SW signal in the tran- sition zone have only a minor inﬂuence. The main loss is caused by the intrinsic damping of the spin system which can be estimated to 9.1 dB for this particular case.37–39 Hence, a signiﬁcant reduction of the losses could be achieved by employing low-damping magnetic materials. The beam propagation completely changes if a DC-current is applied, as exemplarily shown in Fig. 2(b) for IDC¼–100 mA. In this case, the additional Oersted ﬁeld leads to a tilted local ﬁeld and a consequent inhomogeneous distribution of the magnetization direction inside the mag- netic ﬁlm. As a result, a curvilinear propagation of the SW FIG. 1. Sketch of the simulated structure. (a) Magnetic ﬁlm with adjacent input waveguide. An insulation layer separates an overlaying DC-microstrip from the magnetic structure. (b) In-plane component BDC,xof the Oersted magnetic ﬁeld at the position of the magnetic layer in case of a DC-current ofIDC¼–100 mA. This ﬁeld component leads to a change of the in-plane angle bof the local bias ﬁeld Bloc. The upper curve shows the resulting angle in case of an external ﬁeld of Bext¼50 mT. (c) Top view of the simulated magnetic structure with dimensions. The positions of the overlaying DC-microstrip and the local excitation ﬁeld are depicted by the orange and purple areas, respectively. FIG. 2. Results of the micromagnetic simulations. (a) Focused SW beams are created by opening the 1D input waveguide into the 2D magnetic ﬁlm. Without a DC-current applied to the microstrip, the external ﬁeld is homogeneous and a straight propagation of the SW beams can be observed at angles of hB1¼75/C14 andhB2¼105/C14in relation to the y-axis. (b) Curvilinear beam propagation of the SW beams due to an inhomogeneous magnetization distribution generated by the applied DC-current of IDC¼–100 mA and the resulting non-homogeneous local ﬁeld Bloc. This leads to an offset Dy. (c) Offset Dyand maximal intensity of the SW beams at the position x¼5lm as a function of an applied, negative DC-current IDC.122401-2 Heussner et al. Appl. Phys. Lett. 111, 122401 (2017)beams occurs since the anisotropy axes of the SW dispersion are locally rotated together with the magnetization. Thisrotation of the magnetization occurs mainly in the ﬁlm planeand is caused by the in-plane component B DC,xof the addi- tional Oersted ﬁeld. The out-of-plane component BDC,zwith a maximal absolute ﬁeld strength of around 16.7 mT is quiteweak compared to the magnetic ﬁeld of l 0MS¼1018 mT which would be necessary to orient the magnetization out ofplane. Therefore, the B DC,zcomponent leads only to a neglectable out-of-plane tilt. This simulation clearly demon- strates the steering effect on caustic-like SW beams by anexternally controlled inhomogeneity of the bias magneticﬁeld and the consequent inhomogeneous magnetizationdistribution. The curvilinear propagation results in an offset of the SW beams from its initial path. The offset Dy, which is deﬁned as the distance of the intensity maxima with andwithout a DC-current applied, is studied at the positionx¼5lm in dependence on a negative current I DC. As can be seen in Fig. 2(c), the increased offset caused by the rising absolute value of the DC-current IDCis accompanied with a change in the beam intensities. These changes can be relatedto a contraction or an extension of the propagation distancesof the focused SW beams until the evaluation point isreached. To demonstrate the relevance of the aforementioned results for the development of magnonic devices, the intrin-sic splitting of SW energy in the process of caustic-likebeam formation and the studied reconﬁgurable tractability ofthe beam directions are used to design a switchable SW split-ter. As an alternative to previous realizations of SW split-ters 18and SW switches,19–23the here presented device combines two important functionalities needed to realize amagnonic network, namely, the splitting of a SW signalenabling parallel data processing and its guidance throughthe network by controlled toggling between different wave-guides. Hereby, the controllability of the device is based oninhomogeneities of the magnetization distribution in the ﬁlmcaused by local bias ﬁeld inhomogeneities. The switchingdemands only the reversal of the direction of the ﬁeld gener-ating DC-current. Furthermore, the inhomogeneous magneti-zation distribution can also be created by other, more energyefﬁcient methods, e.g., stray ﬁelds of a nearby magnetic tun- nel junction, additionally enabling the creation of interfacesto technologically different networks. To design the switch-able SW splitter, in addition to the input waveguide and theadjacent unpatterned ﬁlm, output waveguides are added tothe structure at the position x¼4lm (see Fig. 3). The transi- tion zones between the magnetic ﬁlm and these output wave-guides are specially tailored to realize an efﬁcientchanneling of the SW energy into the output waveguides at a DC-current of I DC¼6100 mA. This value of the DC-current is chosen in accordance with the previous investigations toreach a sufﬁcient offset of the beams without serious loss inthe SW intensities. All other parameters of the simulationare equivalent to those mentioned earlier. Figure 3shows the resulting distribution of the SW intensity for three different values of the DC-current I DC, namely –100 mA, þ100 mA, and 0 mA. In Fig. 3 (a), a nega- tive current leads to the bending of the beams towards thenegative y-direction. This results in a channeling of the SW intensity into the middle and bottom output waveguide. Ifthe direction of the DC-current is reversed, the SW beamsbend into the opposite direction and a channeling of SWenergy into the middle and the upper output waveguideoccurs, as can be seen in Fig. 3(b). After the splitting and switching processes, the spin waves are channeled into thespeciﬁed output waveguides and subsequent magnonic devi-ces could be selected and provided with SW signals. A fast switching of the device is ensured by the SW intensity life- time of around 0.6 ns/rad 37,38in the presented structure. Spurious SW signals of the switching process are dampedout within this timeframe leading to a switching time of nomore than 4 ns, as has been veriﬁed in further simulations(not shown here). To realize short switching times in mag-netic materials with lower damping parameters, damping-like spin-transfer-torque 40,41could be used to suppress the spurious signals. In addition to the channeling of the spinwaves into selected output waveguides, a further functional-ity arises if no DC-current is applied. This case is shown inFig.3(c) and reveals a blocking of the incoming SW energy forI DC¼0 mA. To utilize the switchable SW signal splitter in magnonic networks, essential requirements have to be fulﬁlled. One FIG. 3. Design of a switchable SW signal splitter based on focused SW beams and its curvilinear propagation in inhomogeneous magnetized ﬁlms. (a) and ( b) Channeling of SW energy into different output waveguides depending on the applied DC-current. (c) Suppression of the output signal, if no DC-current is applied.122401-3 Heussner et al. Appl. Phys. Lett. 111, 122401 (2017)approach for SW based logic devices is to carry the information in the phase of the spin waves and to perform logic opera- tions by employing interference effects.15,16,27As a conse- quence, any device inside such a magnonic network has to preserve the coherence of the transmitted spin wave. It has been shown for metallic systems such as Permalloy that the coherence of spin waves is preserved during their whole propagation time.38In the case of the presented switchable splitter, the focus is on the spatial coherence in terms of a well-deﬁned wave vector of the SW signal in the output waveguides with a deﬁned phase evolution. This is of special interest since the preceding caustic-like SW beams consist of a large number of plane spin-wave modes with differentwave vectors. Additionally, it is mandatory that the phase of the output signal is determined by the phase of the input sig- nal. To verify that these requirements are fulﬁlled, the phase evolution of the output signals for the case of a negative DC- current of I DC¼–100 mA applied to the microstrip is studied in more detail. For this purpose, the dynamic component of the magnetization in the x-direction is extracted from the simulation 5 ns after the start of the SW excitation since the SW propagates through the device within around 3 ns. Figure 4shows the amplitudes of the SW signals integrated over the width of the output waveguides starting at the point (x¼6.2lm) where the regular width of 1 lm of the wave- guides is reached. Oscillations along the waveguide width, which can appear due to spurious edge modes, are averaged out by this integration just as it would be the case if inductive detection methods utilizing microwave antennas would be used. The nearly vanishing SW amplitude in the upper wave-guide in comparison with the strong signals in the middle and lower output waveguide demonstrates the efﬁciency of the switch functionality of the device. Moreover, the ampli- tude modulations are shown for three different initial phases / 0of the excitation ﬁeld. In Figs. 4(b) and4(c), damped sinusoidal curves of the form y¼y0þAexpð/C0x=d0Þsinð2pðx/C0x0Þ=kÞ, ﬁtted the data in the case of /0¼p/2, are added. These ﬁtting curvesreveal an averaged wavelength of k¼1.9060.08lm and an averaged decay length of d0¼4.1560.11lm in the out- put waveguides. The expected wavelength can be calculated according to the analytical theory of Ref. 39. These calcula- tions37,42yield a value of kcal,1¼1.8660.13lm for the 1st transversal waveguide mode and kcal,2¼1.3460.12lm, kcal,3¼1.0860.11lm for higher modes. The comparison of the calculated values and the observed wavelength demon- strates a dominant propagation of the 1st waveguide mode in the output waveguides. Hence, the developed device design enables the transformation of multi-mode focused SW beams back to a SW signal with a well-deﬁned wave vector and a deﬁned phase evolution in the output waveguides. The fulﬁllment of the second requirement can be seen from Fig. 4by means of the shift of the maxima of curves corresponding to different initial phases /0within one wave- guide. This shift demonstrates that the information, which is encoded in a phase shift in the input waveguide, is preserved during the splitting and channeling process of the focused SW beams and leads to an analogous phase shift in the out- put waveguides. When comparing the data curves of corre- sponding initial phases /0in the middle and lower waveguides, a spatial shift of the maxima is visible as well. This can be understood by the different propagation distan- ces of the SW beams in the unstructured ﬁlm area until the output waveguides are reached and the consequential differ- ent phase accumulation during the propagation. In summary, we reveal by micromagnetic simulations the particular role which caustic-like SW beams propagating in unstructured magnetic ﬁlm areas can play in the emerging ﬁeld of magnonics. We have demonstrated the ability of the local manipulation of the SW beam direction by an inhomo- geneous magnetization distribution and its controllability by locally applied magnetic ﬁelds. Based on these results, such an essential element for magnonic networks as a switchableSW signal splitter was developed. The presented design of the device makes it possible to channel spin waves into dif- ferent output waveguides or to block them depending on the local ﬁeld generating DC-current. Furthermore, a coherent propagation of the output SW signal with a well-deﬁned wave vector in the output waveguides was demonstrated, which is an important requirement for devices designed for magnonic networks. The authors thank A.V. Chumak for valuable discussions. Financial support from DFG within project B01 of SFB/TRR 173, Spin þX - Spin in its collective environment, is gratefully acknowledged. 1A. Khitun, M. Bao, and K. Wang, IEEE Trans. Magn. 44, 2141 (2008). 2A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015). 3V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys. 43, 264001 (2010). 4A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl. Phys. 43, 264002 (2010). 5G. Csaba, A. Papp, and W. Porod, J. Appl. Phys. 115, 17C741 (2014). 6G. Csaba, A. Papp, and W. Porod, Phys. Lett. A 381, 1471 (2017). 7O. Zografos, P. Raghavan, L. Amar /C18u, B. Sor /C19ee, R. Lauwereins, I. Radu, D. Verkest, and A. Thean, System-level assessment and area evaluation of spin wave logic circuits, in 2014 IEEE/ACM International Symposium on Nanoscale Architectures (NANOARCH) (2014).FIG. 4. SW amplitude of the output signal integrated over the waveguide width 5 ns after the SW excitation. Three cases with different initial phases /0of the excitation ﬁeld are shown, always with a DC-current of IDC¼–100 mA applied to the microstrip. Exemplary for all initial phases, the integrated SW amplitude inside (b) the middle output waveguide and (c) the lower output waveguide is ﬁtted by a damped sinusoidal curve for /0¼p/2.122401-4 Heussner et al. Appl. Phys. Lett. 111, 122401 (2017)8A. Khitun, J. Appl. Phys. 111, 054307 (2012). 9E. Egel, C. Meier, G. Csaba, and S. Breitkreutz-von Gamm, AIP Adv. 7, 056016 (2017). 10T. Br €acher, F. Heussner, P. Pirro, T. Meyer, T. Fischer, M. Geilen, B. Heinz, B. L €agel, A. A. Serga, and B. Hillebrands, Sci. Rep. 6, 38235 (2016). 11A. Khitun, Energy dissipation in magnonic logic circuits in IEEE 12th International Conference on Nanotechnology (IEEE-NANO) (2012). 12F. Garcia-Sanchez, P. Borys, R. Soucaille, J.-P. Adam, R. L. Stamps, andJ.-V. Kim, Phys. Rev. Lett. 114, 247206 (2015). 13K. Wagner, A. K /C19akay, K. Schultheiss, A. Henschke, T. Sebastian, and H. Schultheiss, Nat. Nanotechnol. 11, 432 (2016). 14O. Dzyapko, I. V. Borisenko, V. E. Demidov, W. Pernice, and S. O. Demokritov, Appl. Phys. Lett. 109, 232407 (2016). 15S. Klingler, P. Pirro, T. Br €acher, B. Leven, B. Hillebrands, and A. V. Chumak, Appl. Phys. Lett. 106, 212406 (2015). 16T. Fischer, M. Kewenig, D. A. Bozhko, A. A. Serga, I. I. Syvorotka, F. Ciubotaru, C. Adelmann, B. Hillebrands, and A. V. Chumak, Appl. Phys. Lett. 110, 152401 (2017). 17K.-S. Lee and S.-K. Kim, J. Appl. Phys. 104, 053909 (2008). 18A. V. Sadovnikov, C. S. Davies, S. V. Grishin, V. V. Kruglyak, D. V. Romanenko, Y. P. Sharaevskii, and S. A. Nikitov, Appl. Phys. Lett. 106, 192406 (2015). 19C. S. Davies, A. Francis, A. V. Sadovnikov, S. V. Chertopalov, M. T.Bryan, S. V. Grishin, D. A. Allwood, Y. P. Sharaevskii, S. A. Nikitov, and V. V. Kruglyak, Phys. Rev. B 92, 020408(R) (2015). 20C. S. Davies, A. V. Sadovnikov, S. V. Grishin, Y. P. Sharaevsky, S. A. Nikitov, and V. V. Kruglyak, IEEE Trans. Magn. 51, 3401904 (2015). 21K. Vogt, H. Schultheiss, S. Jain, J. E. Pearson, A. Hoffmann, S. D. Bader, and B. Hillebrands, Appl. Phys. Lett. 101, 042410 (2012). 22K. Vogt, F. Y. Fradin, J. E. Pearson, T. Sebastian, S. D. Bader, B. Hillebrands, A. Hoffmann, and H. Schultheiss, Nat. Commun. 5, 3727 (2014). 23T. Schneider, A. A. Serga, B. Hillebrands, and M. Kostylev,J. Nanoelectron. Optoelectron. 3(1), 69 (2008). 24T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps, and M. P. Kostylev, Appl. Phys. Lett. 92, 022505 (2008). 25T. Br €acher, F. Heussner, P. Pirro, T. Fischer, M. Geilen, B. Heinz, B. L€agel, A. A. Serga, and B. Hillebrands, Appl. 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B 95, 224412 (2017). 35Material parameters: saturation magnetization MS¼810 kA/m, exchange constant Aex¼13 pJ/m, Gilbert damping constant a¼8/C210–3. 36A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, AIP Adv. 4, 107133 (2014). 37T. Br €acher, O. Boulle, G. Gaudin, and P. Pirro, Phys. Rev. B 95, 064429 (2017). 38P. Pirro, T. Br €acher, K. Vogt, B. Obry, H. Schultheiss, B. Leven, and B. Hillebrands, Phys. Status Solidi B 248(10), 2404–2408 (2011). 39B. A. Kalinikos and A. N. Slavin, J. Phys. C: Solid State Phys. 19, 7013 (1986). 40V. E. Demidov, S. Urazhdin, E. R. J. Edwards, M. D. Stiles, R. D. McMichael, and S. O. Demokritov, Phys. Rev. Lett. 107, 107204 (2011). 41T. Meyer, T. Br €acher, F. Heussner, A. A. Serga, H. Naganuma, K. Mukaiyama, M. Oogane, Y. Ando, B. Hillebrands, and P. Pirro, IEEE Magn. Lett. 8, 318005 (2017). 42The calculations consider the effective magnetic ﬁeld Beff¼31.2560.65 mT and the effective waveguide width weff¼0.860.1lm of the output waveguides, which are extracted from the simulation. These effective val-ues and their uncertainties take into account demagnetization ﬁelds which lead to strongly decreased internal magnetic ﬁelds at the edges of the waveguides and a variation of the internal ﬁeld during the transition from the unstructured ﬁlm area to the waveguides.122401-5 Heussner et al. Appl. Phys. Lett. 111, 122401 (2017)
1.5129996.pdf	AIP Advances 10, 015013 (2020); https://doi.org/10.1063/1.5129996 10, 015013 © 2020 Author(s).Dynamic magnetic properties of amorphous Fe80B20 thin films and their relation to interfaces Cite as: AIP Advances 10, 015013 (2020); https://doi.org/10.1063/1.5129996 Submitted: 21 October 2019 . Accepted: 05 December 2019 . Published Online: 07 January 2020 U. Urdiroz , B. M. S. Teixeira , F. J. Palomares , J. M. Gonzalez , N. A. Sobolev , F. Cebollada , and A. Mayoral AIP Advances ARTICLE scitation.org/journal/adv Dynamic magnetic properties of amorphous Fe80B20thin films and their relation to interfaces Cite as: AIP Advances 10, 015013 (2020); doi: 10.1063/1.5129996 Presented: 6 November 2019 •Submitted: 21 October 2019 • Accepted: 5 December 2019 •Published Online: 7 January 2020 U. Urdiroz,1,2B. M. S. Teixeira,3 F. J. Palomares,1 J. M. Gonzalez,1 N. A. Sobolev,3,4 F. Cebollada,2,a) and A. Mayoral5 AFFILIATIONS 1Instituto de Ciencia de Materiales de Madrid (CSIC), Sor Juana Inés de la Cruz, 3, 28049 Madrid, Spain 2POEMMA-CEMDATIC, ETSI de Telecomunicación, Universidad Politécnica de Madrid, 28040 Madrid, Spain 3Departamento de Física and I3N, Universidade de Aveiro, 3810-193 Aveiro, Portugal 4National University of Science and Technology “MISiS”, 119049 Moscow, Russia 5Advanced Microscopy Laboratory - Nanoscience Institute of Aragon (LMA-INA), Mariano Esquillor, s/n, 50018 Zaragoza, Spain Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials. a)Corresponding author: Federico cebollada, e-mail: fcebollada@etsit.upm.es ABSTRACT We present a ferromagnetic resonance study of the dynamic properties of a set of amorphous Fe-B films deposited on Corning Glass ®and MgO (001) substrates, either with or without capping. We show that the in plane anisotropy of the MgO grown films contains both uniaxial and biaxial components whereas it is just uniaxial for those grown on glass. The angular dependence of the linewidth strongly differs in terms of symmetry and magnitude depending on the substrate and capping. We discuss the role of the interfaces on the magnetization damping and the generation of the anisotropy. We obtained values of the intrinsic damping parameters comparable to the lowest ones reported for amorphous films of similar compositions. ©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5129996 .,s INTRODUCTION Designing magnetic materials for high frequency applications is crucial for emerging magnetic technologies such as spintron- ics and magnonics.1,2Relevant to those applications is under- standing the magnetization relaxation mechanisms of thin films. The damping parameter αin the Landau-Lifshitz-Gilbert (LLG) equation is directly related to the ferromagnetic resonance (fmr) peaks’s linewidth ΔH=ΔHo+ΔHm+ΔHG+ΔHTMS.3The first two terms, frequency independent, correspond to inhomogeneities (ΔHo) and mosaicity ( ΔHm); the isotropic, intrinsic Gilbert term ΔHGresults from the energy transfer from magnetization to lat- tice; finally, ΔHTMS gives the “two magnon scattering” (TMS), due to the energy transfer from the fmr uniform mode (wavevector⃗k=0) to degenerate magnons with⃗k≠0.4–8Many works have analized the role of the structure on the damping in magnetic films. Studies of the effects of interfaces, dislocation networks or specific surfacefeatures provide examples of the extrinsic character of the relaxation mechanisms.9–11Until recently, little attention has been paid to the damping mechanisms of amorphous transition metal-metalloid thin films,12–14which are good candidates for low damping materials due to their homogeneity and to the possibility of tailoring their mag- netic properties by thermal treatments.15In this paper we study the dynamic magnetic properties of amorphous Fe 80B20alloys deposited on Corning Glass ®and MgO (001) substrates, either Au capped or uncapped. EXPERIMENTAL Amorphous F 80B20thin films were grown by means of a Nd- YAG Pulsed Laser Deposition (PLD) system ( λ=532 nm, 4 ns pulses of 180 mJ, 10 Hz rate), under ultrahigh vacuum conditions. Two films, 20 nm thick, were deposited on square 5x5 mm2Corning Glass®substrates, one of them uncapped (C0), the other capped AIP Advances 10, 015013 (2020); doi: 10.1063/1.5129996 10, 015013-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 1 . Easy and hard axis loops of fhe MAu film; angular dependence of its coercivity (inset). with a 7 nm Au layer (CAu). Another two films with the same thick- ness were deposited on MgO (001) substrates with the same dimen- sions, one uncapped (M0), the other capped with a 7 nm Au layer (MAu). Their amorphicity was checked by X-ray difraction (XRD) using an 8 circles Bruker diffractometer and Transmission Electron Microscopy (TEM), using a FEI TITAN low base and a FEI high res- olution TITAN. Their magnetic hysteresis was studied by transverse magnetooptic Kerr effect (MOKE), under maximum applied fields of 0.5 T. A Bruker E-500 electron paramagnetic resonance spectrom- eter (X-band, f = 9.87 GHz) was employed to study their magnetiza- tion dynamics, through the in-plane (IP) angular dependence of the fmr spectra down from saturation at 1.4 T, obtained measuring thederivative of the imaginary part of the dynamic susceptibility in a Lock-in arrangement. RESULTS AND DISCUSSION Figure 1 presents two hysteresis loops measured in the MAu film with the applied field parallel to each diagonal of the substrate. One of the loops is square, with a coercivity close to 5 Oe and a reduced remanence approximately equal to 1, corresponding to a magnetic easy axis (e.a). A magnetization rotation loop is observed along the second diagonal, corresponding to a hard axis (h.a), with little hysteresis and a saturation field of about 15 Oe. The angular evolution of the coercivity (Figure 1, inset) and the remanence show a two-fold, butterfly shape characteristic of uniaxial anisotropy, with minimum values along the h.a. diagonal. All other films present sim- ilar features: uniaxial anisotropy with e.a coercivity of a few Oe and h.a. saturation field between 15 and 35 Oe, with the relevant differ- ence that whereas the easy and hard axes of the M films are parallel to the diagonals, those of the C films are parallel to the substrate sides. The angular dependence of the resonance field H r(Figures 2(a) and (b)) exhibits two-fold symmetry in all cases. The main differ- ences between the C and the M samples are: (i) the spectra of M films present a single peak along the full angular range and the angu- lar evolution of H ris not purely symmetric around the maxima and minima; (ii) the spectra of the C films present two overlap- ping peaks in the angular range 45○-120○and 225○-300○, approxi- mately, (Figure 2(a), inset) and the angular dependence of the res- onance field is highly symmetric around the maxima and minima. The fits of the resonance field (red lines) to the Smit-Beljers formal- ism16included an IP unaxial and a cubic anisotropy contribution given by FIG. 2 . Angular dependence of the reso- nance field: C (a) and M (b) films (inset: split peaks measured in C0 at the indi- cated angles). Angular dependence of the linewidth: C (c) and M (d) films. Red lines: fits indicated in the text. AIP Advances 10, 015013 (2020); doi: 10.1063/1.5129996 10, 015013-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv F=1 2μ0M2 Scos2θ−μ0MsHsinθcos(η−ϕ)−kusin2θcos2(ϕ−ξ) +kc 4[sin22θ+sin4θsin22ϕ] (1) whereθandϕare the magnetization polar and azimuthal angles, respectively, K u(Kc) is the uniaxial (cubic) anisotropy constant, M s is the spontaneous magnetization, and the cubic (100) and (010) directions are taken as the x and y axes, also respectively. The angles ξ andη, from the x axis, correspond to the uniaxial e.a. and the applied field H direction (both IP). Table I summarizes the fitting parameters and the h.a. saturation field H sat. As it can be seen, the anisotropy of the C films is purely uniaxial whereas that of the M films has a non-negligible cubic contribu- tion, in addition to the different orientations of the e.a. However, the anisotropy of the M samples is much weaker than that of the C samples. The good agreement between the uniaxial anisotropy field HKUcalculated from the anisotropy and the h.a. saturation field is remarkable. The usual sources of anisotropy in amorphous ferro- magnets are (i) the magnetoelastic coupling between magnetization and internal stresses, which gives rise to easy (hard) axes in the regions with tensile (compressive) stresses if the magnetostriction is positive, and (ii) the presence of magnetic fields during the fabrica- tion process or annealings. Both produce just uniaxial anisotropies, in no case biaxial anisotropy schemes.15The presence of stray fields in our fabrication setup can be ruled out since the different orien- tations of the axes are not compatible with the identical orientation of all substrates on the sample holder. The anisotropy of Fe-B bulk alloys, about 2 kJm-3, has been calculated from their domain patterns and wall nucleation and pinning magnetization mechanisms.17The much lower anisotropy in our films and the well defined orienta- tions of the easy and hard axes are a clear indication of their weak internal stresses (compared to bulk) and, more important, of their spatial homogeneity. The stronger anisotropy of the C films suggests that the stresses induced by the substrate are much stronger than in the M films. Their origin is unclear, a plausible mechanism might be related to the holder-substrate fixation system, which might bend slightly the glass. If the film accomodates to it during the deposition, it will become subjected to the inverse effort after the substrate is relieved from the holder. The weaker anisotropy of the M films indi- cates that the stresses introduced during the fabrication are lower, probably due to the higher MgO stiffness. The relevant point is the source of the biaxial component of the anisotropy, which is unusual in amorphous alloys. The TEM studies carried out on an uncapped film deposited on MgO under similar conditions have revealed the formation of a bcc Fe layer, TABLE I . Fitting parameters from equation (1) and saturation field obtained from the h.a. loops. KU KCμ0MS HKU HKC Hsat Film (Jm-3) (Jm-3) (T) (Oe) (Oe) (Oe) C0 1640 - 1.40 29 - 35 CAu 1030 - 1.44 18 - 20 M0 530 260 1.66 8.0 4.0 9 MAu 750 160 1.51 12.4 2.7 13 FIG. 3 . TEM image of a film deposited on MgO. Inset: high resolution image of the region marked with a yellow square and its Fourier Transform. about 1 nm thick, at the amorphous-substrate interface and of an oxide layer on the free surface. Figure 3 shows the Fe layer, with a high resolution image (inset) corresponding to the yellow square in the figure. The Fourier transform of this image demonstrates its crystalline nature, the distances calculated for neighboring (100) and (110) planes agreeing with those of bcc Fe, which usually grows epi- taxially, with 4% misfit, on MgO (001) with the (100) and (010) axes rotated 45○with respect to those of MgO.18The Fe layer can be related to the cubic anisotropy detected by fmr and to the orienta- tion of the uniaxial e.a. along one diagonal. A plausible mechanism for the formation of the e.a. is the orientation of the magnetiza- tion of the Fe layer along one of its easy axes during the deposition. The dipolar and/or exchange coupling of the Fe magnetization with that of the layer growing on top of the crystalline layer could act as anisotropy inducing agents. The cubic anisotropy contribution probably results from the interfacial exchange coupling between the Fe layer and film, similar to that ocurring in exchange biased sys- tems. The interfacial exchange is likely to extend its influence to the full amorphous layer since its exchange length is of a few tens of nanometers.19 Up to now, the effect of the Au capping or the free surface oxide has not been discussed. It is evident that it plays no major role in the orientation of the anisotropy axes or the intensity of the inter- nal stresses. In fact, the uncapped C0 film has roughly 50% higher anisotropy than its capped counterpart whereas the anisotropy of the uncapped M0 film is weaker than that of MAu. However, its influence is quite noticeable in the peak linewidth (Figures 2(c) and (d)). Both C0 and CAu films have similar linewidth angular evolu- tion, two-fold with the maxima shifted ca. 45○with respect to the resonance field, where the resonance peak splits. This indicates the presence of a common underlying broadening mechanism. Yet, the magnitude of the linewidth increases in the uncapped sample. The eventual presence of an oxide layer in the uncapped film could be the reason for the large damping increase, likely related to increased inhomogeneities at the amorphous/oxide interface. TMS has been proposed as a source of increased damping in films with linear AIP Advances 10, 015013 (2020); doi: 10.1063/1.5129996 10, 015013-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv structural features.6,9,10Spin waves with wave vector⃗k≠0,⃗kperpen- dicular to the linear structures, appear when the angle between mag- netization and⃗kis below a critical angle ϕC=arcsin[(μ0H0/B0)1/2], where H 0includes the applied and anisotropy fields and B0=μ0(H0 +MS). When the degenerate states can no longer be treated as a per- turbation, a resonance peak splitting occurs. We propose the pres- ence of linear tensile stresses parallel to the e.a. as the sole source of anisotropy in the C films. The tensile stresses, taken as linear per- turbations with defined orientations, could eventually increase the TMS, leading even to a peak splitting.9However, the critical angle around the perpendicular to the stresses calculated for the C films is roughly ±14○, much narrower than the measured value. Another plausible explanation might be related to inhomogeneous tensile stresses (variation in their orientation) confined in small regions but large enough to provide separate resonances. The oxide layer of M0 does not increase the linewidth as dramatically as in C0. The complex two-fold structure of the M0 linewidth angular evolution breaks the symmetry of both the uniaxial and biaxial structural fea- tures responsible for its anisotropy. In contrast, the MAu linewidth is dominated by a four-fold component, likely related to the Fe layer. The IP angular evolution of the linewidth TMS contribution repro- duces the symmetry of the scattering centers, if they centers are linked to the crystal structure, and it can be expressed as a func- tion of the orientation of the crystal axes:8,11The linewidth is then proportional to αTMS=∑XiΓ(Xi)f(ϕH−ϕ(Xi)) (2) Γ(Xi)is the scattering factor along the main crystal directions and f(ϕH−ϕ(Xi)) depends on the applied field direction with respect to them. The linewidth of the capped films can be fitted to an isotropic valueΔHisoplus a function corresponding to equation (2). In the case of amorphous films, those directions could be associated with the internal stress lines and, in MAu, the crystal directions of the Fe interfacial layer. The fits of the CAu and MAu linewidths to ΔHiso plus a function ΔH=A sin2(ϕH−ϕ1)+B sin2(2(ϕH−ϕ2)) (continu- ous red lines in Figure 2) yield close ΔHisovalues (23.5 and 27.5 Oe, respectively) and the following parameters for MAu (CAu): A=3.5 (10.1) Oe;φ1=7.4○(8.2○); B=10.8 Oe; φ2=7.4○(no four-fold compo- nent in CAu). ΔHisorepresents an upper limit of the intrinsic Gilbert Damping, which can be written as8,11 μ0ΔHiso=ΔH0+α4πf γ(3) whereγis the gyromagnetic factor (the mosaicity term can be excluded due to the amorphous nature of our films, at least for the C films). The upper limits for the intrinsic damping coefficient αare 3.3⋅10-3and 4.0 ⋅10-3for CAu and MAu, respectively, comparable to the lowest values reported for amorphous films of Fe and Fe-Co base.12,13,20 CONCLUSIONS We studied the role of the film-substrate and film-capping interfaces on the dynamic properties of amorphous Fe-B films. We showed that the films deposited on glass present stronger IPanisotropy than those deposited on MgO (001), probably due to higher residual stresses, and that the formation of a thin Fe layer on MgO induces a four-fold anisotropy, not usual in amorphous alloys. The damping of the uncapped films is increased due to the oxide layer on top. The damping of the capped samples can be interpreted as a combination of an isotropic and an angle dependent contribu- tion, probably related to TMS. The role of the linear stresses in the amorphous phase and of the exchange Fe-FeB in the MgO/FeB/Au film was discussed. ACKNOWLEDGMENTS We thank the financial support by Spanish MINECO, Grant Nos. MAT2013-47878-C2-R and MAT2016-80394-R. U.U. acknowledges FPI grant BES-2014-070387. B.M.S.T. and N.A.S. acknowledge financial support of the FCT of Portugal through the Project No. I3N/FSCOSD (Ref. FCT UID/CTM/50025/2019) and through the bursary PD/BD/113944/2015. N.A.S. was supported by the Ministry of Education and Science of the Russian Federation in the framework of the Increase Competitiveness Program of NUST “MISiS” (no. K2-2019-015). REFERENCES 1C. Chappert, A. Fert, and F. N. V. Dau, Nat. Mater. 6, 813 (2007). 2V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D Appl. Phys. 43, 264001 (2010). 3S. Akansel, A. Kumar, N. Behera, S. Husain, R. Brucas, S. Chaudhary, and P. Svedlindh, Phys. Rev. B 97, 134421 (2018). 4R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999). 5D. L. Mills and S. M. Rezende, in Spin Dynamics in Confined Magnetic Struc- tures II , (B. Hillebrands and K. Ounadjela eds.) Topics in Appl. Phys. 87, 27 (2003). 6R. Arias and D. L. Mills, J. Appl. Phys. 87, 5455 (2000). 7Kh. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M. Farle, U. von Hörsten, H. Wende, W. Keune, J. Rocker, S. S. Kalarickal, K. Lenz, W. Kuch, K. Baberschke, and Z. Frait, Phys. Rev. B 76, 104416 (2007). 8A. Conca, S. Keller, M. R. Schweizer, E. Th. Papaioannou, and B. Hillebrands, Phys. Rev. B 98, 214439 (2018). 9R. D. McMichael, D. J. Twisselmann, J. E. Bonevich, A. P. Chen, and W. F. Egelhoff, Jr., J. Appl. Phys. 91, 8647 (2002). 10M. Körner, K. Lenz, R. A. Gallardo, M. Fritzsche, A. Mücklich, S. Facsko, J. Lindner, P. Landeros, and J. Fassbender, Phys. Rev. B 88, 054405 (2013). 11G. Woltersdorf and B. Heinrich, Phys. Rev. B 69, 184417 (2004). 12M. Konoto, H. Imamura, T. Taniguchi, K. Yakushiji, H. Kubota, A. Fukushima, K. Ando, and S. Yuasa, Appl. Phys. Express 6, 073002 (2013). 13M. Bersweiler, H. Sato, and H. Ohno, IEEE Mag. Lett. 8, 1 (2017). 14J. Wang, C. Dong, Y. Wei, X. Lin, B. Athey, Y. Chen, A. Winter, G. M. Stephen, D. Heiman, Y. He, H. Chen, X. Liang, C. Yu, Y. Zhang, E. J. Podlaha-Murphy, M. Zhu, X. Wang, J. Ni, M. McConney, J. Jones, M. Page, K. Mahalingam, and N. X. Sun, Phys. Rev. Applied 12, 034011 (2019). 15T. Egami, Rep. Prog. Phys. 47, 1601 (1984). 16J. Smit and H. G. Beljers, Philips Res. Rep. 10, 113 (1955). 17G. Schroeder, R. Schäfer, and H. Kronmüller, Phys. Stat. Sol. A 50, 475 (1978). 18E. Paz, F. Cebollada, F. J. Palomares, F. Garcia-Sanchez, and J. M. González, Nanotechnology 21, 255301 (2010). 19J. Stöhr and H. C. Siegmann, Magnetism (Springer, Berlin, 2006), p. 514. 20T. Devolder, P.-H. Ducrot, J.-P. Adam, I. Barisic, N. Vernier, J.-V. Kim, B. Ockert, and D. Ravelosona, Appl. Phys. Lett. 102, 022407 (2013). AIP Advances 10, 015013 (2020); doi: 10.1063/1.5129996 10, 015013-4 © Author(s) 2020
1.2734118.pdf	Parametric instability of the helical dynamo Marine Peyrota/H20850and Franck Plunianb/H20850 Laboratoire de Géophysique Interne et Tectonophysique, CNRS, Université Joseph Fourier, Maison des Géosciences, B.P . 53, 38041 Grenoble Cedex 9, Franceand Laboratoire des Ecoulements Géophysiques et Industriels, CNRS, Université Joseph Fourier, INPG,B.P . 53, 38041 Grenoble Cedex 9, France Christiane Normandc/H20850 Service de Physique Théorique, CEA/DSM/SPhT, CNRS/URA 2306, CEA/Saclay, 91191 Gif-sur-Yvette Cedex, France /H20849Received 4 December 2006; accepted 3 April 2007; published online 30 May 2007 /H20850 We study the dynamo threshold of a helical ﬂow made of a mean plus a ﬂuctuating part. Two ﬂow geometries are studied: /H20849i/H20850solid body and /H20849ii/H20850smooth. Two well-known resonant dynamo conditions, elaborated for stationary helical ﬂows in the limit of large magnetic Reynolds numbers, are testedagainst lower magnetic Reynolds numbers and for ﬂuctuating ﬂows with zero mean. For a ﬂowmade of a mean plus a ﬂuctuating part, the dynamo threshold depends on the frequency and thestrength of the ﬂuctuation. The resonant dynamo conditions applied on the ﬂuctuating /H20849respectively, mean /H20850part seems to be a good diagnostic to predict the existence of a dynamo threshold when the ﬂuctuation level is high /H20849respectively, low /H20850.©2007 American Institute of Physics . /H20851DOI: 10.1063/1.2734118 /H20852 I. INTRODUCTION In the context of recent dynamo experiments,1–3an im- portant question is to identify the relevant physical param-eters that control the dynamo threshold and eventually mini-mize it. In addition to the parameters usually considered,such as the geometry of the mean ﬂow 4,5or the magnetic boundary conditions,6,7the turbulent ﬂuctuations of the ﬂow seem to have an important inﬂuence on the dynamothreshold. 8–11Some recent experimental results12,13suggest that the large spatial scales of these ﬂuctuations could play adecisive role. In this paper we consider a ﬂow of large spatial scale, ﬂuctuating periodically in time, such that its geometry atsome given time is helical. Such helical ﬂows have beenidentiﬁed to produce dynamo action. 14,15Their efﬁciency has been studied in the context of fast dynamo theory16–21and they have led to the realization of several dynamoexperiments. 3,22–24 The dynamo mechanism of a helical dynamo is of stretch-diffuse type. The radial component Brof the mag- netic ﬁeld is stretched to produce a helical ﬁeld /H208490,B/H9258,Bz/H20850, where /H20849r,/H9258,z/H20850are the cylindrical coordinates. The magnetic diffusion of the azimuthal component B/H9258produces some ra- dial component Brdue to the cylindrical geometry of the problem.17In this paper we shall consider two cases, depend- ing on the type of ﬂow shear necessary for the Brstretching. In case /H20849i/H20850, the helical ﬂow is solid body for r/H110211 and at rest for r/H110221/H20849the same conductivity is assumed in both do- mains /H20850. The ﬂow shear is then inﬁnite and localized at the discontinuity surface r=1. Gilbert17has shown that this dy-namo is fast /H20849positive growth rate in the limit of large mag- netic Reynolds number /H20850and is thus very efﬁcient to generate a helical magnetic ﬁeld of same pitch as the ﬂow. In case /H20849ii/H20850, the helical ﬂow is continuous, and equal to zero for r/H113501. The ﬂow shear is then ﬁnite at any point. Gilbert17has shown that such a smooth helical ﬂow is a slow dynamo and that thedynamo action is localized at a resonant layer r=r 0such that 0/H11021r0/H110211. Contrary to case /H20849i/H20850, having a conducting external medium is not necessary here. In both cases some resonant conditions leading to dy- namo action have been derived.16–18,20,21Such resonant conditions can be achieved by choosing an appropriate ge- ometry of the helical ﬂow, such as changing its geometricalpitch. They have been derived for a stationary ﬂow U/H20849r, /H9258,z/H20850 and can be generalized to a time-dependent ﬂow of the form U/tildewidest/H20849r,/H9258,z/H20850·f/H20849t/H20850, where f/H20849t/H20850is a periodic function of time. Now taking a ﬂow composed of a mean part Uplus a ﬂuctuating part U/tildewidest·f/H20849t/H20850, we expect the dynamo threshold to depend on the geometry of each part of the ﬂow accordingly to the resonant condition of each of them and to the ratio of the intensities /H20841U/tildewidest/H20841//H20841U/H20841. However, we shall see that in some cases even a small intensity of the ﬂuctuating part may have adrastic inﬂuence. The results also depend on the frequencyoff/H20849t/H20850. The Ponomarenko dynamo /H20851case /H20849i/H20850/H20852ﬂuctuating periodi- cally in time and with a ﬂuctuation of inﬁnitesimal magni-tude had already been the object of a perturbativeapproach. 25Here we consider a ﬂuctuation of arbitrary mag- nitude. Comparing our results for a small ﬂuctuation magni-tude with those obtained with the perturbative approach, wefound signiﬁcant differences. We then realized that there wasan error in the computation of the results published in Ref.25/H20849though the perturbative development in itself is correct /H20850. In Appendix E, we give an erratum of these results. a/H20850Electronic mail: Marine.Peyrot@ujf-grenoble.fr b/H20850Electronic mail: Franck.Plunian@ujf-grenoble.fr c/H20850Electronic mail: Christiane.Normand@cea.frPHYSICS OF FLUIDS 19, 054109 /H208492007 /H20850 1070-6631/2007/19 /H208495/H20850/054109/14/$23.00 © 2007 American Institute of Physics 19, 054109-1II. MODEL We consider a dimensionless ﬂow deﬁned in cylindrical coordinates /H20849r,/H9258,z/H20850by U=/H208510,r/H9024/H20849r,t/H20850,V/H20849r,t/H20850/H20852·h/H20849r/H20850 with h/H20849r/H20850=/H208771, r/H110211, 0, r/H110221,/H208491/H20850 corresponding to a helical ﬂow in a cylindrical cavity which is inﬁnite in the zdirection, the external medium being at rest. Each component, azimuthal and vertical, of the dimen-sionless velocity is deﬁned as the sum of a stationary partand of a ﬂuctuating part: /H9024/H20849r,t/H20850=/H20851R ¯m+R/tildewidestmf/H20849t/H20850/H20852/H9264/H20849r/H20850, /H208492/H20850 V/H20849r,t/H20850=/H20851R¯m/H9003¯+R/tildewidestm/H9003/tildewidestf/H20849t/H20850/H20852/H9256/H20849r/H20850, where R ¯mand/H9003¯/H20849R/tildewidestmand/H9003/tildewidest/H20850are the magnetic Reynolds num- ber and a characteristic pitch of the stationary /H20849ﬂuctuating /H20850 part of the ﬂow, respectively. In what follows we consider aﬂuctuation periodic in time, of the form f/H20849t/H20850=cos /H20849 /H9275ft/H20850. De- pending on the radial proﬁles of the functions /H9264and/H9256,w e determine two cases, /H20849i/H20850solid body and /H20849ii/H20850smooth ﬂow, as /H20849i/H20850:/H9264=/H9256=1 , /H208493/H20850 /H20849ii/H20850:/H9264=1− r,/H9256=1− r2. /H208494/H20850 We note here that the magnetic Reynolds numbers are de- ﬁned with the maximum angular velocity /H20849either mean or ﬂuctuating part /H20850and the radius of the moving cylinder. Thinking of an experiment, it would not be sufﬁcient tominimize the magnitude of the azimuthal ﬂow. In particular, if/H9003¯is large /H20849considering a steady ﬂow for simplicity /H20850, one would have to spend too many megawatts in forcing the z velocity. Therefore, the reader interested in linking our re-sults to experiments should bear in mind that our magneticReynolds number is not totally adequate for it. A better deﬁ-nition of the magnetic Reynolds number might be, for ex- ample, Rˆ m=R¯m/H208811+/H9003¯2. For a stationary ﬂow of type /H20849i/H20850, the minimum dynamo threshold Rˆmis obtained for /H9003¯=1.3. Both cases /H20849i/H20850and /H20849ii/H20850differ in the conductivity of the external medium r/H110221. In case /H20849i/H20850, in which the magnetic generation occurs in a cylindrical layer in the neighborhoodofr=1, a conducting external medium is necessary for dy- namo action. For simplicity, we choose the same conductiv-ity as the inner ﬂuid. In the other hand, in case /H20849ii/H20850, in which the magnetic generation is within the ﬂuid, a conducting ex-ternal medium is not necessary for dynamo action; thus, wechoose an insulating external medium. Though the choice ofthe conductivity of the external medium is far from beinginsigniﬁcant for a dynamo experiment, 3,4,6,7we expect that it does not change the overall meaning of the results givenbelow. We deﬁne the magnitude ratio of the ﬂuctuation to the mean ﬂow by /H9267=R/tildewidestm/R¯m. For /H9267=0, there is no ﬂuctuation and the dynamo threshold is given by R ¯m. On the other hand,for/H9267/greatermuch1, the ﬂuctuation dominates and the relevant quantity to determine the threshold is R/tildewidestm=/H9267R¯m. The perturbative ap- proach of Normand25corresponds to /H9267/lessmuch1. The magnetic ﬁeld must satisfy the induction equation /H11509B /H11509t=/H11633/H11003/H20849U/H11003B/H20850+/H116122B, /H208495/H20850 where the dimensionless time tis given in units of the mag- netic diffusion time, implying that the ﬂow frequency /H9275fis also a dimensionless quantity. As the velocity does not de-pend on /H9258or on z, each magnetic mode in /H9258andzis inde- pendent from the others. Therefore, we can look for a solu-tion of the form B/H20849r,t/H20850= exp i/H20849m /H9258+kz/H20850b/H20849r,t/H20850, /H208496/H20850 where mandkare the azimuthal and vertical wave numbers of the ﬁeld, respectively. The solenoidality of the ﬁeld /H11633·B =0 then leads to br r+br/H11032+im rb/H9258+ikbz=0 . /H208497/H20850 With the new variables b±=br±ib/H9258, the induction equa- tion can be written in the form /H11509b± /H11509t+/H20851k2+i/H20849m/H9024+kV/H20850h/H20849r/H20850/H20852b± =±i 2r/H9024/H11032h/H20849r/H20850/H20849b++b−/H20850+L±b±, /H208498/H20850 with L±=/H115092 /H11509r2+1 r/H11509 /H11509r−/H20849m±1/H208502 r2, /H208499/H20850 except in case /H20849ii/H20850, where in the insulating external domain /H20849r/H110221/H20850, the induction equation takes the form /H20849L±−k2/H20850b±=0 . /H2084910/H20850 At the interface r=1, both Band the zcomponent of the electric ﬁeld E=/H11633/H11003B−U/H11003Bare continuous. The continu- ity of BrandB/H9258imply that of b±. The continuity of Band /H208497/H20850 imply the continuity of br/H11032, which, combined with the conti- nuity of Ezimplies /H20851Db±/H208521+1−±i/H9024r=1− 2/H20849b++b−/H20850r=1=0 /H2084911/H20850 with D=/H11509//H11509rand /H20851h/H208521+1−=h/H20849r=1− /H20850−h/H20849r=1+ /H20850. We note that in case /H20849ii/H20850,a s/H9024r=1−=0, /H2084911/H20850implies the continuity of Db±atr=1. In summary, we calculate for both cases /H20849i/H20850and /H20849ii/H20850the growth rate /H9253=/H9253/H20849m,k,/H9003¯,/H9003/tildewidest,R¯m,R/tildewidestm,/H9275f/H20850/H20849 12/H20850 of the kinematic dynamo problem and look for the dynamo054109-2 Peyrot, Plunian, and Normand Phys. Fluids 19, 054109 /H208492007 /H20850threshold /H20849either R ¯mor R/tildewidestm/H20850such that the real part /H20849Re/H9253/H20850of/H9253 is zero. In our numerical simulations we shall take m=1 for it leads to the lowest dynamo threshold. A. Case „i…: Solid body ﬂow In case /H20849i/H20850,w es e t m/H9024+kV=R¯m/H9262¯+R/tildewidestm/H9262/tildewidestf/H20849t/H20850, with /H9262¯=m+k/H9003¯and /H9262/tildewidest=m+k/H9003/tildewidest, /H2084913/H20850 and /H208498/H20850changes into /H11509b± /H11509t+/H20853k2+i/H20851R¯m/H9262¯+R/tildewidestm/H9262/tildewidestf/H20849t/H20850/H20852h/H20849r/H20850/H20854b±=L±b±. /H2084914/H20850 For mathematical convenience, we take /H9262/tildewidest=0. Thus, the non- stationary part of the velocity no longer occurs in /H2084914/H20850.I t occurs only in the expression of the boundary conditions /H2084911/H20850 that can be written in the form /H20851Db±/H208521+1−±i 2/H20851R¯m+R/tildewidestmf/H20849t/H20850/H20852/H20849b++b−/H20850r=1=0 . /H2084915/H20850 Taking /H9262/tildewidest=0 corresponds to a pitch of the magnetic ﬁeld equal to the pitch of the ﬂuctuating part of the ﬂow − m/k =/H9003/tildewidest. On the other hand, it is not necessarily equal to the pitch of the mean ﬂow /H20849except if /H9003¯=/H9003/tildewidest/H20850. In addition, we shall con- sider two situations depending on whether the mean ﬂow is zero /H20849R¯m=0/H20850or not. The method used to solve Eqs. /H2084914/H20850and /H2084915/H20850is given in Appendix A. At this stage we can make two remarks. First, according to boundary layer theory results16,17and for a stationary ﬂow, in the limit of large R ¯m, the magnetic ﬁeld that has the high- est growth rate satisﬁes /H9262¯/H110150. This resonant condition means that the pitch of the magnetic ﬁeld is roughly equal to thepitch of the ﬂow. We shall see in Sec. III A that this staystrue even at the dynamo threshold. Though the case of a ﬂuctuating ﬂow of type U /tildewidest·f/H20849t/H20850may be more complex with possibly a skin effect, the resonant condition is presumably analogous; i.e., /H9262/tildewidest/H110150. This means that setting /H9262/tildewidest=0 implies that if the ﬂuctuations are sufﬁciently large /H20849/H9267/greatermuch1/H20850, dynamo action is always possible. This is indeed what will be found in our results. In other words, setting /H9262/tildewidest=0, we cannot tackle the situation of a stationary dynamo ﬂow to which a ﬂuctua-tion acting against the dynamo would be added. This aspectwill be studied with the smooth ﬂow /H20849ii/H20850. Our second remark is about the effect of a phase lag between the azimuthal and vertical components of the ﬂowﬂuctuation. Though we did not study the effect of an arbi-trary phase lag, we can predict the effect of an out-of-phase lag. This would correspond to take a negative value of /H9003 /tildewidest. Solving numerically Eqs. /H2084914/H20850and /H2084915/H20850for the stationary ﬂow and m=1, we ﬁnd that dynamo action is possible only if k/H9003¯/H110210. For the ﬂuctuating ﬂow with zero mean, m=1 and /H9262/tildewidest=0 necessarily imply that k/H9003/tildewidest=−1. Let us now consider a ﬂow containing both a stationary and a ﬂuctuating part. Set- ting/H9003/tildewidest/H110210 necessarily implies that k/H110220. For /H9003¯/H110220, the sta- tionary ﬂow, then, is not a dynamo. Therefore, in that casewe expect the dynamo threshold to decrease for increasing /H9267. For/H9003¯/H110210, together with /H9003/tildewidest/H110210 and k/H110220, it is equivalent to take/H9003/tildewidest/H110220 and /H9003¯/H110220 for k/H110210, and it is then covered by our subsequent results. B. Case „ii…: Smooth ﬂow For case /H20849ii/H20850, we can directly apply the resonant condi- tion made up for a stationary ﬂow,17,18to the case of a ﬂuc- tuating ﬂow. For given mandk, the magnetic ﬁeld is gener- ated in a resonant layer r=r0, where the magnetic ﬁeld lines are aligned with the shear and thus minimize the magneticﬁeld diffusion. This surface is determined by the followingrelation: 17,18 m/H9024/H11032/H20849r0/H20850+kV/H11032/H20849r0/H20850=0 . /H2084916/H20850 The resonant condition is satisﬁed if the resonant surface is embedded within the ﬂuid: 0/H11021r0/H110211. /H2084917/H20850 As/H9024and Vdepend on time, this condition may only be satisﬁed at discrete times. This implies successive periods ofgrowth and damping, the dynamo threshold corresponding toa zero mean growth rate. We can also deﬁne two distinctresonant surfaces r ¯0andr/tildewidest0corresponding to the mean and ﬂuctuating part of the ﬂow: m/H9024¯/H11032/H20849r¯0/H20850+kV¯/H11032/H20849r¯0/H20850=0 , m/H9024/tildewidest/H11032/H20851r/tildewidest0/H20849t/H20850,t/H20852+kV/tildewidest/H11032/H20851r/tildewidest0/H20849t/H20850,t/H20852=0 , /H2084918/H20850 with appropriate deﬁnitions of /H9024¯,V¯,/H9024/tildewidestandV/tildewidest. In addition, if /H9024/tildewidestandV/tildewidesthave the same time dependency, as in /H208492/H20850, then r/tildewidest0 becomes time independent. We can then predict two different behaviors of the dynamo threshold versus the ﬂuctuation rate /H9267=R/tildewidestm/R¯m.I f0/H11021r¯0/H110211 and r/tildewidest0/H110221, then the dynamo thresh- old will increase with /H9267. In this case, the ﬂuctuation is harm- ful to dynamo action. On the other hand, if 0 /H11021r/tildewidest0/H110211, then the dynamo threshold will decrease with /H9267. From the deﬁnitions /H2084918/H20850and for a ﬂow deﬁned by /H208491/H20850, /H208492/H20850, and /H208494/H20850, we have r¯0=− /H20849m/k/H20850//H208492/H9003¯/H20850and r/tildewidest0=− /H20849m/k/H20850//H208492/H9003/tildewidest/H20850. /H2084919/H20850 Form=1 and k/H110210, taking /H9003/tildewidest/H110210 implies r/tildewidest0/H110210 and then the impossibility of dynamo action for the ﬂuctuating part of theﬂow. Therefore, we expect that the addition of a ﬂuctuatingﬂow with an out-of-phase lag between its vertical and azi-muthal components will necessarily be harmful to dynamoaction. This will be conﬁrmed numerically in Sec. III C. To solve /H208498/H20850,/H2084910/H20850, and /H2084911/H20850, we used a Galerkin approxi- mation method in which the trial and weighting functions arechosen in such a way that the resolution of the inductionequation is reduced to the conducting domain r/H113491. 5The method of resolution is given in Appendix D. For the timeresolution we used a Runge-Kutta scheme of order 4.054109-3 Parametric instability of the helical dynamo Phys. Fluids 19, 054109 /H208492007 /H20850III. RESULTS A. Stationary ﬂow „R/tildewidestm=0 … We solve Re/H9253/H20849m=1 , k,/H9003¯,0 ,R¯m,0 ,0 /H20850=0 /H2084920/H20850 with k=/H20849/H9262¯−1/H20850//H9003¯for case /H20849i/H20850andk=−1/ /H208492r¯0/H9003¯/H20850for case /H20849ii/H20850. In case /H20849i/H20850, the dispersion relation /H20849A6/H20850in Appendix A be- comes F0=0. In Fig. 1, the threshold R ¯mand the ﬁeld fre- quency Im /H20849/H9253/H20850are plotted versus /H9262¯/H20849r¯0/H20850for case /H20849i/H20850/H20851 /H20849ii/H20850/H20852, and for different values of /H9003¯. Though we do not know how these curves asymptote, and though the range of /H9262¯/H20849r¯0/H20850for which dynamo action occurs changes with /H9003¯, it is likely that the resonant condition /H20841/H9262¯/H20841/H110211/H208490/H11021r¯0/H110211/H20850is fulﬁlled for the range of /H9003¯corresponding to a dynamo experiment /H20849/H9003¯/H110151/H20850.B. Periodic ﬂow with zero mean „R¯m=0 … We solve Re/H9253/H20849m=1 , k,0 ,/H9003/tildewidest,0 ,R/tildewidestm,/H9275f/H20850=0 . /H2084921/H20850 In Fig. 2, the threshold R/tildewidestmis plotted versus /H9275ffor both cases /H20849i/H20850and /H20849ii/H20850. In both cases we take /H9262/tildewidest=0 corresponding to k =−1//H9003/tildewidest. For case /H20849ii/H20850, it implies from /H2084919/H20850that r/tildewidest0=1/2, meaning that the resonant surface is embedded in the ﬂuid and thus favorable to dynamo action. In each case /H20849i/H20850/H9003/tildewidest =1,1.78 and /H20849ii/H20850/H9003/tildewidest=1,2, we observe two regimes: one at low frequencies for which the threshold does not depend on /H9275f and the other at high frequencies for which the threshold behaves like R/tildewidestm/H11008/H9275f3/4. To understand the existence of these two regimes, we pay attention to the time evolution of the magnetic ﬁeld for FIG. 1. The dynamo threshold R ¯m/H20849left column /H20850and Im /H20849/H9253/H20850/H20849right column /H20850versus /H20849i/H20850/H9262¯,/H20849ii/H20850r¯0, for the stationary case, m=1 and /H9003¯=0.5, 0.8, 1, 1.3, 2, 4, 10.054109-4 Peyrot, Plunian, and Normand Phys. Fluids 19, 054109 /H208492007 /H20850different frequencies /H9275f. In Fig. 3, the time evolution of b− /H20849real and imaginary parts /H20850for case /H20849ii/H20850/H9003/tildewidest=1 /H20851case /H20849d/H20850in Fig. 2/H20852is plotted for several frequencies /H9275f. 1. Low frequency regime For low frequencies /H20849/H9275f=1/H20850, we observe two time scales: periodic phases of growth and decrease of the ﬁeld, with a time scale equal to the period of the ﬂow as expectedby Floquet’s theory. In addition, the ﬁeld has an eigenfre-quency much higher than /H9275f. In fact, the slow phases of growth and decrease seem to occur every half-period of theﬂow. This can be understood from the following remarks. First of all, the growth /H20849or decrease /H20850of the ﬁeld does not depend on the sign of the ﬂow. Indeed, from /H208498/H20850, we show that if b ±/H20849m,k/H20850is a solution for /H20849/H9024,V/H20850, then its complex conjugate b±/H20849m,k/H20850is a solution for /H20849−/H9024,−V/H20850. Therefore, we have b±/H20849t+T/2/H20850=b±/H20849t/H20850, where T=2/H9266//H9275fis the period of the ﬂow. Now from Floquet’s theory /H20849see Appendix A /H20850,w em a y write b/H20849r,t/H20850of the form b/H20849r,/H9270/H20850exp/H20849/H9253t/H20850, with b/H20849r,/H9270/H20850 2/H9266-periodic in /H9270=/H9275ft. This implies that changing /H20849/H9024,V/H20850in /H20849−/H9024,−V/H20850changes the sign of Im /H20849/H9253/H20850. This is consistent with the fact that for given mandk, the direction of propagation ofBchanges with the direction of the ﬂow. Therefore chang- ing the sign of the ﬂow changes the sign of propagation ofthe ﬁeld but does not change the magnetic energy, neither the dynamo threshold R /tildewidestm, which are then identical from one half-period of the ﬂow to another. This means that the dy-namo threshold does not change if we consider f/H20849t/H20850 =/H20841cos/H20849 /H9275ft/H20850/H20841instead of cos /H20849/H9275ft/H20850. It is then sufﬁcient to con- centrate on one half-period of the ﬂow, such as, for example /H20851/H9266/2/H9275f,3/H9266/2/H9275f/H20852/H20849modulo /H9266/H20850. The second remark uses the fact that the ﬂow geometry that we consider does not change in time /H20849only the ﬂow magnitude changes /H20850. For such a geometry we can calculate the dynamo threshold R ¯mcorresponding to the stationarycase. Coming back to the ﬂuctuating ﬂow, we then under- stand that R/tildewidestm/H20841f/H20849t/H20850/H20841/H11022R¯m/H20851R/tildewidestm/H20841f/H20849t/H20850/H20841/H11021R¯m/H20852corresponds to a growing /H20849decreasing /H20850phase of the ﬁeld. Assuming that the dynamo threshold R/tildewidestmis given by the time average /H20855·/H20856of the ﬂow magnitude leads to the following estimation for R/tildewidestm: R/tildewidestm/H11015/H9266 2R¯m /H2084922/H20850 as/H20855/H20841cos/H20849/H9275ft/H20850/H20841/H20856=2//H9266. For the four cases /H20849a/H20850,/H20849b/H20850,/H20849c/H20850, and /H20849d/H20850 in Fig. 2, we give in Table Ithe ratio 2R/tildewidestm//H9266R¯m, which is found to be always close to unity. In this interpretation of theresults, the frequency /H9275fdoes not appear, provided that it is sufﬁciently weak in order that the successive phases ofgrowth and decrease have sufﬁcient time to occur. This canexplain why for low frequencies in Fig. 2, the dynamo threshold R /tildewidestmdoes not depend on /H9275f. Finally the frequencies /H9275¯of the stationary case for /H9003¯ =/H9003/tildewidestare also reported in Table I. For a geometry identical to case /H20849d/H20850, we ﬁnd, in the stationary case, /H9275¯=33, which indeed corresponds to the eigenfrequency of the ﬁeld occurring inFig.3for /H9275f=1. The previous remarks assume that the ﬂow frequency is sufﬁciently small compared to the eigenfre-quency of the ﬁeld, in order to have successive phases ofgrowth and decrease of the ﬁeld. We can check that the val-ues of /H9275¯given in Table Iare indeed reasonable estimations of the transition frequencies between the low and high fre-quency regimes in Fig. 2. 2. High frequency regime In case /H20849ii/H20850and for high frequencies /H20849Fig. 3,/H9275f=100 /H20850, the signal is made of harmonics without growing or decreas-ing phases. We note that the eigenfrequencies of the real andimaginary parts of b −are different, the one being twice the other. In case /H20849i/H20850, relying on the resolution of Eqs. /H2084914/H20850and /H2084915/H20850given in Appendix A, we can show that R/tildewidestm/H11008/H9275f3/4.W e also ﬁnd that some double frequency as found in Fig. 3for case /H20849ii/H20850can emerge from an approximate 3 /H110033 matrix sys- tem. As these developments necessitate the notations intro-duced in Appendix A, they are postponed until Appendix B. 3. Further comments about the ability for ﬂuctuating ﬂows to sustain dynamo action We found and explained how a ﬂuctuating ﬂow /H20849zero mean /H20850can act as a dynamo. We also understood why the dynamo threshold for a ﬂuctuating ﬂow is higher than thatfor a stationary ﬂow with the same geometry. It is becausethe time-average of the velocity norm of the ﬂuctuating ﬂowis on the mean lower than that of the stationary ﬂow. Thiscan be compensated with other deﬁnitions of the magneticReynolds number. Our deﬁnition is based on max t/H20841/H9024/H20849r,t/H20850/H20841. Another deﬁnition based on /H20855/H20841/H9024/H20849r,t/H20850/H20841/H20856twould exactly com- pensate the difference. FIG. 2. Dynamo threshold R/tildewidestmversus /H9275ffor case /H20849i/H20850with/H9262/tildewidest=0 and /H20849a/H20850/H9003/tildewidest =1.78, and /H20849b/H20850/H9003/tildewidest=1; for case /H20849ii/H20850with r/tildewidest0=0.5, /H20849c/H20850/H9003/tildewidest=2, and /H20849d/H20850/H9003/tildewidest=1.054109-5 Parametric instability of the helical dynamo Phys. Fluids 19, 054109 /H208492007 /H20850Recently, a controversy appeared about the difﬁculty for a ﬂuctuating ﬂow /H20849zero mean /H20850to sustain dynamo action at low P m,26whereas a mean ﬂow /H20849nonzero time average /H20850ex- hibits a ﬁnite threshold at low P m10,27/H20849the magnetic Prandtl number P mbeing deﬁned as the ratio of the viscosity to the diffusivity of the ﬂuid /H20850. This issue is important not only fordynamo experiments but also for natural objects such as the Earth’s inner core or the solar convective zone in which theelectroconducting ﬂuid is characterized by a low P m. Though we did not study this problem, our results suggest that thedynamo threshold should not be much different betweenﬂuctuating and mean ﬂows, provided an appropriate deﬁni- FIG. 3. Time evolution of Re /H20849b−/H20850/H20849solid lines /H20850and Im /H20849b−/H20850/H20849dotted lines /H20850for several values of /H9275f/H20849from top to bottom /H9275f=1; 2, 5, 10, 100 /H20850, for case /H20849ii/H20850with /H9003/tildewidest=1. Time unity corresponds here to 2 /H9266//H9275f.054109-6 Peyrot, Plunian, and Normand Phys. Fluids 19, 054109 /H208492007 /H20850tion of the magnetic Reynolds number is taken. In that case, why does it seem so difﬁcult to sustain dynamo action at lowP mfor a ﬂuctuating ﬂow,26whereas it seems much easier for a mean ﬂow?10,27 In the simulations with a mean ﬂow,10,27two dynamo regimes have been found, one with a threshold much lowerthan the other. In the lowest threshold regime, the magneticﬁeld is generated at some inﬁnite scale in two directions. 27 There is then an inﬁnite scale separation between the mag-netic and the velocity ﬁeld, and the dynamo action is prob-ably of mean-ﬁeld type and might be understood in terms of /H9251-effect, /H9252-effect, etc. In that case, removing the periodic boundary conditions would cancel the scale separation andimply the loss of the dynamo action. In the highest thresholdregime, the magnetic ﬁeld is generated at a scale similar tothe ﬂow scale, the periodic boundary conditions are forgottenand the dynamo action can no longer be understood in termsof an /H9251-effect. In order to compare the mean ﬂow results27 with those for a ﬂuctuating ﬂow,26we have to consider only the highest threshold regime in Ref. 27, the lowest one rely- ing on mean-ﬁeld dynamo processes due to the periodicboundary conditions and which are absent in the ﬂuctuatingﬂow calculations. 26 Now when comparing the threshold of the highest threshold regime for a mean ﬂow with the threshold obtainedfor a ﬂuctuating ﬂow and with appropriate deﬁnitions of R m, a strong difference remains at low P m. A speculation made by Schekochihin et al.28is that the highest threshold regime obtained for the mean ﬂow at low P mwould correspond in fact to the large P mresults for the ﬂuctuating ﬂow. Their arguments rely on the fact that the mean ﬂow in Ref. 27is peaked at large scale and is thus spatially smooth for thegenerated magnetic ﬁeld. It would then belong to the sameclass as the large-P mﬂuctuation dynamo. Both dynamo thresholds are found to be similar indeed, and thus the dis-crepancy vanishes. Though the helical ﬂow that we consider here is notice- ably different /H20849no chaotic trajectories /H20850it may have some con- sistency with the simulations at large P mmentioned above and at least supports the speculation by Schekochihin et al.28 C. Periodic ﬂow with nonzero mean We are now interested in the case in which both R/tildewidestm /HS110050 and R ¯m/HS110050. The ﬂow is then the sum of a nonzero mean part and a ﬂuctuating part. We have considered two ap-proaches depending on which part of the ﬂow geometry isﬁxed, either the mean or the ﬂuctuating part. 1./H9003¯=1 Here we ﬁx /H9003¯=1,m=1, and k=−1, and vary /H9003/tildewidest,/H9267, and /H9275ffor case /H20849ii/H20850. We then solve the equation Re/H9253/H20849m=1 , k=−1 , /H9003¯=1 ,/H9003/tildewidest= 1/2 r/tildewidest0,R¯m,R/tildewidestm=/H9267R¯m,/H9275f/H20850=0 /H2084923/H20850 to plot R ¯mas a function of /H9267in Fig. 4for values of r/tildewidest0and/H9275f. From /H2084919/H20850, we have r¯0=1/2, which corresponds to a mean ﬂow geometry with a dynamo threshold about 100. The curves are plotted for several values of /H9003/tildewidestleading to values of r/tildewidest0not necessarily between 0 and 1. We consider two ﬂuctua-TABLE I. Dynamo thresholds R/tildewidestmfor a ﬂuctuating ﬂow at low frequency, R¯m/H20849and/H9275¯/H20850for a stationary ﬂow with the same geometry. The labels /H20849a/H20850–/H20849d/H20850 have the same meaning as in Fig. 2. R/tildewidestm R¯m 2R/tildewidestm//H9266R¯m /H9275¯ /H20849a/H20850 21 13 1.03 4.4 /H20849b/H20850 33 21 1 3.1 /H20849c/H20850 143 84 1.08 28.8 /H20849d/H20850 170 100 1.08 33 FIG. 4. Dynamo threshold R ¯mversus /H9267for case /H20849ii/H20850, for two frequencies /H9275f=50 and /H9275f=1 and r¯0=1/2 /H20849/H9003¯=1,m=1,k=−1 /H20850. The different curves correspond tor/tildewidest0=/H20849a/H208501/2; /H20849b/H208502/3; /H20849c/H208501;/H20849d/H20850/H11009;/H20849e/H208501/4; /H20849f/H20850−1; /H20849g/H20850−1/2 /H20851/H9003/tildewidest=/H20849a/H208501;/H20849b/H208500.75; /H20849c/H208500.5; /H20849d/H208500;/H20849e/H208502;/H20849f/H20850−0.5; /H20849g/H20850−1/H20852.054109-7 Parametric instability of the helical dynamo Phys. Fluids 19, 054109 /H208492007 /H20850tion frequencies /H9275f=1 and /H9275f=50. We ﬁnd that the dynamo threshold R ¯mincreases asymptotically with /H9267unless the reso- nant condition 0 /H11021r/tildewidest0/H110211 is satisﬁed; see curves /H20849a/H20850,/H20849b/H20850, and /H20849e/H20850. For these three curves we checked that in the limit of large/H9267,R¯m=O/H20849/H9267−1/H20850. For r/tildewidest0=1/4 /H20851curve /H20849e/H20850/H20852and for /H9267/H110151w e do not know if a dynamo threshold exists. 2./H9003/tildewidest=1 Here we ﬁx /H9003/tildewidest=1,m=1, and k=−1 and vary /H9003¯,/H9267, and /H9275f. We then solve the equation Re/H9253/H20849m=1 , k=−1 , /H9003¯,/H9003/tildewidest,R¯m,R/tildewidestm=/H9267R¯m,/H9275f/H20850=0 /H2084924/H20850 with/H9003¯=1−/H9262¯in case /H20849i/H20850and/H9003¯=1/2 r¯0in case /H20849ii/H20850. In Fig. 5, R¯mis plotted versus /H9267for values of /H9262¯/H20849r¯0/H20850in case /H20849i/H20850/H20851 /H20849ii/H20850/H20852 and/H9275f. Taking /H9003/tildewidest=1,m=1, and k=−1 implies /H9262/tildewidest=0 in case /H20849i/H20850and r/tildewidest0=0.5 in case /H20849ii/H20850. In both cases /H20849i/H20850and /H20849ii/H20850the ﬂuctuating part of the ﬂow satisﬁes the resonant conditionfor which dynamo action is possible. This implies that R ¯m should scale as O/H20849/H9267−1/H20850provided that /H9267is sufﬁciently large. In each case we consider two ﬂow frequencies /H9275f=1 and /H9275f =10 for case /H20849i/H20850;/H9275f=1 and /H9275f=50 for case /H20849ii/H20850. The curves are plotted for different values of /H9003¯corresponding to /H20841/H9262¯/H20841 /H110211 for case /H20849i/H20850and 0/H11021r¯0/H110211 for case /H20849ii/H20850. For large /H9267,w e checked that R ¯m=O/H20849/H9267−1/H20850. The main difference between the curves is that R ¯mversus /H9267may decrease monotonically or not. In particular, in case /H20849i/H20850for/H9262¯=0.4, R ¯mdecreases by 40% when /H9267goes from 0 to 1 showing that even a small ﬂuctua- tion can strongly decrease the dynamo threshold. In most ofthe curves there is a bump for /H9267around unity showing a strong increase of the threshold before the ﬁnal decrease atlarger /H9267. 3./H9003¯=/H9003/tildewidest=1 Here we ﬁx /H9003¯=/H9003/tildewidest=1,m=1, and k=−1 for the case /H20849ii/H20850 and vary /H9275fand/H9267. We then solve the equation FIG. 5. The dynamo threshold R ¯mversus /H9267fork=−1, m=1, and /H9003/tildewidest=1 /H20851/H9262/tildewidest=0 in case /H20849i/H20850andr/tildewidest0=0.5 in case /H20849ii/H20850/H20852and/H9275f=1, 10, or 50. The labels correspond to/H9262¯in case /H20849i/H20850andr¯0in case /H20849ii/H20850.054109-8 Peyrot, Plunian, and Normand Phys. Fluids 19, 054109 /H208492007 /H20850Re/H9253/H20849m=1 , k=−1 , /H9003¯=1 ,/H9003/tildewidest=1 , R¯m,R/tildewidestm=/H9267R¯m,/H9275f/H20850=0 /H2084925/H20850 to plot R ¯mversus /H9267in Fig. 6for various frequencies /H9275f. Taking /H9003¯=/H9003/tildewidest=1,m=1, and k=−1 implies r¯0=r/tildewidest0=0.5. For /H9267 larger than 1, R ¯mdecreases as O/H20849/H9267−1/H20850as mentioned earlier. For/H9267smaller than unity, R ¯mdecreases versus /H9267monotoni- cally only if /H9275fis large enough. In fact the transition value of /H9275f, above which R ¯mdecreases monotonically is exactly the ﬁeld frequency /H9275¯/H20849here,/H9275¯=33 /H20850corresponding to /H9267=0. This shows that a ﬂuctuation of small intensity /H20849/H9267/H113491/H20850helps the dynamo action only if its frequency is sufﬁciently high. This is shown in Appendix C for case /H20849i/H20850. However, the frequency above which a small ﬂuctuation intensity helps the dynamomay be much larger than /H9275¯. For example, in case /H20849i/H20850for/H9262¯ =0.4 and /H9262/tildewidest=0 represented in Fig. 5, we have /H9275¯=0.51. For /H9275f=1 small ﬂuctuation helps, for /H9275f=10 they do not help, and for higher frequencies they help again. IV. DISCUSSION In this paper we studied the modiﬁcation of the dynamo threshold of a stationary helical ﬂow by the addition of alarge scale helical ﬂuctuation. We extended a previousasymptotic study 25to the case of a ﬂuctuation of arbitrary intensity /H20849controlled by the parameter /H9267/H20850. We knew from pre- vious studies17,18that the dynamo efﬁciency of a helical ﬂow is characterized by some resonant condition at large R m. First we veriﬁed numerically that such resonant condition holds atlower R mcorresponding to the dynamo threshold, for both a stationary and a ﬂuctuating /H20849no mean /H20850helical ﬂow. For a helical ﬂow made of a mean part plus a ﬂuctuating part weshowed that, in the asymptotic cases /H9267/lessmuch1/H20849dominating mean /H20850and/H9267/greatermuch1/H20849dominating ﬂuctuation /H20850, it is naturally the resonant condition of the mean /H20849for the ﬁrst case /H20850or the ﬂuctuating /H20849for the second case /H20850part of the ﬂow that governs the dynamo efﬁciency and then the dynamo threshold. Inbetween, for /H9267of order unity and if the resonant condition of each ﬂow part /H20849mean and ﬂuctuating /H20850is satisﬁed, the thresh- old ﬁrst increases with /H9267before reaching an asymptotic be- havior in O/H20849/H9267−1/H20850. However, there is no systematic behavior as depicted in Fig. 5, case /H20849i/H20850, for/H9275f=1 and /H9262¯=0.4, in which a threshold decrease of 40% is obtained between /H9267=0 and /H9267=1. If the ﬂuctuation part of the ﬂow does not satisfy the resonant condition, then the dynamo threshold increasesdrastically with /H9267. Contrary to the case of a cellular ﬂow,11there is no sys- tematic effect of the phase lag between the different compo-nents of the helical ﬂow. For the helical ﬂow geometry itmay imply an increase or a decrease of the dynamo thresh-old, depending how it changes the resonant condition men-tioned above. There is some similarity between our results and those obtained for a noisy /H20849instead of periodic /H20850ﬂuctuation. 8In par- ticular, in Ref. 8it was found that increasing the noise level the threshold ﬁrst increases due to geometrical effects of themagnetic ﬁeld lines and then decreases at larger noise. Thiscould explain why at /H9267/H110151 we generally obtain a maximum of the dynamo threshold. Finally, these results show that the optimization of a dy- namo experiment depends not only on the mean part of theﬂow but also on its nonstationary large scale part. If theﬂuctuation is not optimized then the threshold may increasedrastically with /H20849even small /H20850 /H9267, ruling out any hope of pro- ducing dynamo action. In addition, even if the ﬂuctuation isoptimized /H20849resonant condition satisﬁed by the ﬂuctuation /H20850, our results suggest that there is generally some increase ofthe dynamo threshold with /H9267when /H9267/H113491. If the geometry of the ﬂuctuation is identical to that of the mean part of theﬂow, there can be some slight decrease of the threshold athigh frequencies but this decrease is rather small. When /H9267 /H110221, the dynamo threshold decreases as O/H20849/H9267−1/H20850, which at ﬁrst sight seems interesting. However, we have to keep in mind that as soon as /H9267/H110221, the driving power spent to maintain the ﬂuctuation is larger than that to maintain the mean ﬂow. The relevant dynamo threshold is, then, no longer R ¯m, but R/tildewidestm =/H9267R¯minstead. In addition monitoring large scale ﬂuctuations in an experiment may not always be possible, especially ifthey occur from ﬂow destabilization. In that case it is betterto try cancelling them as was done in the von Karman so-dium experiment in which an azimuthal belt has beenadded. 29 ACKNOWLEDGMENTS We acknowledge B. Dubrulle, F. Pétrélis, R. Stepanov, and A. Gilbert for fruitful discussions. APPENDIX A: RESOLUTION OF EQS. „14…AND „15… FOR CASE „I…: SOLID BODY FLOW Asf/H20849t/H20850is time-periodic of period 2 /H9266//H9275f, we look for b/H20849r,t/H20850in the form b/H20849r,/H9270/H20850exp/H20849/H9253t/H20850with b/H20849r,/H9270/H20850being 2/H9266-periodic in /H9270=/H9275ft. Thus, we look for the functions b±/H20849r,/H9270/H20850in the form FIG. 6. Dynamo threshold R ¯mversus /H9267forr¯0=r/tildewidest0=0.5 /H20849/H9003¯=/H9003/tildewidest=1/H20850. The labels correspond to different values of /H9275f. The eigenfrequency for /H9267=0 is /H9275¯=33.054109-9 Parametric instability of the helical dynamo Phys. Fluids 19, 054109 /H208492007 /H20850b±/H20849r,/H9270/H20850=/H20858bn±/H20849r/H20850exp/H20849in/H9270/H20850/H20849 A1/H20850 where, from /H2084914/H20850and for /H9262/tildewidest=0, the Fourier coefﬁcients bn±/H20849r/H20850 must satisfy /H20853/H9253+k2+i/H20851R¯m/H9262¯h/H20849r/H20850+n/H9275f/H20852/H20854bn±=L±bn±. /H20849A2/H20850 In addition, the boundary condition /H2084915/H20850with f/H20849t/H20850=cos /H20849/H9270/H20850 implies /H20851Dbn±/H208521+1−±i 2R¯m/H20849bn++bn−/H20850r=1 ±i 4R/tildewidestm/H20849bn−1++bn−1−+bn+1++bn+1−/H20850r=1=0 . /H20849A3/H20850 The solutions of /H20849A2/H20850, which are continuous at r=1, can be written in the form bn±=Cn±/H9274n±, with /H9274n±=/H20877I±/H20849qnr/H20850/I±/H20849qn/H20850, r/H110211, K±/H20849snr/H20850/K±/H20849sn/H20850, r/H110221, /H20849A4/H20850 with qn2=k2+/H9253+i/H20849R¯m/H9262¯+n/H9275f/H20850,sn2=k2+/H9253+in/H9275f. /H20849A5/H20850 Substituting /H20849A4/H20850in/H20849A3/H20850, we obtain the following system: Cn±Rn±±iR¯m 2/H20849Cn++Cn−/H20850±iR/tildewidestm 4/H20849Cn−1++Cn−1−+Cn+1++Cn+1−/H20850=0 /H20849A6/H20850 withRn±=qnIn±/H11032/In±−snKn/H11032±/Kn±and where In±=Im±1/H20849qn/H20850and Kn±=Km±1/H20849sn/H20850are modiﬁed Bessel functions of ﬁrst and sec- ond kind. The system /H20849A6/H20850implies the following matrix dispersion relation: FnCn−iR/tildewidestm 4/H20849Rn+−Rn−/H20850/H20849Cn−1+Cn+1/H20850=0 /H20849A7/H20850 with Cj=Cj++Cj−and Fn=Rn+Rn−−i/H20849R¯m/2/H20850/H20849Rn+−Rn−/H20850. /H20849A8/H20850 Solving the system /H20849A7/H20850is equivalent to setting to zero the determinant of the matrix Adeﬁned by Ann=Fn,Ann−1=Ann+1=−iR/tildewidestm 4/H20849Rn+−Rn−/H20850/H20849 A9/H20850 and with all other coefﬁcients being set to zero. APPENDIX B: HIGH FREQUENCY REGIME FOR THE PERIODIC FLOW „I…WITH ZERO MEAN Following the notation of Appendix A, and considering a periodic ﬂow /H20849i/H20850with zero mean, we have /H9262¯=0. From /H20849A5/H20850 this implies that qn=sn. Using the identityIn±/H11032Kn±−Kn/H11032±In±=1 sn, /H20849B1/H20850 we obtain Rn±=/H20849In±Kn±/H20850−1.A sR¯m=0, Eq. /H20849A8/H20850becomes Fn =Rn+Rn−. We can then rewrite the system /H20849A7/H20850in the form Cn+iR/tildewidestm 4/H20849In+Kn+−In−Kn−/H20850/H20849Cn−1+Cn+1/H20850=0 . /H20849B2/H20850 From the asymptotic behavior of the Bessel functions for high arguments, we have /H9251n/H11013In−Kn−−In+Kn+/H110151/sn3. /H20849B3/H20850 For the high values of n, these terms are negligible and in ﬁrst approximation we keep in the system /H20849B2/H20850only the terms corresponding to n=0, ±1. This leads to a 3 /H110033 matrix system whose determinant is 1+/H20849R/tildewidestm/H208502 16/H92510/H20849/H9251−1+/H92511/H20850=0 . /H20849B4/H20850 At high forcing frequencies /H9275f, we have s±1/H11015/H20881/H9275f. Together with /H20849B4/H20850, it implies R/tildewidestm/H11015/H9275f3/4. /H20849B5/H20850 In addition, from the approximate 3 /H110033 matrix system, the double-frequency 2 /H9275fdepicted in Fig. 3emerges for n=±1 . APPENDIX C: HIGH FREQUENCY REGIME AND SMALL MODULATION AMPLITUDEFOR THE PERIODIC FLOW „I…WITH NONZERO MEAN For small amplitude modulation /H9267/lessmuch1, the system /H20849A6/H20850 is truncated so as to keep the ﬁrst Fourier modes n=0 and n= ±1. The dispersion relation F0+/H92672/H20873R¯m 4/H208742 /H20849R0+−R0−/H20850/H20873R−1+−R−1− F−1+R+1+−R+1− F+1/H20874=0 /H20849C1/H20850 is then solved perturbatively setting R ¯m=R0+/H9254R and /H9275¯ =/H92750+/H9254/H9275and expanding F0/H20849R¯m,/H9275¯/H20850to ﬁrst order in /H9254R and /H9254/H9275given that F0/H20849R0,/H92750/H20850=0 and with the constants C0/H11007 =±R0±. The dispersion relation /H20849C1/H20850becomes /H9254R/H11509F0 /H11509R¯m+/H9254/H9275/H11509F0 /H11509/H9275¯=−/H92672/H20873R0 4/H208742 C0/H20873/H9252−1 F−1+/H9252+1 F+1/H20874 /H20849C2/H20850 with/H9252n=Rn+−Rn−. The threshold and frequency shifts that behave like /H92672are written /H9254R=/H92672R2and/H9254/H9275=/H92672/H92752.I nt h e left-hand side of /H20849C2/H20850the partial derivatives are given by /H11509F0 /H11509/H9275¯=−1 C0/H20875/H20849C0+/H208502/H11509R0+ /H11509/H9275¯−/H20849C0−/H208502/H11509R0− /H11509/H9275¯/H20876, /H20849C3/H20850 /H11509F0 /H11509R¯m=−1 C0/H20875/H20849C0+/H208502/H11509R0+ /H11509R¯m−/H20849C0−/H208502/H11509R0− /H11509R¯m/H20876−i 2C0. /H20849C4/H20850 One can show that the partial derivatives of R0±are related to integrals calculated in Ref. 25through the relations054109-10 Peyrot, Plunian, and Normand Phys. Fluids 19, 054109 /H208492007 /H20850/H11509R0± /H11509/H9275/H11013i/H20885 0/H11009 /H20849/H90230±/H208502rdr,/H11509R0± /H11509Rm/H11013i/H9262¯/H20885 01 /H20849/H90230±/H208502rdr. /H20849C5/H20850 In the following, we shall focus on the case /H9262¯=0 and we introduce the notations /H11509F0 /H11509R¯m=−iC0/H20849f1+if2/H20850,/H11509F0 /H11509/H9275¯=−iC0/H20849g1+ig2/H20850, /H20849C6/H20850 /H9252−1 F−1+/H9252+1 F+1=X+iY. Solutions of /H20849C2/H20850are R2=/H20873R0 4/H208742Xg1+Yg2 f1g2−f2g1,/H92752=/H20873R0 4/H208742Xf1+Yf2 f1g2−f2g1/H20849C7/H20850 recovering results similar to those obtained in Ref. 25using a different approach. We have in mind that for some values of /H9275f, resonance can occur. An oscillating system forced at a resonant fre-quency is prone to instability and a large negative thresholdshift is expected. However, inspection of /H20849C7/H20850reveals no clear relation between the sign of R 2and the forcing fre- quency, which appears in the quantities XandY. We only know that f1g2−f2g1/H110210, since near the critical point /H20849/H9254R =R¯m−R0/H20850the denominator in /H20849C7/H20850is proportional to the growth rate of the dynamo driven by a steady ﬂow. When /H9267=0, we shall consider Eq. /H20849C2/H20850for an imposed /H9254R and complex values of /H9254/H9275=/H92751+i/H92681, where /H92751is the frequency shift and Re /H20849/H9253/H20850=−/H92681is the growth rate, given by /H92681=/H9254Rf1g2−f2g1 g12+g22. /H20849C8/H20850 Above the dynamo threshold /H20849/H9254R/H110220/H20850the ﬁeld is ampliﬁed /H20851Re/H20849/H9253/H20850/H110220/H20852; thus, /H92681/H110210 and f1g2−f2g1/H110210. In the high frequency limit /H20849/H9275f/greatermuch/H92750/H20850, expressions for X andYcan be derived explicitly using the asymptotic behav- ior of the Bessel functions for large arguments. For /H9262¯=0 with q±1=s±1/H11015/H20849/H9275f±/H92750/H208501/2/H208491±i/H20850//H208812 and using the asymptotic behavior, i.e., /H9252±1/F±1→/H20849s±1/H20850−3, one gets X+iY=−/H208812/H9275f−3/2/H208731−i3/H92750 2/H9275f/H20874. /H20849C9/H20850 When /H9262¯=0, we have also f1=1/2 and f2=0, leading to the expression for R 2: R2/H11015−R02 4/H208812/H9275f−3/2/H20873g1 g2−3/H92750 2/H9275f/H20874. /H20849C10 /H20850 For the wave numbers m=−k=1, numerical calculations of g1andg2, which depend only on the critical parameters R 0 and/H92750, give g1/g2=1.626, and thus R 2/H110210 when /H9275f→/H11009. When /H9262¯=0, there are several reasons to consider the particular value of the forcing; i.e., /H9275f=2/H92750. One of them is that for Hill or Mathieu equations it is a resonant frequency.Moreover, in the present problem it leads to simpliﬁed cal-culations. In particular the asymptotic behavior of /H9252n/Fncan still be used for n= +1 since /H9275f+/H92750is large, while the ap-proximation is no longer valid for n=−1. Nevertheless, the mode n=−1 is remarkable since it corresponds to s−1=s0/H11569, from which it follows that /H9252−1=/H92520andF−1=−iR0/H92520. Finally, one gets the exact result: /H9252−1/F−1=i/R0, which leads to X+iY/H11015i R0+1 s13with s13/H11015−2/H208733/H92750 2/H208743/2 /H208491−i/H20850. /H20849C11 /H20850 For the values R 0=20.82 and /H92750=4.35 corresponding to Fig. 8/H20849f/H20850, one gets X=−1.5 /H1100310−2andY=3.3/H1100310−2. The thresh- old shift is R2/H11015R02 8/H208491.62X+Y/H20850= 0.48, /H20849C12 /H20850 showing that the sign of R 2changes when /H9275fdecreases from inﬁnity to 2 /H92750. This result is in qualitative agreement with the exact results reported in Fig. 8/H20849f/H20850, where R 2=0 for /H9275f =8.3 /H112292/H92750. When the forcing frequency is exactly twice the eigenfrequency /H92750, we had rather expected a large negative value of R 2on the basis that it is a resonant condition for ordinary differential system under temporal modulation. InFig. 8/H20849f/H20850, the maximum negative value of R 2occurs for /H9275f /H112294/H92750, which cannot be explained by simple arguments. For/H9262¯/HS110050, we have not been able to ﬁnd resonant con- ditions such as: n/H9275f+m/H92750=0/H20849n,mintegers /H20850between /H9275fand /H92750such that /H9275fwould be associated with a special behavior of the threshold shift. Contrary to the Hill equation, the in-duction equation is a partial differential equation with theconsequence that the spatial and temporal properties of thedynamo are not independent. The wave numbers kandmare linked to the frequencies /H92750and/H9275fthrough q±nand s±n, which appears as arguments of Bessel functions having rulesof composition less trivial than trigonometric functions. Ex-hibiting resonant conditions implies to ﬁnd relationship be-tween q ±n,s±n, and q0,s0for speciﬁc values of /H9275f. We have shown above for /H9262¯=0 that a relation of complex conjugation exists for n=−1 when /H9275f=2/H92750, but we have not yet found how to generalize to other values of /H9262¯, and have left this part for a future work. APPENDIX D: RESOLUTION OF CASE „II…: SMOOTH FLOW We deﬁne the trial functions /H9274j± =Km±1/H20849k/H20850Jm±1/H20849/H9251jr/H20850/Jm±1/H20849/H9251j/H20850, where the /H9251jare the roots of the equation /H9251j/H20875Km+1/H20849k/H20850 Jm+1/H20849/H9251j/H20850−Km−1/H20849k/H20850 Jm−1/H20849/H9251j/H20850/H20876+2kKm/H20849k/H20850 Jm/H20849/H9251j/H20850=0 , /H20849D1/H20850 and where JandKare, respectively, the Bessel functions of ﬁrst kind and the modiﬁed Bessel functions of second kind.Forr/H113491, we look for solutions in the form b ±=/H20858 j=1N bj/H20849t/H20850/H9274j±/H20849/H9251jr/H20850, /H20849D2/H20850 where Ndeﬁnes the degree of truncature. For r/H113501, the so- lutions of /H2084910/H20850are thus of the form054109-11 Parametric instability of the helical dynamo Phys. Fluids 19, 054109 /H208492007 /H20850b±=Km±1/H20849kr/H20850/H20858 j=1N bj/H20849t/H20850/H20849 D3/H20850 and, from /H20849D1/H20850, these solutions satisfy the conditions /H2084911/H20850at the interface r=1. To determine the functions bj/H20849t/H20850, it is suf- ﬁcient to solve the induction equation /H208498/H20850forr/H113491. For that, we replace the expression of b±given by /H20849D2/H20850into the in- duction equation /H208498/H20850, in order to determine the residual R±=/H20858 j=1N /H20853b˙j+/H20851k2+/H9251j2+i/H20849m/H9024+kV/H20850/H20852bj/H20854/H9274j± /H11007i 2r/H9024/H11032/H20858 j=1N bj/H20849/H9274j++/H9274j−/H20850. /H20849D4/H20850 We then solve the following system: /H20885 01 R+/H9278i+rdr+/H20885 01 R−/H9278i−rdr=0 , i=1, ..., N, /H20849D5/H20850 where the weighting functions are deﬁned by /H9278j± =Jm±1/H20849/H9251jr/H20850/Jm±1/H20849/H9251j/H20850. Using the orthogonality relation /H20885 01 /H9278i+/H9274j+rdr+/H20885 01 /H9278i−/H9274j−rdr=/H9254ijGij /H20849D6/H20850 with Gii=Km+1/H20849k/H20850 Jm+12/H20849/H9251i/H20850/H20885 01 Jm+12/H20849/H9251ir/H20850rdr +Km−1/H20849k/H20850 Jm−12/H20849/H9251i/H20850/H20885 01 Jm−12/H20849/H9251ir/H20850rdr, /H20849D7/H20850 we write the system /H20849D5/H20850in the following matrix form: X˙=MXwith X=/H20849b1, ..., bN/H20850/H20849 D8/H20850 withMij=/H9254ij/H20849k2+/H9251j2/H20850+i Gii/H20885 01 /H20849/H9278i+/H9274j++/H9278i−/H9274j−/H20850/H20849m/H9024+kV/H20850rdr −i 2Gii/H20885 01 /H20849/H9278i+−/H9278i−/H20850/H20849/H9274j++/H9274j−/H20850/H9024/H11032r2dr. /H20849D9/H20850 The numerical resolution of this system is done with a fourth-order Runge-Kutta time-step scheme. We took a whitenoise as an initial condition for the b j. APPENDIX E: ERRATUM OF NORMAND „2003 … RESULTS „REF. 25… For very small values of the ﬂuctuation rate /H9267and for an inﬁnite shear /H20851case /H20849i/H20850/H20852, a comparison can be made between the results obtained for /H9262/tildewidest=0 by the method based on Floquet theory /H20849see Appendix A /H20850and those obtained by a perturba- tive approach,25which consists in expanding R ¯mand the fre- quency Im /H20849p/H20850in powers of /H9267according to Im/H20849p/H20850=/H92750+/H9267/H92751+/H92672/H92752+¯, /H20849E1/H20850 R¯m=R 0+/H9267R1+/H92672R2+¯, /H20849E2/H20850 where R 0and/H92750are, respectively, the critical values of the Reynolds number and the frequency in the case of a station-ary ﬂow. At the leading order, it appears that /H92751=R1=0. At the next order, the expressions of R 2and/H92752are given in Ref. 25; however, their numerical values are not correct due to an error in their computation. After correction, the new values of R 2and/H92752are given in Fig. 7for the set of parameters considered in Ref. 25:m =1,k=−0.56, and /H9003¯=1 /H20849/H9262¯=0.44 /H20850. The different curves cor- respond to values of /H9003/tildewidest, which are not necessarily the same as those taken in Ref. 25. For /H20841/H9262/tildewidest/H20841sufﬁciently small /H20849curves e and f /H20850,R 2changes its sign twice versus the forcing fre- quency. We ﬁnd that R 2is negative for low and high frequen- FIG. 7. Results obtained by the perturbative approach for m=1,k=−0.56, and /H9262¯=0.44 /H20849/H9003¯=1/H20850. The dynamo threshold R2is plotted versus /H9275ffor several values of/H9262/tildewidest=/H20849a/H208501.56; /H20849b/H208501.28; /H20849c/H208501;/H20849d/H208500.72; /H20849e/H208500.44; /H20849f/H208500.16; and /H20849g/H208500.054109-12 Peyrot, Plunian, and Normand Phys. Fluids 19, 054109 /H208492007 /H20850cies, implying a dynamo threshold smaller than the one for the stationary ﬂow. At intermediate frequencies, R 2is posi- tive with a maximum value, implying a dynamo thresholdlarger than the one for the stationary ﬂow. For larger valuesof/H20841 /H9262/tildewidest/H20841/H20849curves a, b, c, and d /H20850R2is positive for all forcing frequencies with a maximum value at a low frequency, whichincreases with /H20841 /H9262/tildewidest/H20841. This implies that for /H20841/H9262/tildewidest/H20841sufﬁciently large, the dynamo threshold is larger than the one obtained for thestationary ﬂow, as was already mentioned in Sec. III A. For /H9003 /tildewidest=1.78, we have /H9262/tildewidest=0. In this case, we have checked that the values of R 2and/H92752are in good agreement with the values of R mand/H9275obtained by the method of Appendix A, provided /H9267/H113490.1. For completeness, we have also calculated the values of R2and/H92752form=1,k=−1, and /H9003¯=1, /H20849/H9262¯=0/H20850, as considered in the body of the paper. The results are plotted in Fig. 8. Qualitatively, the results are in good agreement with those ofFig.7. For /H9262/tildewidest=0 again, we have checked that the values of R 2 and/H92752are in good agreement with the values of R mand/H9275¯ obtained by the method of Appendix A, provided /H9267/H113490.1. For higher values of the modulation amplitude /H9267, the relative difference between the results obtained by the two methodscan reach 10% on R 2for/H9267=0.4. Finally, it must be noticed that our parameters /H9003/tildewidestand/H9275f are strictly equivalent to, respectively, /H92551and/H9268in Ref. 25. 1P. Cardin, D. Jault, H.-C. Nataf, and J.-P. Masson, “Towards a rapidly rotating liquid sodium dynamo experiment,” Magnetohydrodynamics 38, 177 /H208492002 /H20850. 2M. Bourgoin, L. Marié, F. Pétrélis, C. Gasquet, A. Guigon, J.-B. Luciani, M. Moulin, F. Namer, J. Burgete, A. Chiffaudel, F. Daviaud, S. Fauve, P.Odier, and J.-F. Pinton, “Magnetohydrodynamics measurements in the vonKarman sodium experiment,” Phys. Fluids 14, 3046 /H208492002 /H20850. 3P. Frick, V. Noskov, S. Denisov, S. Khripchenko, D. Sokoloff, R. Stepanov, and A. Sukhanovsky, “Non-stationary screw ﬂow in a toroidalchannel: way to a laboratory dynamo experiment,” Magnetohydrodynam-ics 38, 143 /H208492002 /H20850. 4F. Ravelet, A. Chiffaudel, F. Daviaud, and J. Léorat, “Towards an experi- mental von Karman dynamo: numerical studies for an optimized design,”Phys. Fluids 17, 117104 /H208492005 /H20850.5L. Marié, C. Normand, and F. Daviaud, “Galerkin analysis of kinematic dynamos in the von Kármán geometry,” Phys. Fluids 18, 017102 /H208492006 /H20850. 6R. Avalos-Zunñiga and F. Plunian, “Inﬂuence of electro-magnetic bound- ary conditions onto the onset of dynamo action in laboratory experi-ments,” Phys. Rev. E 68, 066307 /H208492003 /H20850. 7R. Avalos-Zunñiga and F. Plunian, “Inﬂuence of inner and outer walls electromagnetic properties on the onset of a stationary dynamo,” Eur.Phys. J. B 47, 127 /H208492005 /H20850. 8N. Leprovost and B. Dubrulle, “The turbulent dynamo as an instability in a noisy medium,” Eur. Phys. J. B 44, 395 /H208492005 /H20850. 9S. Fauve and F. Pétrélis, “Effect of turbulence on the onset and saturation of ﬂuid dynamos,” in Peyresq Lectures on Nonlinear Phenomena , edited by J. Sepulchre /H20849World Scientiﬁc, Singapore, 2003 /H20850, pp. 1–64. 10J-P. Laval, P. Blaineau, N. Leprovost, B. Dubrulle, and F. Daviaud, “In- ﬂuence of turbulence on the dynamo threshold,” Phys. Rev. Lett. 96, 204503 /H208492006 /H20850. 11F. Petrelis and S. Fauve, “Inhibition of the dynamo effect by phase ﬂuc- tuations,” Europhys. Lett. 76, 602 /H208492006 /H20850. 12F. Ravelet, “Bifurcations globales hydrodynamiques et magnétohydrody- namiques dans un écoulement de von Kármán turbulent,” Ph.D. thesis,Ecole Polytechnique, Palaiseau, France, 2005. 13R. Volk, Ph. Odier, and J.-F. Pinton, “Fluctuation of magnetic induction invon Kármán swirling ﬂows,” Phys. Fluids 18, 085105 /H208492006 /H20850. 14D. Lortz, “Exact solutions of the hydromagnetic dynamo problem,” Plasma Phys. 10, 967 /H208491968 /H20850. 15Yu. Ponomarenko, “Theory of the hydromagnetic generator,” J. Appl. Mech. Tech. Phys. 14, 775 /H208491973 /H20850. 16P. H. Roberts, “Dynamo theory,” in Irreversible Phenomena and Dynami- cal Systems Analysis in Geosciences , edited by C. Nicolis and G. Nicolis /H20849D. Reidel, Dordrecht, 1987 /H20850, pp. 73–133. 17A. D. Gilbert, “Fast dynamo action in the Ponomarenko dynamo,” Geo- phys. Fluid Dyn. 44, 241 /H208491988 /H20850. 18A. A. Ruzmaikin, D. D. Sokoloff, and A. M. Shukurov, “Hydromagnetic screw dynamo,” J. Fluid Mech. 197,3 9 /H208491988 /H20850. 19A. Basu, “Screw dynamo and the generation of nonaxisymmetric magnetic ﬁelds,” Phys. Rev. E 56, 2869 /H208491997 /H20850. 20A. D. Gilbert and Y. Ponty, “Slow Ponomarenko dynamos on stream sur- faces,” Geophys. Fluid Dyn. 93,5 5 /H208492000 /H20850. 21A. D. Gilbert, “Dynamo theory,” in Handbook of MathematicalFluid Dy- namics , edited by S. Friedlander and D. Serre /H20849Elsevier, New York, 2003 /H20850, Vol. 2, pp. 355–441. 22A. Gailitis and Ya. Freiberg, “Non uniform model of a helical dynamo,”Magnetohydrodynamics /H20849N.Y. /H2085016,1 1 /H208491980 /H20850. 23A. Gailitis, O. Lielausis, S. Dementiev, E. Platacis, A. Cifersons, G. Ger- beth, Th. Gundrum, F. Stefani, M. Christen, H. Hänel, and G. Will, “De-tection of a ﬂow induced magnetic ﬁeld eigenmode in the Riga dynamofacility,” Phys. Rev. Lett. 84, 4365 /H208492000 /H20850. FIG. 8. Same as Fig. 7, but for m=1,k=−1, and /H9262¯=0 /H20849/H9003¯=1/H20850. The labels correspond to /H9262/tildewidest/H11005/H20849a/H208501.5; /H20849b/H208501.25; /H20849c/H208501;/H20849d/H208500.5; /H20849e/H20850−0.5; and /H20849f/H208500.054109-13 Parametric instability of the helical dynamo Phys. Fluids 19, 054109 /H208492007 /H2085024A. Gailitis, O. Lielausis, E. Platacis, S. Dementiev, A. Cifersons, G. Ger- beth, Th. Gundrum, F. Stefani, M. Christen, and G. Will, “Magnetic ﬁeldsaturation in the Riga dynamo experiment,” Phys. Rev. Lett. 86, 3024 /H208492001 /H20850. 25C. Normand, “Ponomarenko dynamo with time-periodic ﬂow,” Phys. Fluids 15, 1606 /H208492003 /H20850. 26A. A. Schekochihin, S. C. Cowley, J. L. Maron, and J. C. McWilliams, “Critical magnetic Prandtl number for small-scale dynamo,” Phys. Rev.Lett. 92, 054502 /H208492004 /H20850. 27Y. Ponty, P. D. Mininni, H. Politano, J.-F. Pinton, and A. Pouquet, “Dy- namo action at low magnetic Prandtl numbers: mean ﬂow vs. fully turbu-lent motion,” New J. Phys. /H20849in press /H20850.28A. A. Schekochihin, A. B. Iskakov, S. C. Cowley, J. McWilliams, M. R. E. Proctor, and T. A. Jousef, “Fluctuation dynamo and turbulent inductionat low magnetic Prandtl numbers,” New J. Phys. /H20849in press /H20850; see also Fig. 2 of A. B. Iskakov, A. A. Schekochihin, S. C. Cowley, J. McWilliams, andM. R. E. Proctor, “Numerical demonstration of ﬂuctuation dynamo at lowmagnetic Prandtl numbers,” Phys. Rev. Lett. 98, 208501 /H208492007 /H20850. 29R. Monchaux, M. Berhanu, M. Bourgoin, P. Odier, M. Moulin, J.-F. Pin- ton, R. Volk, S. Fauve, N. Mordant, F. Pétrélis, A. Chiffaudel, F. Daviaud,B. Dubrulle, C. Gasquet, L. Marié, and F. Ravelet, “Generation of mag-netic ﬁeld by a turbulent ﬂow of liquid sodium,” Phys. Rev. Lett. 98, 044502 /H208492007 /H20850.054109-14 Peyrot, Plunian, and Normand Phys. Fluids 19, 054109 /H208492007 /H20850
1.3583599.pdf	Effects of film thickness and mismatch strains on magnetoelectric coupling in vertical heteroepitaxial nanocomposite thin films H. T. Chen, L. Hong, and A. K. Soh Citation: Journal of Applied Physics 109, 094102 (2011); doi: 10.1063/1.3583599 View online: http://dx.doi.org/10.1063/1.3583599 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/109/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Adjustable magnetoelectric effect of self-assembled vertical multiferroic nanocomposite films by the in-plane misfit strain and ferromagnetic volume fraction J. Appl. Phys. 115, 114105 (2014); 10.1063/1.4868896 Local probing of magnetoelectric coupling and magnetoelastic control of switching in BiFeO3-CoFe2O4 thin-film nanocomposite Appl. Phys. Lett. 103, 042906 (2013); 10.1063/1.4816793 Magnetoelectric and multiferroic properties of variously oriented epitaxial BiFeO 3 – CoFe 2 O 4 nanostructured thin films J. Appl. Phys. 107, 064106 (2010); 10.1063/1.3359650 Electric-field-induced magnetization reversal in 1–3 type multiferroic nanocomposite thin films J. Appl. Phys. 106, 014902 (2009); 10.1063/1.3158069 Strain-mediated magnetoelectric coupling in Ba Ti O 3 - Co nanocomposite thin films Appl. Phys. Lett. 92, 062908 (2008); 10.1063/1.2842383 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.35.40.137 On: Fri, 21 Nov 2014 19:39:50Effects of film thickness and mismatch strains on magnetoelectric coupling in vertical heteroepitaxial nanocomposite thin films H. T. Chen, L. Hong, and A. K. Soha) Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, People’s Republic of China (Received 18 February 2011; accepted 26 March 2011; published online 3 May 2011) The phase ﬁeld model is adopted to study the magnetoelectric coupling effects in vertical heteroepitaxial nanocomposite thin ﬁlms. Both the lateral epitaxial strains between the ﬁlm and the substrate and the vertical epitaxial strains between the ferroelectric and ferromagnetic phases areaccounted for in the model devised. The effects of the ﬁlm thickness on the magnetic-ﬁeld-induced electric polarization (MIEP) are investigated. The results obtained show that the MIEP is strongly dependent on the ﬁlm thickness, as well as on the vertical and lateral epitaxial strains. VC2011 American Institute of Physics . [doi: 10.1063/1.3583599 ] I. INTRODUCTION Multiferroic materials1–3that combine two or more of the ferroic properties ferroelectricity, ferromagnetism, andferroelasticity have attracted considerable interest for their potential applications as multifunctional devices. 4–6Magne- toelectric (ME) coupling effects, i.e., the variation of thepolarization induced by an applied magnetic ﬁeld or of the magnetization induced by an applied electric ﬁeld, can be realized in multiferroics with the coexistence of ferroelec-tricity and ferromagnetism. Weak ME effects at low temper- atures have been observed in some single-phase materials, 7,8 and this limits their practical applications considerably. Comparatively, composite multiferroics9,10can produce much larger ME coupling effects due to the incorporation of magnetostrictive and electrostrictive effects. Although largeME coupling effects have been observed in bulk bilayer mul- tiferroics, the ME coupling effects that exist in artiﬁcially assembled bilayer epitaxial nanocomposite thin ﬁlms areweak due to the clamping effects from the substrate. 11How- ever, the reported enhancement of elastic coupling resulting from the larger interfacial area between the two phases in avertical heteroepitaxial nanocomposite thin ﬁlm with ferro- magnetic (FM) nanopillars embedded in a ferroelectric (FE) matrix, 12in which the clamping effect from the substrate is reduced, has triggered great enthusiasm for the study of ME effects in such nanocomposite thin ﬁlms. Compared with bulk composite multiferroics, nanocomposite thin ﬁlms pos-sess more degrees of freedom in tuning the ME coupling effects due to their three-dimensional epitaxial properties. Some theoretical works have been carried out in order tostudy the ME coupling effects in such multiferroics; for example, the magnetic-ﬁeld-induced electric polarization (MIEP) was studied 13using the Green’s function technique, and the inﬂuence of the elastic stresses induced by the FE– FM and ﬁlm–substrate interfaces on the ME coupling was investigated using the time-dependent Ginzburg–Landauequation. 14Because the stress states were only approxi- mately determined in those works, they are unable to providea good understanding of the behavior of complex nanostruc- tures. The phase ﬁeld approach is a promising method forthe study of ME coupling effects, as not only the ferroelec- tric and ferromagnetic domain states can be determined dur- ing the evolution process but also the long-range elasticinteractions in a multiferroic. However, in the phase ﬁeld study carried out by Zhang et al. , 15only the lateral epitaxial misﬁt strains were accounted for, not the corresponding ver-tical strains, which have been observed experimentally as the domineering strains in ﬁlms of relatively large thicknesses (say, more than 20 nm). 16As the large lattice mismatch between the FE and FM phases cannot be fully relaxed by the embedded pillars with small diameters in the nanocom- posite ﬁlm, signiﬁcant vertical epitaxial strains would existand would affect the ME coupling in the ﬁlm. Therefore, both the lateral and vertical epitaxial misﬁt strains should be taken into account in phase ﬁeld modeling. In the present study, a three-dimensional (3D) phase ﬁeld model is devised in which both the lateral and vertical epitaxial misﬁt strains are included in the strain state. Theeffects of the thickness on the magnetic-ﬁeld-induced elec- tric polarization will be investigated. II. PHASE FIELD MODELING In the present study, BaTiO 3–CoFe 2O4vertical heteroepi- taxial nanostructure multiferroic thin ﬁlms are investigated; aschematic diagram of these ﬁlms is depicted in Fig. 1(a).N o t e FIG. 1. (a) 3D schematic illustration of a vertical heteroepitaxial nanocom- posite ﬁlm in which CoFe 2O4nanopillars are embedded in a BaTiO 3matrix. (b) Cross-section of the simulation system along the x1/C0x3plane.a)Author to whom correspondence should be addressed. Electronic mail: aksoh@hkucc.hku.hk. FAX: þ852-58585415. 0021-8979/2011/109(9)/094102/5/$30.00 VC2011 American Institute of Physics 109, 094102-1JOURNAL OF APPLIED PHYSICS 109, 094102 (2011) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.35.40.137 On: Fri, 21 Nov 2014 19:39:50that a circular nanopillar was selected for the investigation in view of the experimental work carried out by Zheng et al. ,12 in which the multiferroic composite fabricated was composed of nearly circular CoFe 2O4nanopillars embedded in a BaTiO 3 matrix. Three ﬁeld parameters are introduced to characterizethe total free energy of the nanocomposite system: a localpolarization ﬁeld P¼(P 1,P2,P3); a local magnetization ﬁeld M¼Msm¼Ms(m1,m2,m3), where Msandmdenote the sat- uration magnetization and the unit magnetization vector,respectively; and an order parameter g, which describes the spatial distribution of the FE and FM phases in the nanocom-posite ﬁlm, where g¼1 and 0 represent the FM and FE phases, respectively. The total free energy of the system is given by F¼ð V1/C0g ðÞ fpþgfmþfelas/C2/C3 dV; (1) where fprepresents the free energy in the FE phase, includ- ing the ferroelectric bulk free energy, ferroelectric domain wall energy, and electrostatic energy, which can be expressed as fp¼a1P2 1þP2 2þP2 3/C0/C1 þa11P4 1þP4 2þP4 3/C0/C1 þa12P2 1P22þP2 2P23þP2 3P21/C0/C1 þa111P6 1þP6 2þP6 3/C0/C1 þa112P4 1P22þP2 3/C0/C1 þP4 2P21þP2 3/C0/C1 þP4 3P21þP2 2/C0/C1 /C2/C3 þa123P2 1þP2 2þP2 3/C0/C1 þa1111P8 1þP8 2þP8 3/C0/C1 þa1122P4 1P42þP4 2P43þP4 3P41/C0/C1 þa1112P6 1P22þP2 3/C0/C1 þP6 2P21þP2 3/C0/C1 þP6 3P21þP2 2/C0/C1 /C2/C3 þa1123P4 1P22P23þP4 2P23P21þP4 3P21P22/C0/C1 þ1 2G11P2 1;1þP2 2;2þP2 3;3/C16/C17 þG12P1;1P2;2þP2;2P3;3þP3;3P1;1/C0/C1 þ1 2G44P1;2þP2;1/C0/C12þP2;3þP3;2/C0/C12þP1;3þP3;1/C0/C12hi þ1 2G0 44P1;2/C0P2;1/C0/C12þP2;3/C0P3;2/C0/C12þP1;3/C0P3;1/C0/C12hi /C0Edip/C1P;(2) where a1,a11,a12,a111,a112,a123,a1111,a1122,a1112,a n d a1123 are the phenomenological Landau expansion coefﬁcients; G11, G12,G44,a n d G440are the gradient energy coefﬁcients; and Edip is the electric ﬁeld generated by the long-range dipole–dipole interaction that can be estab lished in Fourier space. The com- mas in the subscripts denote spatial differentiation. The ferromagnetic energy term fm, which includes the magnetocrystalline anisotropy energy, magnetic exchange energy, magnetostatic energy, and external magnetic ﬁeld energy, can be expressed as fm¼K1m2 1m22þm2 1m23þm2 2m23/C0/C1 þK2m2 1m22m23 þAm2 1;1þm2 1;2þm2 1;3þm2 2;1þm2 2;2þm2 2;3þm2 3;1/C16 þm2 3;2þm2 3;3/C17 /C01 2l0MsHd/C1m/C0l0MsHex/C1m; (3) where K1andK2are the anisotropy constants, and A,l0,Ms, Hd, and Hexdenote the exchange stiffness constant, perme- ability of vacuum, saturation magnetization, demagnetiza- tion ﬁeld, and exterior magnetic ﬁeld, respectively. The elastic energy felascan be expressed as felas¼1 2cijkleifekl¼1 2cijkl/C0 eij/C0e0 ijÞekl/C0e0 kl/C0/C1 ; (4) where cijkl,eij,eij,a n d eij0represent the elastic stiffness tensor, elas- tic strain, total strain, and stress-free strain, respectively. With ref- erence to Fig. 1(b), the elastic energy in the heterogeneous ﬁlm/ substrate bilayer system is calculated by reducing the system to anelastic homogeneous system with an appropriately chosen effec-tive stress-free strain e ij0.17Thus, the effective stress-free strain in this nanocomposite ﬁlm is the sum of the strain eij*related to the electrostrictive/magnetostrictiv e effect, the epitaxial misﬁt strain eijepitax, and the virtual stress-free strain eijvirtual,t h a ti s , e0 ij¼e/C3 ijþeepitax ijþevirtual ij : (5) The electrostrictive/magnetostrictive related stress-free strain is given by e/C3 ij¼g3 2k100mimj/C01 3/C18/C19 /C20/C21 þ1/C0g ðÞ QijklPkPl/C0/C1 i¼jðÞ g3 2k111mimj/C18/C19 þ1/C0g ðÞ QijklPkPl i6¼j ðÞ ;8 >>< >>: (6) where Qijklis the electrostrictive coefﬁcient; i,j,k,l¼1, 2, 3; and k100andk111are the magnetostrictive constants. The epitaxial misﬁt strain eijepitax, which arises from the lateral lattice mismatch between the ﬁlm and the substrate and the vertical lattice mismatch between the FE and FM phases, can be expressed as eepitax ijet 11 00 0et 22 0 00 et 332 43 5; (7) in which the in-plane epitaxial misﬁt strains e11t ¼e22t¼(1/C0g)e11pþge11mare described as a function of the ﬁlm thickness18as follows: ep 11¼ep 22¼1/C01/C0ep0 11/C16/C17 1/C0ep0 111/C0hp c=h ðÞ; (8a)094102-2 Chen, Hong, and Soh J. Appl. Phys. 109, 094102 (2011) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.35.40.137 On: Fri, 21 Nov 2014 19:39:50em 11¼em 22¼1/C01/C0em0 11/C0/C1 1/C0em0 111/C0hm c=h/C0/C1 ; (8b) where his the ﬁlm thickness; hcpandhcmare the critical thick- ness for the formation of dislocation in the FE and FM phases, respectively;19ande11p0ande11m0are the corresponding pseudo- morphic in-plane misﬁt strains, deﬁned as follows: ep0 11¼ep0 22¼ap/C0as/C0/C1 =ap; (9a) em0 11¼em0 22¼am/C0as ðÞ =am; (9b) where apand amare the in-plane parameters of the FE and FM phases, respectively, and asis the lattice parameter of the cubic substrate. The vertical epitaxial misﬁt strainis expressed as e 33t¼(1/C0g)e33pþge33m, where e33p¼/C0e33m ¼/C00.8% is adopted.12,13The virtual stress-free strain eijvirtual should be distributed only inside the vacuum region, and the equilibrium strain produces vanishing stress in the vacuum region, which automatically satisﬁes the free surface bound- ary conditions. The dynamic evolution of the virtual stress-free strain and the polarization are described by the time-dependent Ginzburg–Landau equations @evirtual ij r;tðÞ @t¼/C0KdF devirtual ij r;tðÞ; (10) @Pir;tðÞ @t¼/C0LdF dPir;tðÞ; (11) where KandLare kinetic coefﬁcients. The dynamic evolution of the magnetization is obtained by solving the Landau–Lifshitz–Gilbert equation using the Gauss–Seidel projection method:20,21 1þa2/C0/C1 @M @t¼/C0 c0M/C2Heff/C0c0a MsM/C2M/C2Heff ðÞ ; (12) where c0is the gyromagnetic ratio, ais the damping con- stant, and Heff¼/C0 ð 1=l0Þð@F=@MÞis the effective mag- netic ﬁeld. III. SIMULATION PARAMETERS AND MATERIAL PROPERTIES The three-dimensional simulation system is composed of 64 /C264/C2Nzdiscrete grids, where Nzgrids encompass the vacuum layer, multiferroic nanocomposite thin ﬁlm, andsubstrate. With reference to Fig. 1(b), the top 6 layers are assumed to be the vacuum region, the bottom 40 layers are assumed to be the substrate, and the layers in between simu-late the multiferroic nanocomposite thin ﬁlm. The periodic boundary conditions are applied along the x 1andx2axes. The cell size in real space is chosen to be l0¼1 nm. The coefﬁcients used in the simulation are listed below.15 For BaTiO 3, a1¼4:124T/C0115 ðÞ /C2 105C/C02m2N;a11 ¼/C02:097/C2188C/C04m6N;a12¼7:974/C2188C/C04m6N;a111¼1:294/C2109C/C06m10N; a112¼/C01:950/C2109C/C06m10N;a123 ¼/C02:500/C2109C/C06m10N; a1111¼3:863/C21010C/C08m14N;a1112 ¼2:529/C21010C/C08m14N; a1122¼1:637/C21010C/C08m14N;a1123 ¼1:367/C21010C/C08m14N; Q11¼0:10C/C02m4;Q12¼/C00:034C/C02m4;Q44 ¼0:029C/C02m4;T¼25/C14C; c11¼1:78/C21011Nm/C02;c12¼0:96/C21011Nm/C02; c44¼1:22/C21011Nm/C02: For CoFe 2O4, Ms¼4/C2105A=m;k100¼/C0590/C210/C06;k111 ¼120/C210/C06;K1¼3/C2105J=m3;K2¼0J=m3; A¼7/C210/C012J=m: The in-plane lattice parameters for BaTiO 3and CoFe 2O4are ap¼0.399 nm and am¼0.419 nm, and the cubic parameters for the substrates SrTiO 3and DyScO 3areas(SrTiO 3) ¼0.3905 nm and as(DyScO 3)¼0.3943 nm.12,22The critical thickness for the formation of dislocation is on the order ofseveral nanometers. The values h cp¼hcm¼4 nm are adopted in the present study. The volume fraction of CoFe 2O4is set as 35%. For simplicity, elastic homogeneity is assumed inthe material whose elastic constants are taken as those of BaTiO 3. Considering the case where there is an in-plane compressive stress ﬁeld in the ferroelectric phase, the initialpolarization is set to be along the x 3direction. Upon switch- ing the direction of the applied magnetic ﬁeld from the x1to thex3axis, the MIEP is given by D/C22P3¼/C22P3(H//x1)/C0/C22P3(H//x3), where /C22P3is the average polarization of the nanocomposite ﬁlm. IV. RESULTS AND DISCUSSION Figure 2presents the variation of the MIEP with respect to the thickness of a nanocomposite ﬁlm. In a multiferroiccomposite, the ME coupling is essentially due to the stress- mediated interaction between the ferromagnetic and ferro- electric phases. In the present study, the magnetostrictiveproperties of the nanocomposite signiﬁcantly affect the MIEP. By rotating the applied magnetic ﬁeld from the x 1to thex3direction, the CoFe 2O4nanopillars with a negative magnetostrictive constant k100tend to contract in the x3 direction and extend along the x1axis. Consequently, the neighboring BaTiO 3phase is subject to both lateral and ver- tical contraction. Through the electrostrictive effects, the out-of-plane contraction reduces the polarization P3while the in-plane contraction enhances P3. Thus, the MIEP is attributed to the result of the competition between the lateral and vertical elastic interactions. In the present case, the verti- cal elastic interaction is more signiﬁcant, which leads to thedecrease of the out-of-plane polarization of the ﬁlm. With increasing ﬁlm thickness, the inﬂuences from the vertical094102-3 Chen, Hong, and Soh J. Appl. Phys. 109, 094102 (2011) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.35.40.137 On: Fri, 21 Nov 2014 19:39:50elastic interaction would become more and more dominant. As a result, the magnitude of the MIEP increases with increasing ﬁlm thickness. Moreover, in order to study the inﬂuence of the lateral interfacial strains, two substrates (i.e.,SiTiO 3and DyScO 3) are considered. With reference to Fig. 2, the MIEP is enhanced in the case of DyScO 3due to the smaller lattice mismatch between the nanocomposite thinﬁlm and the substrate. Thus, one could infer that comparable lattice parameters of the substrate and the ﬁlm are desirable in order to produce large magnetoelectric coupling effects. Figure 3presents the contours of the vertical stress r 33 with respect to the thickness of the ﬁlm deposited on the SrTiO 3substrate. It can be seen clearly from Fig. 3(a) that when the ﬁlm thickness is small, the vertical stress is greatly inﬂuenced by the substrate and the free surface, which causes rapid variation of vertical stresses in the direction of thethickness. With reference to Figs. 3(b)–3(d), the inﬂuence of the vertical epitaxial strains on the vertical stress state is increased with increasing ﬁlm thickness, which leads to amore homogeneously distributed vertical stress state along the direction of the thickness. Thus, it can be inferred that when the ﬁlm is sufﬁciently thick, the inﬂuence of the sub-strate becomes negligible, and the vertical strains play an im- portant role in manipulating the ME coupling effects. Consequently, the inﬂuence of the vertical strains on MEcoupling effects cannot be neglected in the nanocomposite ﬁlms in which large vertical strains would exist. Recently, some experimental results 23have illustrated that the vertical strain state could be controlled by tuning ei- ther the deposition frequency or the ﬁlm composition. Thus, a coherent coefﬁcient fis deﬁned to describe the tunable vertical epitaxial strain, which is expressed as e33t¼f[(1/C0g)e33pþge33m]. Note that f¼1 stands for full co- herence between the FE and FM phases, and f¼0 denotes full relaxation of the lattice mismatch between the two phases. Figure 4presents the relation between the MIEP and the coherent coefﬁcient, which clearly shows that the MIEPcan be enhanced by relaxing the vertical lattice mismatch. However, the small diameter of the nanopillars in the vertical heteroepitaxial nanocomposite thin ﬁlm leads to little relaxa-tion of the vertical strains in the ﬁlm, which would greatly suppress the ME coupling effects. This may provide some clue as to why only weak ME coupling was observed innanocomposite multiferroic ﬁlms. It is worth noting that for thin ﬁlms, the top and bottom electrodes should have some effect on the distribution of strains. However, for relativelythicker ﬁlms, the inﬂuences from the electrodes as well as the substrate are not dominant, as demonstrated above. In summary, the ME coupling effects in vertical heteroe- pitaxial nanocomposite thin ﬁlms have been studied using the phase ﬁeld method. It has been found that a better under- standing of the ME coupling effects in such thin ﬁlms can beachieved by incorporating the epitaxial strains in all three directions and the free surface and substrate effects. The MIEP is found to be strongly dependent on ﬁlm thickness, aswell as on vertical and lateral epitaxial strains. FIG. 3. (Color online) Variations of the vertical stress r33with the thickness of a ﬁlm deposited on a SrTiO 3substrate: (a) 16 nm, (b) 32 nm, (c) 48 nm, and (d) 64 nm. FIG. 4. Plot of the magnetic-ﬁeld-induced electric polarization vs the coher-ent coefﬁcient for the case of a ﬁlm deposited on a SrTiO 3substrate. FIG. 2. The dependence of the magnetic-ﬁeld-induced electric polarization on the ﬁlm thickness.094102-4 Chen, Hong, and Soh J. Appl. Phys. 109, 094102 (2011) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.35.40.137 On: Fri, 21 Nov 2014 19:39:50ACKNOWLEDGMENTS Support from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project Nos. HKU716508E and HKU716007E) is acknowledged. 1M. Fiebig, J. Phys. D 38, R123 (2005). 2W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature (London) 442, 759 (2006). 3C. W. Nan, J. Appl. Phys. 103, 031101 (2008). 4M. Gajek, M. Bibes, S. Fusil, K. Bouzehouane, J. Fontcuberta, A. Barthe- lemy, and A. Fert, Nature Mater. 6, 296 (2007). 5J. F. Scott, Nature Mater. 6, 256 (2007). 6M. Bibes and A. Barthelemy, Nature Mater. 7, 425 (2008). 7T. Kimura, T. Goto, H. Shinani, K. Ishizaka, T. Arima, and Y. Tokura, Nature (London) 426, 55 (2003). 8Y. Tokunaga, N. Furukawa, H. Sakai, Y. Taguchi, T. Arima, and Y. Tokura, Nature Mater. 8, 558 (2009). 9J. Zhai, S. Dong, Z. Xing, J. Li, and D. Viehland, Appl. Phys. Lett. 89, 083507 (2006). 10J. Ma, Z. Shi, and C. W. Nan, Adv. Mater. 19, 2571 (2007). 11M. K. Lee, T. K. Nath, C. B. Eom, M. C. Smoak, and F. Tsui, Appl. Phys. Lett.77, 3547 (2000).12H. Zheng, J. Wang, S. E. Loﬂand, Z. Ma, L. Mohaddes-Ardabili, T. Zhao, L. Salamanca-Riba, S. R. Shinde, S. B. Ogale, F. Bai, D. Viehland, Y. Jia, D. G. Gchlom, M. Wuttig, A. Roytburd, and R. Ramesh, Science 303, 661 (2004). 13C. W. Nan, G. Liu, Y. H. Lin, and H. Chen, Phys. Rev. Lett. 94, 197203 (2005). 14C. G. Zhong, Q. Jiang, and J. H. Fang, J. Appl. Phys. 106, 014902 (2009). 15J. X. Zhang, Y. L. Li, D. G. Schlom, L. Q. Chen, F. Zavaliche, R. Ramesh, and Q. X. Jia, Appl. Phys. Lett. 90, 052909 (2007). 16J. L. MacManus-Driscoll, P. Zerrer, H. Wang, H. Yang, J. Yoon, A. Fouchet, R. Yu. M. G. Blamire, and Q. X. Jia, Nature Mater. 7, 314 (2008). 17Y. U. Wang, Y. M. Jin, and A. G. Khachaturyan, Acta Mater. 51, 4209 (2003). 18Q. Y. Qiu, S. P. Alpay, and V. Nagarajan, J. Appl. Phys. 107, 114105 (2010). 19J. S. Speck and W. Pompe, J. Appl. Phys. 76, 466 (1994). 20X. P. Wang and C. J. Garcia-Cervera, J. Comput. Phys. 171, 357 (2001). 21J. X. Zhang and L. Q. Chen, Acta. Mater. 53, 2845 (2005). 22K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert, R. Uecker, P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan, L. Q. Chen, D. G. Schlom, and C. B. Eom, Science 306, 1005 (2004). 23Z. X. Bi, J. H. Lee, H. Yang, Q. X. Jia, J. L. MacManus-Driscoll, and H. Y. Wang, J. Appl. Phys. 106, 094309 (2009).094102-5 Chen, Hong, and Soh J. Appl. Phys. 109, 094102 (2011) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.35.40.137 On: Fri, 21 Nov 2014 19:39:50
1.333693.pdf	Effects of critical fluctuations on Gilbert damping in iron near T C B. Heinrich and A. S. Arrott Citation: Journal of Applied Physics 55, 2455 (1984); doi: 10.1063/1.333693 View online: http://dx.doi.org/10.1063/1.333693 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/55/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Pulsed laser deposition of epitaxial yttrium iron garnet films with low Gilbert damping and bulk-like magnetization APL Mat. 2, 106102 (2014); 10.1063/1.4896936 Gilbert ferromagnetic damping theory and the fluctuation-dissipation theorem J. Appl. Phys. 108, 073924 (2010); 10.1063/1.3330646 Origin of low Gilbert damping in half metals Appl. Phys. Lett. 95, 022509 (2009); 10.1063/1.3157267 Effect of interfaces on Gilbert damping and ferromagnetic resonance linewidth in magnetic multilayers J. Appl. Phys. 90, 4632 (2001); 10.1063/1.1405824 The effect of critical fluctuations on chemical equilibrium J. Chem. Phys. 86, 3602 (1987); 10.1063/1.451964 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.86.233 On: Mon, 22 Dec 2014 17:53:45Effects of critical fluctuations on Gilbert damping in iron near Tc B. Heinrich and A. S. Arrott Department of Physics. Simon Fraser University. Burnaby. British Columbia. Canada V5A IS6 Low field ferromagnetic resonance (FMR) is measured near Tc. The whisker is magnetized along its long axis (00 1), but not uniformly so because of demagnetizing effects and the nonlinear response of the magnetization to the internal field. A current at fm frequencies is passed along the axis creating a fm magnetic field which is transverse to and circulating about the axis in a skin depth of 0.01 mm. The condition ofFMR is reflected in the measured impedance. The frequency and temperature dependence of the Gilbert damping is extracted using a five parameter parametric equation of state taking into account the FMR equation of motion and the magnetostatics. The Gilbert damping is greater than that measured in high field microwave FMR and it peaks near Tc. The results are consistent with extrapolation of neutron measurements to extract the wave vector zero loss. PACS numbers: 76.50. + g I. INTRINSIC DAMPING Though the dynamic response of a ferromagnetic metal is dominated by eddy currents, once Maxwell's equations and Ohm's law are taken into account, measurements of the ac susceptibility yield an intrinsic damping parameter, for which Kamberskyl has given a theoretical interpretation us ing the interaction of the spins of the itinerant electrons with lattice vibrations mediated by the spin-orbit interaction. We have carried out several types of susceptibility experiments to obtain the intrinsic damping parameter for iron whiskers near the critical temperature in order to determine the ef fects of magnetic fluctuations and to see if the spin-orbit interactions are changed significantly as the spontaneous magnetization vanishes. II. EQUATIONS OF MOTION Gilbert has described the dissipation of energy for the precession of the magnetization in the local field by dM/dt = -yoMX[H -f3(dM/dt)), (1) where Yo is the gyromagnetic ratio and P is the damping coefficient, usually expressed as P= (I/AGHAG/yoMf, (2) where AG is the Gilbert damping parameter. The bracketed expression is dimensionless. Though f3 is proportional to A G , it has the dimensions of time while AG is a relaxation rate. Landau and Lifshitz have described the relaxation process using dM/dt= -y'(MXH)-A£dMIXl -H), where (3) (4) Presumably y' should be less than or equal to Yo, for it does not make sense for the precession rate to increase with damp ing. As often discussed2 the Gilbert and Landau-Lifshitz equations are formally equivalent for a magnetization of fixed magnitude. Both can be expressed as dMldt= -yMX[H-P(dMldt))' (5) For Gilbert's isotropic frictional model, y is not a parameter. For Landau-Lifshitz both y and P are parameters, given by y = y'[1 + (ALL/y'Mn (6) P = (I/ALL )(ALL/y'M)2/[1 + (ALL ly'M n (7) The Landau-Lifshitz form can describe any motion permit ted by a choice ofthe single Gilbert parameter AG by setting ALL =AG/[l + (AG/yoMf] , (8) y' = Yol[l + (AG/yoMf]· (9) As one goes through the Curie temperature, the Landau Lifshitz formulation makes a natural contact with paramag netic relaxation. To the extent that we can account for the low field reso nance experiments reported here, it is sufficient to use the single Gilbert coefficient. This is because the dimensionless parameter AG/YoM is sufficiently less than unity. On the other hand, our previous experiments on the susceptibility in zero field3 gave indications that AG/YoM reached values greater than unity. III. LOW FIELD RESONANCE The experimental arrangement by which we achieve ferromagnetic resonance in low fields is described in a recent report.4 An ac current at 100 MHz is used to create an ac field which curls about the central axis of an iron whisker magnetized by a longitudinal field sufficient for saturation of its central portion. The peak of Im(K) appears in fields of about 10 Oe. We determine the contribution of the whisker to the impedance of the circuit by measuring the standing wave ratio (Z -Zo)/(Z + Zo), where Zo is the impedance of both the coaxial cable and the Lecher wires that carry the current to the whisker. This is f+l, LiZ = -I, Pedz/21TR{)(z), (10) where Pe is the electrical resistivity, 2/) is the spacing between the Lecher wires contacting the whisker on its side surfaces, and {) (z) is the skin depth calculated from the per meability Jt(z) using (11 ) 2455 J. Appl. Phys. 55 (6).15 March 1984 0021-8979/84/062455-03$02.40 @ 1984 American Institute of Physics 2455 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.86.233 On: Mon, 22 Dec 2014 17:53:45The permeability is calculated from the equation of motion Eq. (5). It depends on z because the demagnetizing field var ies along the length of the whisker. To solve the equation of motion we assume that the whisker is circular. (We etch the whisker to make this assumption more realistic.) The field is H = (h,p¢ -41Tmpp)Jo(Kp)lJ o(KR )exp(iwt) + Hiz.(12) There is a large demagnetizing factor 41T for the radial com ponent mp' but none for the azimuthal component m,p' The internal field Hi = Ho + H D' The calculation of the demag netizing field H D for this nonlinear problem needs to be dis cussed further. The response is of the form M = (m,p¢ + mpp)Jo(Kp)IJo(KR )exp(iwt) + Mzz. (13) The frequency dependent inverse transverse susceptibility is = 1 + iwf3-Z -_w_ 1 + iwfJ-z , (14) Xl M 2 ( M) ··1 Xl(W) Hi "lHiBi Bi where Xl = Mzl Hi and Bi = Hi + 41TMz· The precession is very elliptical, Im,p I/lmp I = BJHi at resonance. The magnetization in iron near Te is described using a Schofield type equation of state5: HJM = cos (J rY; M 1M, = sin (J,t3; ( 15) (T -Te)IT, = (cos (J -sin2(J )r, with T, = 1.45, M, = 254 Oe, y = 1.33, and fJ = 0.368. The demagnetizing field is estimated by taking the total magnetic charge on one half of the cylinder, given by QT = 1TR 2M (0), and assuming it to be distributed linearly along the side of the cylinder starting at a position z, from the center. This charge gives a demagnetizing field at the center which, together with Ho, determine M (0) from the Schofield equation of state. This must be done self-consistently be cause M (0) determines QT' One finds, also self-consistently, the value of z, from the requirement that it be the position along the whisker where the internal field goes to zero. As the approximate nature of this estimate of Ho is a possible source of our difficulties in fitting our results on the low field side of resonance, we are continuing to work on this prob lem.n The results presented here differ from those given re cently.4 The sample is now an etched whisker. Instead of measuring the impedance directly as a function of applied field, we have added a single turn coil wrapped closely about the center of the whisker in order to locally modulate the applied field as it is swept through the resonance fields. This removes some of the effects of the uncertainty of how to model the demagnetizing field as one passes from the region near the center of the whisker where the differential suscepti bility is low to the region near Zl and beyond where the sus ceptibility becomes very large. The greater the ratio of z, to the diameter of the modulation coil, the less the uncertainty of the effect of the demagnetizing field on the modulated signal. IV. RESULTS Figure 1 shows results for the derivative of the predo minantly outphase component of the fm impedance. There are difficulties with respect to the accuracy of the phase set- 2456 J. Appl. Phys., Vol. 55, No.6, 15 March 1984 o 20 40 60 H[Oe] FIG. I. Field dependence of the derivative with respect to field of imaginary part of an averaged reciprocal skin depth at 100 MHz. Zero represents the peak in the imaginary part. The upper curve is for Te -T = 1.34 K and the lower one is for T, -T = 0.23 K calculated for AG = 4 X 10K sec -I and AG = 6x 108 sec-I with appropriate magnetic parameters for each tem perature and field. ting in these experiments. These are not sufficient to effect our conclusions, but we do plan to eliminate them in future experiments. The fits using the analysis outlined above are shown in Fig. 1. The results of the analysis are summarized in the first part of Table I. Note that as long as one stays at a fixed frequency, the closer one gets to Te , the higher the field at resonance. Even though the measurements are quite close to Te, the magnetization is still greater than 1/ 12 the value at absolute zero. The dc differential susceptibility is just into the dipolar region and orders of magnitude smaller than in the zero field measurements.3 Nevertheless there is a sub stantial increase in AG as X increases near Te. The second group of entries in Table I are from the previous report, where the comparison of theory and experiment are illus trated. In all the fittings we have chosen Yo as 1.836x 107 sec - " corresponding to the g value 2.088 found by ferro magnetic resonance below Tc. 7 The experiments from which we first obtained evidence for large values of AG near Tc were carried out in zero dc bias field. The magnetization pattern appears to take a configura tion in which the magnetization curls about the long axis of the whisker. Theory and experiment for this case are shown TABLE I. Temperature dependence ofrelevant physical parameters. Temperature AG Mz Resonant H, (Te -T) sec-I dMz/dH, (Oe) (Oe) 1.34 4X 108 0.25 248 7 0.38 5.5X 108 0.83 143 14 0.23 6X 10" 1.02 155 17 0.084 6X 10K 1.21 140 20 1.58 4.5X 108 0.2 265 11 0.06 7X lOS 1.1 144 25 0.06 3X 109 3x IOJ 80 B. Heinrich and A. S. Arrott 2456 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.86.233 On: Mon, 22 Dec 2014 17:53:45• 1 wB a_1 4nD; 4nD o 10 20 f [kHz] FIG. 2. Frequency dependence of inverse susceptibility 1/ 417DXext = a(w) + iwfJ (w). Tc -T = 0.06 K. X's are [a(w) -I]. + 's are wfJ (w). The solid curves are calculated for AG = 3 X 109 sec -I. in Fig. 2. The.8 deduced is converted to AG = 3 X 109 sec -I using Eq. (2); see last entry in Table I. This may be taken as an upper limit for it presumes that all of the response comes from magnetization pointing in the ifJ direction, whereas the effect of applying a de bias field along the whisker shows that the loss increases as the magnetization rotates toward the axis, as expected from the model with constant .8. As the magnetization rotates, the internal field remains very close to zero until saturation is reached. In the resonance experi ment saturation occurs and Hi increases with Ho. In this sense the resonance experiments are further from the critical 2457 J. Appl. Phys., Vol. 55, No.6, 15 March 1984 point, as measured by the differential susceptibility. Thus it is consistent that A G should be larger in the zero field experi ments . One can also find such large damping parameters in the neutron diffraction results of Mezei.8 He measures the dy namic response (16) where rq is the line width parameter, which seems to have a q independent contribution that goes as ro = r vY -112. At T= T + 1.4 K, where X is unity, ro = rl = 2X 109 sec-I. The neutron diffraction line width for the diffusive mode can be related to the relaxation rate in the Landau-Lifshitz for mulation of the problem by ALL = Xro. The connection to the Gilbert description in terms of magnetic viscosity is not so clear, but the agreement in magnitude is worth noting. These relaxation rates are substantially greater than found at lower temperature from ferromagnetic resonance. Heinrich and Frait9 found AG = 4 X 107 sec -I at 300 K and 2 X 108 sec-I at 800 K. Bhagae reports AG = 1.3 X 108 sec-I from 800 to 1000 K. These values in the noncritical region have been reasonbly accounted for by Kambersky. I 'v. Kambersky, Can. 1. Phys. 48, 2906 (1970); Czech. 1. Phys. B 26, 1366 (1976). 's. Iida, 1. Phys. Chern. Solids 24,631 (1963). ·'A. S. Arrott and B. Heinrich, 1. App!. Phys. 49, 2028 (1978). 4B. Heinrich and A. S. Arrott, 1. Magn. Magn. Mater. 31-34, 669 (1983). sA. Arrott and 1. E. Noakes, 1. App!. Phys. 42, 1288 (1971). "T. L. Templeton, A. S. Arrott and A. Aharoni (these Proceedings). 7S. M. Bhagat and M. S. Rothstein, Solid State Commun. 11, 1535 (1972). "F. Mezei, Phys. Rev. Lett. 49.1096,1537 (1982). 9B. Heinrich and Z. Frait, Phys. Status Solidi 16, Kll (1966). B. Heinrich and A. S. Arrott 2457 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.113.86.233 On: Mon, 22 Dec 2014 17:53:45
1.4811331.pdf	A simplified Tamm-Dancoff density functional approach for the electronic excitation spectra of very large molecules Stefan Grimme Citation: The Journal of Chemical Physics 138, 244104 (2013); doi: 10.1063/1.4811331 View online: http://dx.doi.org/10.1063/1.4811331 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/138/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Determining the appropriate exchange-correlation functional for time-dependent density functional theory studies of charge-transfer excitations in organic dyes J. Chem. Phys. 136, 224301 (2012); 10.1063/1.4725540 Analytical Hessian of electronic excited states in time-dependent density functional theory with Tamm-Dancoff approximation J. Chem. Phys. 135, 014113 (2011); 10.1063/1.3605504 Assessment of the ΔSCF density functional theory approach for electronic excitations in organic dyes J. Chem. Phys. 134, 054128 (2011); 10.1063/1.3530801 Nonadiabatic coupling vectors for excited states within time-dependent density functional theory in the Tamm–Dancoff approximation and beyond J. Chem. Phys. 133, 194104 (2010); 10.1063/1.3503765 Electronic circular dichroism spectra from the complex polarization propagator J. Chem. Phys. 126, 134102 (2007); 10.1063/1.2716660 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12THE JOURNAL OF CHEMICAL PHYSICS 138, 244104 (2013) A simpliﬁed Tamm-Dancoff density functional approach for the electronic excitation spectra of very large molecules Stefan Grimmea) Mulliken Center for Theoretical Chemistry, Institut für Physikalische und Theoretische Chemie der Universität Bonn, Beringstr. 4, D-53115 Bonn, Germany (Received 22 April 2013; accepted 3 June 2013; published online 25 June 2013) Two approximations in the Tamm-Dancoff density functional theory approach (TDA-DFT) to elec- tronically excited states are proposed which allow routine computations for electronic ultraviolet (UV)- or circular dichroism (CD) spectra of molecules with 500–1000 atoms. Speed-ups comparedto conventional time-dependent DFT (TD-DFT) treatments of about two to three orders of magni- tude in the excited state part at only minor loss of accuracy are obtained. The method termed sTDA (“s” for simpliﬁed) employs atom-centered Löwdin-monopole based two-electron repulsion inte- grals with the asymptotically correct 1/ Rbehavior and perturbative single excitation conﬁguration selection. It is formulated generally for any standard global hybrid density functional with givenFock-exchange mixing parameter a x. The method performs well for two standard benchmark sets of vertical singlet-singlet excitations for values of axin the range 0.2–0.6. The mean absolute de- viations from reference data are only 0.2–0.3 eV and similar to those from standard TD-DFT. Inthree cases (two dyes and one polypeptide), good mutual agreement between the electronic spectra (up to 10–11 eV excitation energy) from the sTDA method and those from TD(A)-DFT is obtained. The computed UV- and CD-spectra of a few typical systems (e.g., C 60, two transition metal com- plexes, [7]helicene, polyalanine, a supramolecular aggregate with 483 atoms and about 7000 basis functions) compare well with corresponding experimental data. The method is proposed together with medium-sized double- or triple-zeta type atomic-orbital basis sets as a quantum chemical toolto investigate the spectra of huge molecular systems at a reliable DFT level. © 2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4811331 ] I. INTRODUCTION Kohn-Sham density functional theory (KS-DFT) is now the most widely used method for electronic structure calcu-lations of larger molecules and this holds not only for elec- tronic ground states. In recent years, time-dependent den- sity functional theory (TD-DFT) 1–4has also emerged as the “work-horse” of quantum chemistry for the calculation of excited state properties and electronic spectra (see, e.g., Refs. 5–9for reviews, for TD-DFT calculations of ground state properties such as dispersion coefﬁcients, see, e.g., Refs. 10and11). Because of the moderate computational cost and complexity, TD-DFT is applicable to fairly large sys- tems (about 100 atoms) for which traditional wave function based methods are not routinely feasible (for alternative “low-cost” single-reference wave function based methods, see, e.g., Refs. 12and13, for TD-DFT treatments of a few excited states in very large systems with special hard- and software,see Ref. 14). However, as already simple computational con- siderations show, the theoretical treatment of an entireultraviolet-visible (UV-Vis) electronic spectrum (e.g., in a typical excitation energy range from 2 to 7 eV) for systems with several hundreds of atoms or a small protein with about a)grimme@thch.uni-bonn.de1000 atoms still remains a challenge. This problem is the topic of the present work. Normally, the standard TD-DFT approach yields roughly the same good accuracy for excited states as for ground states. With the usually employed adiabatic approximation,1,2TD- DFT is expected to work well for many of the low-lying va-lence states considered here, 3and one remaining problem is the choice of the time-independent density functional. Cur- rently, it is common practice to employ standard functionals inTD-DFT like those of the generalized gradient approximation (GGA) or global hybrid GGAs, e.g., B3LYP 15,16that origi- nally have been developed for ground states. One basic rea-son for the success is the use of the “correlated” KS orbitals that seem more reliable in an excited state treatment than, e.g., those from Hartree-Fock or semi-empirical alternatives. The inherent problems of TD-DFT are also apparent since excited state methods must include orbital relaxation ef- fects as well as static and dynamical electron correlation ef- fects for states of often very different character in a balanced manner. Moreover, some excited states have at least partialdouble excitation character and pose challenging multiplet problems for all single-reference approaches. Typically, stan- dard TD-DFT provides larger errors in situations where thereis substantial ionic, charge-transfer (CT), double excitation, Rydberg or multiplet character in the excited states which somewhat limits its applicability in electronic spectroscopy. 17 Partial solutions to this list of problems have been suggested. 0021-9606/2013/138(24)/244104/14/$30.00 © 2013 AIP Publishing LLC 138, 244104-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12244104-2 Stefan Grimme J. Chem. Phys. 138, 244104 (2013) We consider here mainly the CT-state problem (see, e.g., Refs. 18–21) which is very important in large systems as it leads to many artiﬁcial (“ghost”) states. The failure is relatedto the wrong asymptotic form for the exchange-correlation (XC) potential, the related self-interaction-error (SIE 22–25), and the integer-discontinuity problem20,25(see below for fur- ther discussion). This leads to the too small KS-orbital energy gap, too high single-particle ionization potentials (IP), and too low electron afﬁnity (EA). Although these states often havevanishing transition moments, they can “cover” the states of interest and must be avoided because of an otherwise extreme increase of the number of roots that have to be computed in order to uncover the physical ones. For a recently developed theoretical tool to identify such states, see Ref. 26. One way to alleviate the CT- and Rydberg-state problem is to increase the admixture of non-local Fock-exchange in hybrid functionals from typically 10% to 25%, which is bestfor ground states, to values up to 50% as, e.g., in the BHLYP functional. 27The dependency of TD-DFT excitation energies on the Fock-exchange admixture has been investigated sys-tematically for large organic molecules. 28This study showed that the optimum value on average is around 40% which is consistent with the ﬁnding, that PBE38 (3/8 =37.5% of Fock- exchange) performs best for frequency-dependent molecular dipole polarizabilities (which are related to UV-spectra).29 Both problems are mostly solved by an exact treatment of the DFT exchange30or by hybrid functionals containing 100% Fock-exchange.31Special treatments32,33as well as asymp- totically correct functionals34for such cases have also been developed. Nowadays, a common technique in TD-DFT cal- culations is to employ long-range corrected (LC, also calledrange-separated, RS) functionals which asymptotically em- ploy 100% non-local Fock-exchange. 35–38Recently, a kind of reformulation of TD-DFT called constricted variationaldensity functional theory (CV-DFT) has been proposed by Ziegler et al. 39and in this approach the CT/SIE-problem is cured by inclusion of exact two-electron integrals which al-ready appear in a second-order expansion of the theory. For a recent thorough discussion of the gap problem and a non- empirical (partial) solution in the LC/RS-framework, see thevery good publication of Kronik et al. 40 In summary, it seems clear that at present one cannot avoid inclusion of “exact” (Fock) exchange (and the result-ing response terms) in a DFT excited state treatment even for relatively simple valence excited states of electronically well- behaved systems (which are the topics of this work). This doesnot pose a major problem for the always required solution of the ground state KS self-consistent ﬁeld (SCF) type equations which is nowadays possible even for 500–1000 atoms using reasonable (double- or triple-zeta, e.g., SV(P) or TZVP 41,42) atomic orbital (AO) basis sets. However, what is really pro-hibitive in terms of computational cost is the solution of the special TD-DFT eigenvalue problem for the full UV-Vis spec- trum of a 500 atom system. Typically, the expansion space ofthe single excitation amplitudes is then of dimension 10 6–107 and importantly, about 102–103“true” (physical) eigenval- ues are required (one valence excited state per “light” atomis a good rule of thumb). It is currently not possible to per- form such calculations routinely when the necessary Fock-exchange is involved and hence, the electronic spectra at medium to higher energies of many interesting supramolec- ular, nanoscopic, or bio-molecular systems (e.g., small pro-teins) are not accessible theoretically at a DFT level. In this context, it is noted that a special TD-DFT treatment has been proposed for spectral applications with high density of statesin which the computation of individual roots is avoided but replaced by many calculations of frequency-dependent imag- inary polarizabilities. 43–45For a related direct time-dependent approach to an UV-spectrum coupled with semi-empirical methods see Ref. 46, for alternative fragment (subsystem) based TD-DFT methods for extended systems see, e.g., Ref. 47. The goal of the present work is to develop a simpliﬁed variant of TD-DFT that is based on the Tamm-Dancoff ap- proximation (TDA).48,49It will be dubbed sTDA (or sTDA- DFT) from now on. The approach makes use of drastic sim-pliﬁcations to the treatment of molecular two-electron inte- grals and massively truncates the single excitation expansion space. This leads to computational savings compared to stan-dard TD-DFT by at least two orders of magnitude at only mi- nor loss of accuracy in typical applications. In the proposed scheme, the solution of the ground state SCF equations is rate-determining and the ensuing computation of the full UV-Vis spectrum of a 500 atom system requires less than 1 h on a nor- mal laptop computer (single Intel-i7 2.7 GHz CPU throughoutthis work). Besides the principle limitations of the current TD-DFT treatment of excited states as mentioned above, high accuracy and including the subtle details of the many individual states is often not necessary in spectra calculations. Instead, a simplebut physically still reasonable description involving the most important (coupled) single excitations should be sufﬁcient to describe the typical situation of a very high density of elec-tronic states. In essence, it seems to be often sufﬁcient if in a broad electronic band composed of dozens of states only the majority is computed adequately. After a brief outline of the basic theory, the two fun- damental approximations involved in the sTDA approach are described. Their accuracy is then tested on two dif-ferent benchmark sets for common valence excitations of (mainly organic) molecules. The scope and limitations of the method are demonstrated for a few examples ranging fromthe CT state in a model system to Rydberg states and ﬁ- nally to electronic excitation (UV-Vis and circular dichroism, CD) spectra of medium-sized and large (up to 483 atoms)molecules. II. THEORY A. General In the following, the term Tamm-Dancoff approximation is used which is in the DFT community well established49 and denotes a simpliﬁcation to the full TD-DFT response problem.1,7In wave function theory, the TDA corresponds to a conﬁguration interaction (CI) problem in which only sin- gle excitations from a Hartree-Fock (or Kohn-Sham) determi- nant are included (usually termed CIS). In the following, the This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12244104-3 Stefan Grimme J. Chem. Phys. 138, 244104 (2013) abbreviation TD(A)-DFT (or TD(A)) is used whenever both, the full TD-DFT and its Tamm-Dancoff approximation is meant while TDA or TD is used speciﬁcally for the corre-sponding methods. The resulting TDA/CIS standard eigenvalue problem reads At=ωt, (1) where t a iare the unknown CIS amplitudes corresponding to an excitation from occupied molecular orbital (MO) ito vir- tual orbital a(ij. . . is used for occupied, ab. . . for virtual, and pqrs. . . for orbitals from any set), and ωrepresents an excita- tion energy vector of length mformdesired excited electronic states (roots). We consider in the following a closed-shell (re- stricted) reference ground state and only spin-adapted singletfunctions. The extension to triplet states, an unrestricted or restricted open-shell reference is straightforward (see, e.g., Refs. 50–52). Hirata and Head-Gordon 49have ﬁrst explored the TDA in the framework of DFT and have shown that it per- forms as well as the parent linear response approach. With large amounts of Fock-exchange mixing in hybrids or for LC/RS-functionals, TDA seems to be more robust regarding so-called triplet instabilities.53For a very recent comparison of TDA and TD treatments for valence states, see Ref. 54,f o r a very early precursor to TDA-DFT (dubbed DFT/SCI), see Ref. 55. In a general notation including CIS as well as TDA-DFT, the elements of the matrix Aread1,3,49 ATDA iajb=δijδab(/epsilon1a−/epsilon1i)+2(ia\|jb)−ax(ij\|ab) +(1−ax)(ia\|f\|jb), (2) where axis the amount of non-local Fock exchange which is included in a hybrid density functional (i.e., ax=1 for CIS), /epsilon1are the single-particle energies, and ( ia\|jb) is a two-electron integral in charge-cloud notation (i.e.,(ia\|jb)=/integraltext/integraltext i(r 1)a(r1)1 r12j(r2)b(r2)dr1dr2) with the corre- sponding MOs. They are obtained by solving the KS-SCF equations including the XC-energy from a standard hybrid ansatz15 Ehybrid XC=(1−ax)EGGA X+axEFock X+EGGA C, (3) where EGGA X andEFock X denote semi-local GGA and non- local Fock terms, respectively, and EGGA C is the GGA correla- tion energy. We will employ here mostly the Perdew-Burke-Ernzerhoff (PBE 56) functional as GGA component and note that excitation spectra are generally very insensitive to the choice of the GGA (but more strongly dependent on ax,s e e below). Because various non-standard values for axare tested, we will employ in addition to common abbreviations (e.g.,PBE0 57for a PBE hybrid with ax=0.25 or PBE3829for one with ax=0.375) the notation PBE( ax) to denote a hybrid with any amount of Fock-exchange. The last term in Eq. (2), which is of DFT origin, is deﬁned as (ia\|f\|jb)=/integraldisplay/integraldisplay i(r1)a(r1)fXC(r1,r2)j(r2)b(r2)dr1dr2. (4)In the adiabatic approximation, the time-dependent exchange- correlation kernel fXCis derived from the time-independent GGA portion of the ground state functional. This term is ne-glected here (or rather replaced) so that computationally ex- pensive numerical quadrature is completely avoided in the excited state part. As already mentioned, a TDA-DFT treatment for a larger system becomes computationally prohibitive for two basic reasons: First, the number of matrix elements grows as N 4 with system size (as measured by N) and an increasing num- ber of roots (on the order of hundreds to thousands) has to be computed in a given energy range. Hence, one step should be concerned with a reduction of the excitation space by se- lection techniques without too much sacriﬁcing the accuracy.Second, the matrix elements in Eq. (2)require manipulation of four-index two-electron integrals in the MO basis (or in the AO basis with transformed excitation vectors) which becomesextremely demanding when 10 3–104AO basis functions are involved. Simpliﬁcation of these two issues is in the focus of this work. B. Integrals The most intriguing ﬁnding of the present work is that a simple monopole (atomic charge) type approximation for the four-index two-electron repulsion integrals in the MO basiscan be used in the framework of TDA even with extended AO basis sets. In general, it reads (pq\|rs)≈N/summationdisplay AN/summationdisplay BqA pqqB rsγ(A,B )( 5 ) with the here generalized Mataga-Nishimoto-Ohno- Klopman58–60damped Coulomb law given by γ(A,B )J=/parenleftbigg1 (RAB)β+(axη)−β/parenrightbigg1/β . (6) The superscript Jindicates a Coulomb-type integral, RABis the interatomic distance, βis a parameter, and ηis the arith- metic average of the chemical hardness of the two atoms A andB,η(A)=∂2E(A) ∂n2, where nis the number of electrons and Eis the total atomic electronic energy. We take tabulated η(A) values consistent for all elements of the periodic table fromRef. 61. In semi-empirical theories, ηis normally identiﬁed with an average of atomic Coulomb and exchange integrals (or orbital based IP −EA values 62) and the exponent βis ﬁxed to integer values of one or two.58–60 TheqA ipare atom-centered point charges for a transition density ip(or charge density for p=i). Asymptotically for large distances, the integrals computed using the formula (6) have the correct 1/ Rbehavior. For RAB→0, the integral be- comes axηwhich is the desired result consistent with the orig- inal TDA-DFT matrix elements (cf. the third term in Eq. (2)). For exchange integrals ( ia\|jb) (superscript K), we employ a similar formula γ(A,B )K=/parenleftbigg1 (RAB)α+η−α/parenrightbigg1/α (7) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12244104-4 Stefan Grimme J. Chem. Phys. 138, 244104 (2013) but with the important difference that the corresponding in- tegrals remain ﬁnite (opposed to those derived from γJ)a l s o for vanishing axas required. The so computed exchange inte- grals have a different distance dependence than the Coulomb integrals which in the context of DFT represent the GGA response.7These physically different origins are reﬂected here in different decay properties with RAB. One further new element of the present method is that the (transition) density charges qA ipare obtained from a Löwdin population analysis63 qA ip=/summationdisplay μ∈AC/prime μiC/prime μp, (8) where the sum is over AO functions (index μ) centered on atom A. The matrix C/primedenotes orthogonalized MO coefﬁ- cients obtained from C/prime=S1/2C, where Care the coefﬁcients in the original basis. Note that/summationtextNbf μ\|C/prime μp\|2=1 for all p(Nbfbeing the number of AOs). If the qare pre-computed and stored in memory and the intermediate products qB pqγ(A,B )i nE q . (5)are also kept in memory, the evaluation of any MO repulsion integral requiresonly N at(the number of atoms in the system) operations. Be- cause the pre-factor is also small and optimized linear algebra routines can be used, the matrix Ain Eq. (1)can be built in a short time for several hundreds of atoms. From another point of view, the integral approximation corresponds to a kind of resolution-of-the-identity (RI64,65) approach with one “func- tion” per atom, i.e., one saves about a factor of 30–50 in com- putation time for typical AO basis sets compared to conven- tional RI. A monopole expansion of molecular (transition) densities is certainly a drastic approximation that has to be validated asdemonstrated below. Simple considerations show that local- ized exchange-type distributions are described badly, e.g., q ia is (incorrectly) vanishing for a single atom. It is thus expected, that localized atom-like excitations (e.g., nπ) for which the corresponding integrals are small but still on the order of about 0.2 eV are treated less accurately than, e.g., more de-localized ππ states (which are more in our focus). The ef- fect of these drastic (but necessary) approximations can be alleviated by introducing global but a x-dependent empirical parameters. However, because the orbitals and states of larger systems are generally more delocalized, the integral approxi- mation will improve for our actual targets. The simplest way to account for some systematic un- derestimation of exchange type integrals ( ia\|jb) and for the amount of Fock-admixture in the functional (i.e., the degree of correlation effects included) is to modify the distance de- pendence of the γfunctions by the values of βandαin Eqs. (6)and(7)separately. It is proposed to use linear re- lations of the form β=β(1)+β(2)ax (9) and analogously for the exchange part α=α(1)+α(2)ax, (10)where β(1–2)andα(1–2)are the only empirical parameters of the method. In summary, the sTDA matrix elements are computed from KS-DFT ground state quantities as AsTDA iajb=δijδab(/epsilon1a−/epsilon1i)+2(ia\|jb)/prime−(ij\|ab)/prime, (11) where the prime indicates the monopole based two-electron integrals. Note that the ( ij\|ab)/primevanish as required for ax=0 and that there is no analogue of the TD(A)-DFT ( ia\|f\|jb) response term. Its effect is small and absorbed here into(ia\|jb) /primeintegrals. The so computed matrix elements without further (conﬁguration selection) approximation as discussed below preserve invariance under unitary orbital transforma-tions and provide exactly degenerate states for non-Abelian symmetries. Overall the method requires as input only the Fock-exchange mixing parameter from the chosen density functional. The four global parameters β (1),β(2),α(1),α(2) are determined once by a least-squares ﬁt to reference exci- tation energies for a few values of ax. Due to this approach, the method can be applied without further adjustment to any existing density functional. It has been tested for global hy-brids with a xup to 0.8 but only values in the range 0.2 <ax ≤0.6 are recommended. The application of the sTDA method to LC/RS-functionals will be investigated in a forthcomingpaper. As noted already, the proposed approach for the TDA Hamiltonian matrix elements uses ideas from older semi-empirical MO theories. More speciﬁcally, a monopole approximation for integrals appearing in linear response type equations has already been employed by Niehaus et al. 66 in the context of the semi-empirical density functional tight-binding (DFTB) method. Because in this case the linear-response problem of a semi-local density functional was approximated, the Coulomb term (which arises from the response of ground state “exact” exchange) in Eq. (2) is not present ( ax=0) and the monopole (called “gamma”) approximation is only applied to ( ia\|jb)+(ia\|f\|jb). A monopole-type treatment for all integrals is ﬁrst proposedhere. The second difference concerns the computation of the charges q. In DFTB, the Mulliken population analysis 67is used which provides reasonable results if small (minimal)basis sets are applied. In general, however, for large and possibly extended (diffuse) basis functions the Mulliken ap- proach leads to artiﬁcial charges and too inaccurate integrals.This problem has been solved by the Löwdin partitioning which yields robust and consistent Coulomb and exchange integrals even for extended AO basis sets as tested below (forfurther improvements of the Löwdin scheme for conventional charges, see Ref. 68). III. TRUNCATION OF THE SINGLE EXCITATION SPACE In order to make the above integral approximation sen- sible, the other computational bottleneck in TDA must alsobe considered. Intelligent truncation of CI spaces has been successfully applied since decades to reduce the computa- tional effort at only minor loss of accuracy. 69–72These ap- proaches are physically based on the fact that the states of in- terest (composed here of so-called primary CIS conﬁguration This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12244104-5 Stefan Grimme J. Chem. Phys. 138, 244104 (2013) state functions, P-CSF) are strongly interacting only with a relatively small number of secondary conﬁgurations (S-CSF). A huge number of remaining ones can either be discardedor their effect is estimated by perturbation theory. The three spaces are termed P-CSF, S-CSF, and N-CSF (N stands for neglected). In the presently proposed approach, the P-CSF are de- ﬁned by an ordered set (according to their diagonal Hamilto- nian matrix element) of single excitations ψwhich are lower in energy than the maximum energy E maxof the spectral range of interest. For example, if one is interested in an UV- spectrum up to 7.5 eV , the P-CSF space initially consists of all CSF with a diagonal element ≤7.5 eV above the ground state. The ground state is represented by the non-interactingKS-determinant which is always set to the zero of energy by implicitly assuming Brillouins’ theorem. In a typical case with a few hundred atoms, the P-space includes about 10 3 CSF while the total dimension is about 106–107. In TD(A)- DFT and related methods (e.g., DFT/MRCI71), inclusion of the huge number of S-CSF/N-CSF is not necessary becausemajor electron correlation effects are already included at the KS-SCF level. The technique applied here could also partially speed up standard TD(A)-DFT. However, selection by diago-nal element only is not fully satisfactory because CSF which are slightly higher in energy than E maxmay contribute sig- niﬁcantly due to a large coupling element. Therefore, otherimportant CSF are selected by second-order-perturbation the- ory. A CSF ψ kis included in the S-CSF space if its cumulative perturbative energy contribution E(2) u=P−CSF/summationdisplay v\|Auv\|2 Eu−Ev(12) to all of the P-CSF is larger than a user-deﬁned threshold tp, where Erefers to the diagonal elements of the Amatrix (Eq. (11)) andAuvis their respective coupling element. If the sum in Eq. (12) is smaller than the threshold, it belongs to the N-CSF space and the second-order energy is summed up for all CSF v. This contribution which is usually small ( <0.2 eV) is added to the diagonal elements in the P-space conﬁgura- tions. The selected S-CSF are merged with those in the P- space and the resulting matrix Ais diagonalized. All roots up to the requested upper energy Emaxare computed. For a typical large molecule using an Emaxvalue of 9 eV , the diagonalized (S+P)-space is about 2 ×104–3×104fortpvalues of 10−4– 10−5(tpis given always in Hartree units in the following). The convergence properties of this approach have been care- fully checked for a few molecules and it is suggested to uset p=10−4as a reasonable default (see below for examples). Taking a few thousand roots from matrices of dimension 2 ×104is a routine task on modern computers and takes only CPU minutes. Note further that the simple form of perturba- tion theory used here works in the CSF but not in a symmetry- adapted state basis and hence breaks level degeneracies. Theeffect, however, is found to be very small for the default se- lection threshold and is <0.05 eV for EandTstates in, e.g., benzene and C 60, respectively. In order to save computation time and in particular core memory, the active space of MOs which are considered insTDA is truncated at a very early stage of the calculation by estimating conservatively (based on a scaled orbital energy gap criterion similar to the treatment in Ref. 71) which MOs altogether can be used to generate signiﬁcantly contributing CSF. IV. TECHNICAL DETAILS OF THE CALCULATIONS The DFT calculations were done with TURBOMOLE73,74 orORCA75,76while for sTDA an in-house written code was used. We employ standard integration grids (m4/m5 in TUR- BOMOLE or grid5 in ORCA ) and SCF convergence criteria. In the SCF step, the RI integral approximation77–79was used. If not stated otherwise, structures were fully optimized at the TPSS-D3(BJ)/def2-TZVP level.29,80–82In the excited state treatments, mostly SV(P)41or TZVP42AO basis sets are used which provide results close enough to the basis set limit con- sidering the inherent accuracy of the method. Because (except in one example) Rydberg states are not of interest here, the basis sets were not augmented with spa- tially diffuse functions. In the large molecules considered in this work, Rydberg states are “quenched” by Pauli-exchange repulsion with the valence electrons.83The TZVP basis con- tains semi-diffuse functions with Gaussian exponents in be- tween those of typical valence and Rydberg orbitals (e.g., for carbon about 0.15–0.2 (valence), 0.05 (Rydberg), and 0.095(smallest in TZVP)) which is sufﬁcient to account for the of- ten spatially extended character of excited states compared to the ground state. Note, that typical augmented basis sets (e.g.,aug-cc-pVTZ 84) already for medium-sized systems like C 60 are inapplicable due to near linear dependencies in the ba-sis resulting in a non-convergent SCF. In larger systems, thisproblem is more acute and similar to the case of solids where even TZVP can be problematic. 85 As standard, density functionals B3LYP,15,16PBE0,57 TPSS0,86PBE38,29and BHLYP27have been employed. All excitation energies refer to vertical singlet states for the op- timized ground state geometry. As default CSF selection threshold, a value of tp=10−4Ehand an Emaxvalue of 8 eV were used. The active occupied and virtual MOs are au-tomatically selected according to the chosen E maxvalue and their number is typically only about 20%–30% of the full MO space. Un-truncated TD(A)-DFT calculations with con-ventional integrals but with the same basis set and density functional for comparison were carried out using the escf module 87from TURBOMOLE . The one-electron transition moments are calculated with- out any further approximation from the sTDA wave func- tion and the exact moment integrals in the AO basis. ForCD-spectra, we recommend the mixed form for the rotatory strength R Mcomputed as RM=RVfL fV, (13) where RVis the gauge-origin independent velocity form for the rotatory strength and fLandfVare the dipole-length and dipole-velocity oscillator strengths, respectively. For the UV-spectra, the dipole-lengths form is used which generallyconverges faster towards the basis set limit than the dipole- velocity form. 6 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12244104-6 Stefan Grimme J. Chem. Phys. 138, 244104 (2013) The vibrationally broadened experimental UV or CD bands are simulated by summing oscillator- or rotatory strengths weighted Gaussian curves with a full width at 1/ e- height of σ=0.4 eV for each calculated electronic transition. V. RESULTS AND DISCUSSION A. Fit set and cross-validation The sTDA method has been ﬁrst tested on a stan- dard set of 26 valence singlet-singlet excitation energies of (mainly) organic molecules that originates from the data inRefs. 71,88–90(dubbed EXC26 set). It has also been used to determine the four empirical parameters αandβof the method. Moreover, for cross-validation 12 typical organicdyes are investigated (dubbed DYE12 set from Ref. 91). Table Ipresents computed excitation energies in com- parison to TD(A)-DFT and experimental data for EXC26. Figure 1shows the mean absolute deviation (MAD) as a func- TABLE I. Comparison of calculatedaand experimentalbsinglet-singlet ex- citation energies (in eV) for the EXC26 benchmark set. The states are orderedaccording to energy in two groups (more local states in entries 1–9 and delo-calized ππ* states in entries 10–26). Entry Molecule State sTDA TDA TD Exptl. 1O 3 11B2 1.89 2.02 1.93 2.0 2C H 2S11A2 2.09 2.26 2.23 2.24 3 Tetrazine 11B1u 2.09 2.36 2.27 2.29 4C 5 11/Pi1u 3.02 3.58 3.47 2.8 5C H 2O11A2 3.20 3.19 3.16 3.88 6 Uracil 21A/prime/prime4.70 4.78 4.76 4.8 7P 4 11T1 5.31 5.29 5.29 5.6 8 Adenine 11A/prime/prime5.02 5.11 5.10 5.12 9 Acetamide 11A/prime/prime5.73 5.67 5.65 5.69 10 Porphyrine 11B2u 2.10 2.35 2.30 2.0 11 Porphyrine 11B3u 2.30 2.54 2.46 2.4 12 Azulene 11B1 2.73 2.79 2.71 2.19 13 Perylene 21B2u 2.89 3.08 2.85 3.44 14 Coumarin153 21A 3.17 3.58 3.28 3.51 15 Anthracene 11B3u 3.37 3.52 3.29 3.7 16 t-azobenzene 11Bu 3.85 3.97 3.75 3.9 17 Naphthalene 11B2u 4.56 4.55 4.53 4.24 18 Naphthalene 11B3u 4.62 4.67 4.47 4.77 19 DMABN 11B1 4.48 4.59 4.50 4.3 20 DMABN 21A1 4.87 4.97 4.73 4.6 21 Octatetraene 11Bu 4.42 4.49 4.05 4.66 22 Hexatriene 11Bu 5.25 5.22 4.75 5.10 23 Adenine 21A/prime5.14 5.24 5.09 5.25 24 Norbornadiene 11A2 5.24 5.19 5.03 5.34 25 Benzene 11B1u 5.52 5.53 5.50 5.08 26 Benzene 11B2u 6.72 6.44 6.21 6.54 MDc−0.04 0.06 −0.08 . . . Max-mind1.22 1.47 1.39 . . . MADe0.23 0.22 0.28 . . . aPBE0 functional using TPSS-D3/def2-TZVP82optimized geometries and TZVP basis set. bFor experimental or best estimate values from high-level wave function calculations, see Refs. 71,88,89,a n d 90. cMean deviation. A negative value corresponds to a systematic underestimation of the excitation energy. dDifference between largest and smallest deviation. eMean absolute deviation.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0. 8ax0.150.20.250.30.350.40.450.5MAD / eVfit set (EXC26) DYE12 set, tp=10-4 DYE12 set, tp=10-5 FIG. 1. Dependence of mean absolute deviation (MAD) from reference val- ues on the Fock exchange mixing parameter axfor the two test sets and for the DYE12 set also for two different selection thresholds (with tp=10−4 being the default). tion of the Fock-exchange mixing parameter with the PBE hy- brids for EXC26. This ﬁgure also includes MAD data for the DYE12 benchmark which has been used in Ref. 91for test- ing various DFT and wave function based excited state meth- ods. For the dye molecules, always the lowest-lying optically bright transition has been considered, for further details seeRef. 91. For related TD-DFT work on organic dyes see, e.g., Refs. 92and93. From the statistical data, it is clear that the sTDA method with the PBE0 standard functional performs already very well and in fact better than one might have anticipated. For EXC26, the MAD is only 0.23 eV which is almost the sameas for standard TDA and even slightly better than the full TD- DFT treatment which yields a MAD of 0.28 eV . A similar MAD for TD-DFT/PBE0 in excitation energy benchmarkshas been reported recently. 94Compared to TD(A)-DFT also the error range (max-min value) is notably reduced in the simpliﬁed treatment. Moreover, in particular the performance for the larger systems is exceptional with most deviations be- low 0.2 eV and only a few “outliers” with errors on the or-der of 0.5 eV (e.g., azulene or perylene which are similar in TD(A)-DFT). The mean deviation (MD) is only signiﬁcant for the ﬁrst nine transitions ( −0.18 eV) which are more local (e.g., of nπ* type) than for the remaining ones for which a MD close to zero is obtained. In contrast, the TD-DFT treat- ment provides almost the same MD ( −0.08 eV) for all transi- tions. Compared to TDA the MD for sTDA is lower by about 0.1 eV mainly due to the results for the nπ* states. This is ex- pected because, as outlined above, the integral approximationis worse for locally excited states than for the ππ* transi- tions in the second group (entries 10–26 in Table I). In this respect, the sTDA performs nicely as theoretically expected.For the ππ* states, one notes in general no signiﬁcant errors from the monopole integrals which for some larger systems (e.g., porphyrine) even seem to correct (probably by their bet- ter distance dependence) the TD(A)-DFT excitation energies. In general, we ﬁnd better results for the ππ* states with TDA This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12244104-7 Stefan Grimme J. Chem. Phys. 138, 244104 (2013) 1234567 8 91 0 1 11 2 energy / eV020000400006000080000ε / cm-1 M-1TD-DFT (150 roots) TDA-DFT (150 roots) sTDA (Emax=11.5 eV) FIG. 2. Comparison of computed (PBE0/TZVP) UV-spectra for indigo at sTDA- (black) and TD(A)-DFT levels (gray). and sTDA compared to TD-DFT. Remarkably, for naphtha- lene the well-known wrong ordering of the two lowest Lb andLastates88,95by TD-DFT is corrected by TDA and this good property is conserved by sTDA. Another striking exam- ple in which sTDA is much better than TD(A)-DFT is the1/Pi1u state of C 5for which the error is reduced from about 0.7 to only 0.2 eV . This excellent performance of sTDA is not limited to the PBE0 functional with a relatively small value of ax=0.25 but also holds for a wide range of global hybrid function- als as can be seen in Fig. 1. For the EXC26 set, we ﬁnd the MAD minimum around ax=0.3–0.4 with a very respectable value of 0.2 eV . In the range of ax=0.1–0.6, also reasonable MAD values <0.3 eV are obtained and only functionals with ax>0.6 or ax=0 (semi-local GGA) cannot be recommended. These ﬁndings are similar for the set of 12 organic dyes which are investigated as a cross-validation. In Ref. 91,T D - DFT together with various density functionals were evaluated on this set and we cite here only MAD values for B3LYP (0.31 eV) and the best method (0.19 eV) represented bythe B2GP-PLYP double hybrid functional. In the recommend range of a xvalues rather accurate excitation energies are ob- tained by sTDA with MAD values always below 0.3 eV anda low minimum value of 0.21 eV for a x=0.5. The sTDA method with a medium-sized TZVP AO basis sets can com- pete in accuracy with much more sophisticated (and compu-tationally expensive) CC2 based methods. 91The DYE12 set was also used to investigate the dependence of the results on the perturbation selection threshold tp(for a test on a com- puted spectrum see below). As can be seen in Fig. 1, the effect is small (always <0.1 eV for individual energies) and negli- gible in practice. For larger axvalues, the excitation energies are typically overestimated (similar to standard TD(A)-DFT) and a smaller value of tpleads to bigger expansion spaces and hence smaller excitation energies, so that the slight reduction of the MAD seen in Fig. 1is understandable.As last examples in this section demonstrating the good accuracy of sTDA-DFT compared to standard TD(A)-DFT, two UV-spectra (indigo in Fig. 2and 9-(N-carbazolyl)- anthracene in Fig. 3) are considered. The number of roots in TD(A)-DFT was 150 and Emax=10/11.5 eV was used in sTDA. For indigo, the spectra overall exhibit a remarkablemutual agreement even for high energies in the vacuum-UV region and we only note a signiﬁcant deviation for an intense band at about 8 eV which is missing in TD-DFT but presentin both TDA spectra. In fact, over the entire energy range, the deviations between sTDA and TDA or TD-DFT are similarly small as between TDA and TD-DFT. The TD-DFT excited state calculation took about 28 000 CPU seconds while the sTDA treatment was ﬁnished in about 4 s! In the same way,a second example (9-(N-carbazolyl)-anthracene 96) with less symmetry and with another density functional (BHLYP) was investigated (Fig. 3) and likewise a reasonable mutual agree- ment between sTDA, TDA and TD-DFT is found. Here, we note very similar intensities of sTDA and TDA reﬂecting that they are based on the same wave function but that the sTDAexcitation energies are lower and more similar to those from TD-DFT. Besides these direct comparisons of sTDA/TDA spectra, we also present an analysis of the single excitation conﬁg- uration contributions in both methods for one example. The rather complicated case of indole with the four lowest dipole-allowed ππ* transitions (often termed L a/bandBa/b) is con- sidered (see Table II). The TDA single excitation expansion coefﬁcients as well as the oscillator strengths are sensitive to the off-diagonal elements of the TDA matrix (conﬁguration mixing) and can reﬂect errors of the sTDA integral approxi-mation for these highly multi-conﬁgurational states. As can be see from Table II, there is good mutual agreement not only for the excitation energies and oscillatorstrengths but also between the corresponding single excitation expansion coefﬁcients. This holds for the mixing between the This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12244104-8 Stefan Grimme J. Chem. Phys. 138, 244104 (2013) 234567 8 91 0 energy / eV050000100000150000200000250000 ε / cm-1 M-1TD-DFT (150 roots) TDA-DFT (150 roots) sTDA (Emax=10 eV) FIG. 3. Comparison of computed (BHLYP/TZVP) UV-spectra for 9-(N-carbazolyl)-anthracene at sTDA- (black) and TD(A)-DFT levels (gray). two lowest La/bstates but also for the relatively high-lying Ba/b states which for indole is not symmetry determined as in simi- lar aromatic systems (e.g., naphthalene). These results are en- couraging for an accurate description of transition moments in complicated cases as discussed below, e.g., for electroniccircular dichroism spectra. Before the discussion of larger molecules for which the present approach is really designed for, two small and poten- TABLE II. Analysis of TDA wave functions in terms of single excitation conﬁguration state function (CSF) coefﬁcients for the four lowest dipole- allowed transitions of indole with sTDA and TDA (PBE0/TZVP). Given are the three largest contributions as well as the excitation energies /Delta1E(in eV) and oscillator strengths f. The letters “H” and “L” denote highest occu- pied and lowest unoccupied orbitals, respectively, and indicate the orbitalsinvolved in the excitation (“from” →“to”). Coefﬁcient for /Delta1Ef CSF 1 CSF 2 CSF 3 State 1 H−1→LH →LH →L+1 TDA 4.99 0.051 0.60 −0.64 0.35 sTDA 4.95 0.028 0.69 −0.52 0.43 State 2 H→LH →L+1H −1→L TDA 5.05 0.077 0.64 0.51 0.50 sTDA 5.01 0.067 0.75 0.45 0.38 State 4 H−2→LH →L+1H →L+3 TDA 6.39 0.235 0.51 0.49 0.42sTDA 6.33 0.181 0.54 0.47 0.44 State 6 H→L+1H →L+3H −1→L TDA 6.71 0.647 0.54 0.39 0.39 sTDA 6.63 0.435 0.54 0.47 0.41tially problematic cases are considered in order to clearly out- line its limitations. B. Rydberg states of acetone The Rydberg states in small and medium-sized molecules are often atom-like in character and their electronic spatial extent varies considerably, particularly compared to valencestates. This property is not reﬂected in the monopole integrals and hence such states represent a very difﬁcult test for the present approach. In Table III,t h el o w e s t n→3spdRydberg states of acetone are given as an example including a compar- ison to experimental excitation energies, computed high-level(reference) oscillator strengths f, and values from a standard TD-DFT treatment. In comparison to the TD-DFT excita- tion energies, the corresponding sTDA values are lower by TABLE III. C alculated (aug-cc-pVTZ,84AO basis for DFT, B3LYP func- tional) and experimental excitation energies /Delta1E(in eV) for vertical singlet excited states of acetone. The oscillator strengths ( f) for sTDA-DFT are given in parentheses. /Delta1E State TD-DFT sTDA-DFT Exptl.afb 1B2(n→3s) 5.76 5.38 (0.017) 6.35 0.037 2A2(n→3px) 6 . 7 4 6 . 2 3... 7 . 3 6 ... 2A1(n→3py) 6.61 6.36 (0.009) 7.41 0.006 2B2(n→3pz) 6.89 6.06 (0.002) 7.45 0.002 3A1(n→3dyz) 7.40 6.73 (0.015) 7.8 0.047 3B2(n→3dx2−y2) 7.90 6.82 (0.034) 8.09 0.048 3A2(n→3dxz) 8 . 0 9 7 . 3 5... ... ... 4B2(n→3dz2) 7.22 6.95 (0.011) . . . 0.004 1B1(n→3dxy) 7.52 7.16 (0.002) 8.17 0.002 aReference 97. bMRDCI values from Ref. 6. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12244104-9 Stefan Grimme J. Chem. Phys. 138, 244104 (2013) 0.2–0.8 eV mainly due to the neglect of the atom-type valence-Rydberg orbital exchange integrals. Compared to the experimental excitation energies, the values are too low by 1–1.5 eV which is a large but not an extensive error and about half of it is attributed to the incorrect asymptotic behavior of the density functional used (i.e., the TD-DFT values are alsotoo low by about 0.2–0.6 eV). The ordering of the states as well as all oscillator strengths are, however, described well by the simpliﬁed method. Although sTDA-DFT is not the recom-mended treatment for such states and small systems in gen- eral (for which numerous very reliable alternatives are avail- able), this example shows that the integral approximations (the Coulomb integrals are non-vanishing here) work reason- ably well even for extended AO basis sets and spatially diffusestates. C. Charge transfer states As mentioned already in the Introduction, too low-lying CT states plague many TD-DFT computations and it is shown here by a numerical example how sTDA performs for a sim-ple model system. We would also like to clarify the CT problem in sTDA following the discussion in Refs. 7and 21. The lowest lying CT transition in the F 2–Be complex (2s2(Be)→σ∗(F2) excitation) is studied which was already used as an example.89Potential curves (excitation energies) for F 2−Be (T-shaped structure, rF−F=1.415 Å) as a func- tion of the Be–F 2center-of-mass distance Rare shown in Fig. 4. As reference curve, we use ωCT=IP/prime−EA/prime−R−1where IP/primeand EA/primecorrespond to the KS-DFT HOMO and LUMO energies of the Be atom and the F 2molecule, respectively. Note, that this well-known98asymptotic expression is shifted compared to the true limiting value because of the SIE andthe gap (integer discontinuity) problem. 20,25,40The functional used (PBE0) severely overestimates the HOMO energy and underestimates the LUMO energy (the true IP −EA difference is about 6 eV89instead of 2.4 eV as in PBE0). Hence, the ma- jor part of what is termed in the literature as the “TD-DFT CT 0 100 200 300 400 500 600 700 R(F2-Be) / Bohr0.811.21.41.61.822.22.4excitation energy / eVsTD-DFT TD-DFT IP’-EA’-1/R FIG. 4. Comparison of computed (PBE0/TZVP) charge-transfer excitation energies for the F 2−Be complex as a function of the intermolecular distance R. The solid gray line refers to an analytical single-particle reference.problem” is introduced already at the KS-SCF level. Because sTDA as TD(A)-DFT is based on the same single-particle en- ergies, CT transitions will similarly be underestimated whenglobal hybrids with small a xare used. This part of the problem can be solved by, e.g., the LC/RS technique. However, apart from this gap problem there is a second failure in standard TD(A)-DFT which is remedied by sTDA. In the CT state, the formed charges (here, Be⊕and F/circleminus 2) attract each other as 1/ Rat large distance and this is described by the−(ii\|aa) Coulomb term where iandacorrespond to the Be(2 s) andσ(F 2) MOs, respectively. In TD(A)-DFT, this in- tegral is scaled by axand if this reduction is not compensated for by the GGA response term in Eq. (2)(which is normally the case), the CT excitation energy is overestimated compared to the limiting value and not underestimated as due to the gap problem. This is clearly seen in Fig. 4where the TD- DFT curve (shown with asterisks) is always above the refer-ence (in gray). Because in sTDA the ( ii\|aa) term is not scaled but asymptotically approaches 1/ Ra better behavior for all distances is obtained. Hence, the sTDA is not just a simpli-ﬁcation that introduces additional errors but a new element in the theory that solves (a small) part of the CT state prob- lem. With “high Fock-exchange” or LC/RS functionals, thesTDA method should provide reasonable results for typical CT states. D. UV-spectrum of C 60 The C 60fullerene represents a nice ﬁrst example for a large delocalized system. It has been used already in Ref. 66 as a test case for TD-DFTB and the importance of includ-ing the correct coupling elements in the Amatrix has been noted. Because these are also subject to approximations in sTDA, we employ the calculation of its UV-spectrum as a test here. The comparison to the experimental spectrum (in n-hexane 99)i nF i g . 5shows that sTDA reproduces all major features of the measurement regarding relative intensity and band position fairly well (absolute intensities were not given in Ref. 99so that the experimental absorbance was scaled to roughly match the theoretical data). In the high energy re- gion (bands D and E), there is considerable inﬂuence of the amount of Fock-exchange on the excitation energy which iswell documented already for ππ states of smaller unsatu- rated hydrocarbons. 100 E. UV-spectrum of transition metal complexes The electronic spectra of transition metal complexes rep- resent a special challenge for theoretical methods because lo- calized d−d,d−π* (metal to ligand CT, MLCT) and lig- andππ* states can be present in the same molecule and due to their different electronic nature involve varying amounts of electron correlation effects. For the sTDA method such sys-tems and in particular the d−dtype transitions might be a worst case scenario and are therefore investigated for the two examples ferrocene and [Ru(bipy) 3]2+(Fig. 6and Fig. 7). The computed UV-spectra for [Ru(bipy) 3]2+show an ex- cellent agreement with experiment101for the ﬁve bands A–E This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12244104-10 Stefan Grimme J. Chem. Phys. 138, 244104 (2013) 200 250 300 350 400 450 500 550 600 650 700 wavelength / nm101001000100001e+05 ε / cm-1 M-1exptl. sTDA/PBE0 sTDA/PBE(0.5) ABCDE FIG. 5. Comparison of computed and experimental UV-spectra for C 60at the sTDA-DFT/TZVP level for two ﬂavors of the PBE hybrid functional.Oscillator strengths for forbidden transitions have been set to f=10 −4in order to simulate vibronic couplings. in position and intensity. The change of the GGA compo- nent from PBE to the meta-GGA TPSS80with the same ax (termed TPSS0 in Ref. 86) has only a very minor effect ex- cept for a slight blueshift of about 0.1 eV compared to PBE0. A less good agreement is obtained for ferrocene as shown in Fig.7. In the high-energy region, one notes a blueshift of the computed bands C–D compared to the experiment. However, this occurs for sTDA as well as TDA and both methods yield similar spectra between 150 and 220 nm. The reason maybe the neglect of solvent effects and/or inherent DFT errors (i.e., the computed spectra are rather sensitive to the value ofa x). For the dipole-forbidden, metal-centred d−dtransi- tions (bands A/B and A/prime/B/prime) sTDA yields a large error (about 1.5 eV blueshift) compared to experiment and TDA which is probably related to a bad description of Coulomb integrals in-volving the iron orbitals. Nevertheless, even in this worst-case scenario, the relative ordering of the bands as well as their in- tensity is described at least qualitatively correct. 200 250 300 350 400 450 500 550 wavelength / nm0200004000060000800001e+05 ε / cm-1 M-1exptl. sTDA/PBE0 sTDA/TPSS0 ABC DE FIG. 6. Comparison of computed and experimental UV-spectra for [Ru(bipy) 3]2+at sTDA-PBE0/TZVP and sTDA-TPSS0/TZVP levels. The calculated spectra have been shifted by −0.23 and −0.35 eV , respectively, to obtain agreement with the experimental position of band A.150 200 250 300 350 400 450 500 550 wavelength / nm101001000100001e+05 ε / cm-1 M-1exptl. TDA/PBE0 sTDA/PBE0 ABCDE A’ B’ FIG. 7. Comparison of computed and experimental102UV-spectra for fer- rocene at the sTDA and TDA levels (PBE0/def2-TZVP). The computed spec-tra are not shifted. Oscillator strengths for forbidden transitions have been set tof=5×10 −4in order to simulate vibronic couplings. F. CD-spectrum of [7]helicene Chiral molecules are important in nucleic acid, peptide, and sugar chemistry and the determination of the absolute conﬁguration of, e.g., natural products is an important taskfor quantum chemistry. The different interaction of the left- and right-handed enantiomers with circularly polarized light, respectively, is a very fundamental process that can be stud-ied by electronic CD measurements (for overviews see, e.g., Refs. 103,104, and 105). The CD-spectra of helicenes with four to 12 rings have been studied using semi-local TD-DFT with some success in Ref. 106. We study here the CD-spectrum of the [7]he- licene which was already a test case for the TD(A)-B2PLYP 89 method. The computation of CD-spectra is challenging be- cause the relative orientation of two (electric and mag- netic) dipole transition moments has to be calculated ac-curately and canceling effects (because the moments are signed quantities) can be present. This and another CD exam- ple presented below were chosen because this spectroscopy presents a sensitive test for the quality of the sTDA wave function. The experimental CD-spectrum of (P)-[7]helicene 107 shown in Fig. 8shows four distinct bands (A–D). As can be clearly seen from the comparison of the theoretical andexperimental spectra, the sTDA approach provides good accuracy over the entire wavelength range and only the inten- sity of band B is not correct. Even in the high-energy region,the computed excitation energies are accurate to ±0.2 eV and overall, without an energy shift the computed spectrum for this non-polar system matches the experiment fairly well.The accuracy is overall similar to that of a full TD-B2PLYP treatment and better (mainly because of the larger a xused here) than for TD-B3LYP.89Figure 8also shows that the choice of the CSF perturbation selection threshold has only a minor impact on the CD-spectrum and almost complete convergence is reached for tp=10−5. The intensity for band B, however, seems to be sensitive to details of the treatment as can be seen from the wrong sign in the tp=10−4calculation. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12244104-11 Stefan Grimme J. Chem. Phys. 138, 244104 (2013) 200 250 300 350 400 450 wavelength / nm-500-400-300-200-1000100200300Δε / cm-1 M-1 sTDA, tp=10-4 sTDA, tp=10-5 sTDA, tp=10-6 exptl.A B CD FIG. 8. Comparison of computed and experimental CD-spectra for (P)- [7]helicene at the sTDA-PBE38/TZVP level for three conﬁguration selectionthresholds. G. CD-spectra of (Ala) npolypeptides Electronic CD is an important tool for the characteriza- tion of the secondary structure of proteins. For an overview of recent theoretical work which usually refers to a semi- empirical exciton-coupling model, see Refs. 108and109. In these calculations, the conductor-like screening con- tinuum solvation model (COSMO110) with a dielectric con- stant of 78 for water has been applied in the SCF step whichprovides a more realistic electronic structure of these highly polar molecules (e.g., the ground state permanent electric dipole moment of (Ala) 10is 42 Debye). All the polypeptide model structures were optimized using the MMX force ﬁeld as implemented in the PCM software.111 We ﬁrst investigate the sTDA in comparison to TD(A)- DFT for exactly the same system and SCF input. Because of the high computational cost of TD(A)-DFT, a small polyala- nine peptide with 10 residues (103 atoms) and only the low-est 40 roots were considered. The three simulated CD-spectra are shown in Fig. 9(a). The mutual agreement of the theo- retical spectra in Fig. 9(a) is very satisfactory regarding the position of the bands and we only note a signiﬁcant differ- ence in absolute intensity (mainly for the two bands between160 and 170 nm) with sTDA/TDA compared to TD-DFT. As noted above, the sTDA excitation energies are closer to those from TD-DFT while the intensities resemble more the TDA-DFT ones. In any case, the very good performance of sTDA also for relatively high-lying states of inherently small pep- tide chromophores seems very promising and hence largerpolyalanines with perfect α-helix secondary structure were investigated. Convergence studies show that already around 30–40 residues the shape of the computed spectra does not change signiﬁcantly anymore and may be compared to experimen- tal data. However, these are normally obtained for muchlarger proteins which are only predominantly composed of α-helical structures but also can contain loops and varying amino acids. Although the spectra of poly- γ-methyl gluta- mate in hexaﬂuoro-propanol 112which is used here for com- parison and, e.g., myoglobin113(in water) look rather simi-150 160 170 1 80 190 200 210 220 230 wavelength / nm-400-300-200-1000100200300400500Δε / cm-1 M-1 TD-DFT TDA-DFT sTDA-DFT 140 150 160 170 1 80 190 200 210 220 230 240 250 260 270 wavelength / nm-100102030Δε per resid ue / cm-1 M-1exptl. sTDA/SV(P) sTDA/TZVP A BC DE (a) (b) FIG. 9. (a) Comparison of computed CD-spectra for (Ala) 10at TD(A)-DFT and sTDA-DFT levels using the BHLYP functional and the SV(P) AO basis set. (b) Comparison of experimental and computed CD-spectra (same level as above and in addition with the TZVP basis set) for a realistic α-helix (the (Ala) 40structure is shown in the inset, for details see text). The intensities are normalized to the number of residues as usual and the sTDA excitation energies have been shifted by −1e V . lar, one should keep these additional (structural) problems in mind. We can thus expect a qualitative agreement between theory and experiment at best and note, that a full treatment of myoglobin with about 2400 atoms (excluding water) seems possible with sTDA and such complete proteins will be in-vestigated in future work in our laboratory. All spectra were simulated with a doubled width of the electronic transitions in order to account for solvent effects as well as broadeningdue to conformational ﬂexibility. In the sTDA calculation for (Ala) 40with the TZVP basis set, an Emaxvalue of 10 eV and tp=3×10−3were chosen leading to about 2 ×104selected CSF. Nevertheless, the agreement between theory (computed for (Ala) 40with 403 atoms) and experiment as documented in Fig.9(b) can be considered as rather good. Note that no em- pirical adjustments (except for an energy shift) were made, that spectra up to very high energies ( Emax=10 eV) are com- pared, and ﬁnally that also the computed absolute intensities match the experiment nicely. The bands A–C are very well described by sTDA with both AO basis sets. Not unexpect- edly, at higher energies (bands D and E) signiﬁcant differ- ences in the two calculations are observed which indicates an This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12244104-12 Stefan Grimme J. Chem. Phys. 138, 244104 (2013) 200 250 300 350 400 450 500 wavelength / nm050000100000150000200000250000 ε / cm-1 M-1exptl. sTDA AB CD FIG. 10. Comparison of computed and experimental UV-spectra for Zn2+ 3(LLL) at the sTDA-DFT level (BHLYP/TZVP for TPSS-D3(BJ)- COSMO(CH 2Cl2)/SV(P) optimized geometries). The COSMO model with a dielectric constant of 20, an energy shift of 0.3 eV , and a bandwidth of σ =0.6 eV was applied. The theoretical intensity is scaled by a factor of 0.5 and in addition individual transitions are shown as sticks. The experimen- tal intensity below about 230 nm is unreliable because of absorption of the solvent (1:1 mixture of CH 2Cl2/CH 3CN). increasing diffuse character of the states even in solution. Ten- tatively, part of the discrepancies in this region of the spec-trum are assigned to a remaining SIE in the BHLYP func- tional and it is expected that LC/RS functionals will improve the description. H. UV-spectrum of a supramolecular system Finally, another realistic problem will be investigated where UV-spectra are used for identiﬁcation of reaction prod-ucts in self-assembly reactions. The trinuclear triple-stranded zinc(II) helicate of a binaphthol-based tris(bipyridine) ligand shown as inset in Fig. 10is used as example 114(for similar systems, see Ref. 115). Approaching this structure consist- ing of unsaturated linkers/ligands (L) and Zn2+ions (dubbed Zn2+ 3(LLL) in the following) by molecular fragmentation techniques seems hopeless due to a small but non-negligible electronic communication between the units. The system con-sists of 483 atoms (2130 electrons, 6879 basis functions) and the reported calculation (including the SCF step) could be per- formed in about one day on a conventional laptop computer(taking about 1 h for the sTDA calculation). The experimental spectrum showing four bands A–D is overall nicely reproduced by the sTDA method. We notesome larger error only for the high-energy band D and that the deviations for the other resolved bands and shoulders are reasonably small ( <0.5 eV). Relative intensities match the experimental ones well enough to allow identiﬁcation of the complex. Note that the total charge of +6 of the complex is rather large and that solvation effects beyond the COSMOcontinuum model used in the SCF also might play a role for the remaining discrepancies between theory and experiment. In passing, it is noted that the low-energy region is dominated by a few intense transitions (e.g., with a f-value of up to 3.6 for the third state) as normally observed for medium-sizeddye molecules while the high-energy part of the spectrum shows hundreds of excitations which add up to bands of similar intensity. In the range up to 6 eV , we ﬁnd in total 1170excited states in the sTDA treatment. Because of a residual SIE in the BHLYP functional, many of these are still artiﬁcial and we tried to estimate their number by an analogous sTDAcalculation based on HF input data as suggested by an anony- mous reviewer. According to this test, a still sizeable number of about 210 states is found in the considered energy range af-ter applying a redshift of 1.4 eV (which brings the sTDA/HF lowest excitation energy in agreement to the sTDA/BHLYP value in order to make the calculations comparable). If one takes the tendency of HF to overestimate higher excitation energies into account (i.e., the computed intensity of bandD is too small for sTDA/HF), the “true” number of physical states in this case is probably about 300–400. 116The SIE issue and the related question on the right density of statesin such systems will be approached by coupling sTDA with LC/RC-functionals and discussed in a forthcoming paper. VI. CONCLUSIONS Molecules or supramolecular aggregates with 500–1000 atoms are often difﬁcult to characterize chemically and spec- troscopic techniques are of utmost importance for an un-derstanding of their structure and eventually their function. Although electronic spectroscopy certainly cannot provide high-resolution data for such systems, the information fromUV and in particular CD-spectra can be useful for struc- tural assignments. It is clear that these experimental tech- niques should be supplied by adequate quantum chemical“ﬁrst-principles” models for understanding the data and elu- cidating structure-property relationships. As outlined in the Introduction, the application of standard TD-DFT in this areais strongly limited by the necessary computation times when an entire spectrum in the commonly accessible energy range is of interest. The extremely high density of electronic statesalready at common energies (e.g., 5 eV or 250 nm) has rarely been considered in the literature. This problem be- comes catastrophic when “cheap” GGA density functionals are applied in TD-DFT which produce a vast amount of ar- tiﬁcial states due to their SIE/gap and concomitant CT stateproblem. The present approach solves the above mentioned com- putational problems by introducing two basic approximationsto the standard TDA-DFT treatment. It seems clear that in- clusion of non-local Fock exchange is essential and therefore the approach is built on that basic consideration (although it isapplicable in principle also with a GGA). The method termed sTDA employs atom-centered Löwdin-monopole based two- electron repulsion integrals with the asymptotically correct1/Rbehavior. Together with well-established perturbative conﬁguration selection approaches, speed-ups by two to three orders of magnitude in the excited state part at only minor lossof accuracy are obtained. In fact, on average for two stan- dard sets of benchmark excitation energies, sTDA performs even slightly better than TD-DFT with the same two common density functionals (B3LYP and PBE0). The obtained mean absolute deviation of 0.2–0.3 eV is relatively low and a few This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.11.242.100 On: Wed, 07 Jan 2015 16:36:12244104-13 Stefan Grimme J. Chem. Phys. 138, 244104 (2013) “outliers” with about 0.5–1 eV error are acceptable in typical applications (and similar to standard TD-DFT). The method is formulated generally for any conventional global hybriddensity functional with given Fock-exchange mixing param- etera xbut future work will also consider it in the framework of long-range corrected/range-separated functionals. This willmore or less solve the SIE/gap problem in excited state DFT calculations for large systems as sTDA employs asymptoti- cally correct response integrals. The calculation of accurateCT state energies for large charge separations is of utmost importance for applications in technologically important ar- eas such as organic photo-voltaics or semi-conductors. The few spectra shown here as obtained with various conventional hybrid functionals and a range of differentFock-exchange mixings already demonstrate the generality, accuracy, and feasibility of the method. For example, the computation of a polypeptide CD-spectrum for 400 atomsruns for less than 1 h on a standard laptop computer. With the proposed simpliﬁcations, DFT calculations can be performed in a size regime where currently semi-empirical orbital andexciton-coupling models are used. The improvement accessi- ble by using DFT as starting point opens up new and bright possibilities for the computation and interpretation of elec-tronic spectra for large supra- and bio-molecular structures. 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1.5131689.pdf	Appl. Phys. Lett. 116, 062401 (2020); https://doi.org/10.1063/1.5131689 116, 062401 © 2020 Author(s).Sub-micrometer near-field focusing of spin waves in ultrathin YIG films Cite as: Appl. Phys. Lett. 116, 062401 (2020); https://doi.org/10.1063/1.5131689 Submitted: 15 October 2019 . Accepted: 23 December 2019 . Published Online: 10 February 2020 B. Divinskiy , N. Thiery , L. Vila , O. Klein , N. Beaulieu , J. Ben Youssef , S. O. Demokritov , and V. E. Demidov COLLECTIONS This paper was selected as Featured Sub-micrometer near-field focusing of spin waves in ultrathin YIG films Cite as: Appl. Phys. Lett. 116, 062401 (2020); doi: 10.1063/1.5131689 Submitted: 15 October 2019 .Accepted: 23 December 2019 . Published Online: 10 February 2020 B.Divinskiy,1,a) N.Thiery,2L.Vila,2O.Klein,2 N.Beaulieu,3J.Ben Youssef,3 S. O. Demokritov,1 and V. E. Demidov1 AFFILIATIONS 1Institute for Applied Physics and Center for NanoTechnology, University of Muenster, 48149 Muenster, Germany 2CNRS, CEA, Grenoble INP, University Grenoble Alpes, IRIG-SPINTEC, F-38000 Grenoble, France 3LabSTICC, CNRS, Universit /C19e de Bretagne Occidentale, 29238 Brest, France a)Author to whom correspondence should be addressed :b_divi01@uni-muenster.de ABSTRACT We experimentally demonstrate tight focusing of a spin wave beam excited in extended nanometer-thick ﬁlms of yttrium iron garnet by a simple microscopic antenna functioning as a single-slit near-ﬁeld lens. We show that the focal distance and the minimum transverse width of the focal spot can be controlled in a broad range by varying the frequency/wavelength of spin waves and the antenna geometry. The exper- imental data are in good agreement with the results of numerical simulations. Our ﬁndings provide a simple solution for the implementationof magnonic nanodevices requiring a local concentration of the spin-wave energy. Published under license by AIP Publishing. https://doi.org/10.1063/1.5131689 The advent of high-quality nanometer-thick ﬁlms of magnetic insulator yttrium iron garnet (YIG) 1–3essentially expanded horizons for the ﬁeld of magnonics4–6utilizing spin waves for the transmission and processing of information on the nanoscale. Thanks to the small thickness and ultra-low magnetic damping, these ﬁlms enable the implementation of magnonic devices with nanometer dimensions,7,8 where the spin-wave losses are by several orders of magnitude smallercompared to those in devices based on metallic ferromagnetic ﬁlms. 9 The large propagation length of spin waves in YIG is particularly bene-ﬁcial for the implementation of spatial manipulation of spin-wavebeams in the real space. It is now well established that the propagation of spin waves can be controlled by using approaches similar to those used in optics. 10–16 However, in contrast to light waves, the wavelength of spin waves canbe as small as few tens of nanometers, 17,18which allows one to imple- ment efﬁcient wave manipulation on the nanoscale. In recent years,particular attention was given to the possibility to controllably focus propagating spin waves. 10–14,19,20Such focusing allows one to concen- trate the spin-wave energy in a small spatial area, which is important, for example, for the implementation of the efﬁcient local detection of spin-wave signals. Provided that the position of the focal point is con-trollable by the spin-wave frequency, the focusing can also be utilized for the implementation of the frequency multiplexing. 21Additionally, the strong local concentration of the spin-wave energy can be used tostimulate nonlinear phenomena, for example, the second-harmonic generation.22 Efﬁcient spin-wave focusing can be achieved relatively easily in conﬁned geometries, such as stripe waveguides,23where it is governed by the interference of multiple co-propagating quantized spin-wavemodes. 24In the case of extended magnetic ﬁlms, the implementation of focusing appears to be less straightforward. In recent years, several approaches have been suggested utilizing a spatial variation of theeffective spin-wave refraction index, 14,16refraction of spin waves at the modulation of the ﬁlm thickness11or the temperature,12diffraction from a Fresnel zone plate,13and excitation of spin-wave beams by curved transducers.19,20All these approaches are rather complex in terms of practical implementation, particularly on the nanoscale. A much simpler approach known in optics25–27relies on the utilization of Fresnel diffraction patterns appearing in the near-ﬁeld region of asingle slit, where the Fresnel number F¼a 2/(kd)i so ft h eo r d e ro r larger than 1 (Ref. 28). Here, ais the length of the slit, kis the wave- length, and dis the distance from the slit to the observation point. As was experimentally shown for light waves26and surface plasmon polaritons,27such a slit functions as an efﬁcient near-ﬁeld lens enabling tight focusing of the incident wave. In this work, we demonstrate experimentally that the principles of near-ﬁeld diffractive focusing are also applicable for spin waves inin-plane magnetized magnetic ﬁlms, which, in contrast to light, exhibit Appl. Phys. Lett. 116, 062401 (2020); doi: 10.1063/1.5131689 116, 062401-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplanisotropic dispersion. By using a 2 lm long spin-wave antenna, which is equivalent to a single one-dimensional slit,29we achieve focusing of the excited spin waves into an area with the transverse width below 700 nm. We show that, in agreement with the theory developed forlight waves, the focal distance increases with the decrease in the wave-length of the spin wave, which allows electronic control of the spin- wave focusing by the frequency and/or the static magnetic ﬁeld. We also perform micromagnetic simulations, which show excellent agree-ment with the experimental data and allow us to analyze the effects ofthe antenna geometry on the focusing characteristics. Figure 1(a) shows the schematic of the experiment. The test devi- ces are based on a 56 nm thick YIG ﬁlm grown by liquid phase epitaxy on a gadolinium gallium substrate. The independently determined sat-uration magnetization of the ﬁlm is 4 pM¼1.78 kG, and the Gilbert damping parameter is a¼1.4/C210 /C04.T h eY I Gﬁ l mi sm a g n e t i z e dt o saturation by the static magnetic ﬁeld Happlied in the ﬁlm plane. The excitation of spin waves is performed by using a lithographically deﬁned 2 lm long, 300 nm wide, and 7 nm thick spin-wave antenna contacted by 2 lm wide and 30 nm thick Au microstrip lines.30The microwave-frequency electrical current IMW ﬂowing through the antenna creates a dynamic magnetic ﬁeld h, which couples to the mag- netization in the YIG ﬁlm and excites spin waves propagating away from the antenna. Figure 1(b) shows the normalized spatial distribu- tion of the amplitude of the dynamic ﬁeld created by the antenna inthe YIG ﬁlm calculated by using COMSOL Multiphysics simulationsoftware ( https://www.comsol.com/comsol-multiphysics ). As seen from these data, due to the large difference in the width of the antenna and the microstrip lines, the amplitude of the dynamic ﬁeld under-neath the antenna is by an order of magnitude larger compared to thatunderneath the lines. Therefore, the efﬁcient spin wave excitation isonly possible in the 2 lm long antenna region. This disbalance is fur- ther enhanced for spin waves with wavelengths comparable or smaller than the width of the microstrip lines due to the reduced coupling efﬁ- ciency of the inductive mechanism. 9 Spatially resolved detection of excited spin waves is performed by micro-focus Brillouin light scattering (BLS) spectroscopy.9The prob- ing light with a wavelength of 473 nm and a power of 0.1 mW pro- duced by a single-frequency laser is focused into a diffraction-limited spot on the surface of the YIG ﬁlm. By analyzing the spectrum of light inelastically scattered from magnetic excitations, we obtain a signal— the BLS intensity—proportional to the intensity of spin waves at the location of the probing spot. By scanning the spot over the sample sur-face, we obtain spatial maps of the spin-wave intensity. Additionally, by using the interference of the scattered light with the reference light modulated at the excitation frequency, 9we record spatial maps of the spin-wave phase. Figures 2(a) and2(b)show the representative intensity and phase maps recorded at H¼500 Oe by applying excitation current with the frequency f¼3.8 GHz. The power of the applied signal is 10 lW, which is proven to provide a linear regime of excitation and propaga- tion of spin waves. In agreement with the above discussion, spin waves are only radiated from the region of the narrow antenna. More impor- tantly, the radiated beam exhibits signiﬁcant narrowing and an increase in the intensity at the distance d¼3.6lm from the center of the antenna, clearly indicating the focusing of the excited spin waves. Qualitatively similar behaviors were also observed for different excita- tion frequencies in the range f¼3.2–4 GHz, although the distance d was found to change strongly with the variation of f. From the phase-resolved measurements [ Fig. 2(b) ], we obtain the wavelength k¼0.6lm of spin waves at f¼3.8 GHz. By repeating these measurements for different excitation frequencies, we obtain the spin-wave dispersion curve [ Fig. 2(c) ], which allows us to relate the excitation frequency to the spin-wave wavelength. Note that the exper-imental data [symbols in Fig. 2(c) ] are in perfect agreement with the results of calculations [curve in Fig. 2(c) ] based on the analytical theory (Ref. 31). On one side, the observed focusing is counterintuitive. Indeed, the excitation of waves by a ﬁnite-length straight antenna, as used in our experiment, is equivalent to a diffraction of a wave with an inﬁniteplane front from a slit, 29which is known to result in a formation of a divergent beam. On the other side, it is also known25–27that, before the beam starts to diverge, a complex focusing-like diffraction pattern is formed in the near ﬁeld just behind a slit. In recent years, it was shown theoretically25and proven experimentally26,27that these near-ﬁeld effects can be used for efﬁcient focusing of waves of different nature. Due to the insufﬁcient spatial resolution, the ﬁne details of the near-ﬁeld spin-wave pattern cannot be seen in the experimental maps[Fig. 2(b) ]. Therefore, we perform micromagnetic simulations using the software package MuMax3 (Ref. 32). We consider a magnetic ﬁlm with dimensions of 20 lm/C210lm/C20.05lm discretized into 10 nm /C210 nm /C250 nm cells. The standard for YIG exchange con- stant of 3.66 pJ/m is used. The spin waves are excited by applying a sinusoidal dynamic magnetic ﬁeld with an amplitude of 1 Oe, which is close to the estimated experimental value of 3 Oe. The spatial distri- bution of the excitation ﬁeld is taken from COMSOL simulations [Fig. 1(b) ]. The angle of the excited magnetization precession is of the order of 0.1 /C14. FIG. 1. (a) Schematic of the experiment. (b) Normalized calculated spatial distribu- tion of the amplitude of the dynamic magnetic ﬁeld created by the antenna in theYIG ﬁlm.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 062401 (2020); doi: 10.1063/1.5131689 116, 062401-2 Published under license by AIP PublishingThe results of simulations for the excitation frequency f¼3.8 GHz and H¼500 Oe are shown in Fig. 3(a) . The simulated map of the normalized spin-wave intensity hMx2i/hMx2maxiexhibits a narrowing of the excited beam and concentration of the spin-waveenergy in exactly the same way, as it is observed in the experiment [compare with Fig. 2(a) ]. Simultaneously, it shows a ﬁne structure, which is reminiscent of that obtained for light diffracted on a slit. 26To further conﬁrm the analogy, we perform simulations for the case ofplane spin waves diffracting from a slit formed by two 300 nm wide rectangular regions with increased magnetic damping ( a¼1) with a 2lm long gap between them [ Fig. 3(b) ]. The close similarity between the obtained patterns shows that the experimental results obtained forspin-wave excitation by the antenna are equally applicable for spin- wave focusing by a slit lens. We emphasize that such a lens can be eas- ily implemented in practice, for example, by using ion implantationinto nanometer-thick YIG ﬁlms. To additionally address the effects of the anisotropy of the spin- wave dispersion, we show in Fig. 3(c) an intensity map calculated forspin waves in an out-of-plane magnetized ﬁlm characterized by an iso- tropic dispersion. 33Comparison of Fig. 3(c) with Fig. 3(a) shows that the anisotropy makes the near-ﬁeld focusing even better pronounced due to the existence of the preferential direction of the energy ﬂowcharacterized by the angle h¼17 /C14[seeFig. 3(a) ] calculated according to Ref. 34. Figure 4 shows the quantitative comparison of the experimental results with those obtained from simulations. In Fig. 4(a) , we plot one- dimensional sections of the experimental [ Fig. 2(a) ]a n dc a l c u l a t e d [Fig. 3(a) ] intensity maps along the axis of the spin-wave beam at FIG. 2. Representative spatial maps of the intensity (a) and phase (b) of radiated spin waves recorded by BLS at H¼500 Oe and f¼3.8 GHz. (c) Measured (symbols) and calculated (solid curve) dispersion curve of spin waves. FIG. 3. (a) Calculated intensity map of spin waves radiated by the antenna. The schematic of the antenna and the connecting microwave lines is superimposed on the map. (b) Calculated intensity map of spin-wave diffraction from a one-dimensional slit. Superimposed rectangles mark the regions with an increaseddamping. (c) Similar to (a) calculated for the case of isotropic spin-wave dispersion. Calculations were performed for H¼500 Oe and f¼3.8 GHz.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 062401 (2020); doi: 10.1063/1.5131689 116, 062401-3 Published under license by AIP Publishingz¼0. Both datasets show perfect agreement. In both cases, the inten- sity increases during the ﬁrst 3 lm of propagation and reaches a maxi- mum at the distance y/C253.6lm, which can be treated as a focal distance. Transverse sections of the intensity maps at this distance[Fig. 4(b) ] also exhibit very similar narrowing of the beam to 670620 nm. As seen from Figs. 4(c) and 4(d), a good agreement between the experimental data and the result of simulations isobserved in a broad range of spin-wave wavelengths. We note that, in contrast to the far-ﬁeld focusing, in our case, the focal distance depends strongly on the wavelength [ Fig. 4(c) ]. This dependence is in agreement with the theory of the near-ﬁeld diffractive focusing of light, 25which predicts that the focal distance should increase with the decrease in the wavelength. This dependence can beparticularly important for magnonic applications since it allows one tofocus spin waves with different frequencies at different spatial loca- tions. Alternatively, the focal position can be controlled by the varia- tion of the static magnetic ﬁeld at the ﬁxed spin-wave frequency. As seen from Fig. 4(d) , the transverse width wof the spin-wave beam at the focal position exhibits a monotonous decrease with thedecrease in the wavelength k. Therefore, similar to the far-ﬁeld focus- ing, one can obtain a stronger concentration of the energy for spin waves with smaller wavelengths. Note, however, that the ratio w/k increases at smaller k, making the focusing of short-wavelength spin w a v e sl e s se f ﬁ c i e n t . To study the effects of the antenna geometry on the focusing efﬁ- ciency, we perform micromagnetic simulations for different lengths of the spin-wave antenna aat the ﬁxed value of the wavelengthk¼0.6lm .T h er e s u l t so ft h e s es i m u l a t i o n ss h o w( Fig. 5 ) that the focal-point width wgenerally reduces with decreasing a.T h e r e f o r e , more tight focusing can be achieved by using smaller antennae or slit lenses. Additionally, as can be seen from Fig. 5 , the reduction of the antenna length aresults in a decrease in the focal distance, which allows one to achieve stronger focusing in devices with smaller dimensions. In conclusion, we have experimentally demonstrated a simple and efﬁcient approach to the focusing of spin waves on the sub-micrometer scale. The obtained results are not only applicable to the excitation- stage focusing but can also be used for the implementation of near-ﬁeld lenses for plane spin waves. Our ﬁndings signiﬁcantly simplify theFIG. 4. (a) One-dimensional sections of the experimental (symbols) and calculated (solid curve) intensity maps along the axis of the spin-wave beam at z¼0. (b) Transverse sections of the experimental (symbols) and calculated (solid curve) intensity maps at the y-position corresponding to the maximum intensity. (c) Dependence of the focal dis- tance on the wavelength. (d) Dependence of the width of the spin-wave beam at the focal position on the wavelength. Curves in (c) and (d) are guides to the eye. FIG. 5. Dependences of the beam width at the focal position (squares) and of the focal distance (diamonds) on the length of the antenna calculated for spin waveswith the wavelength of 0.6 lm. Dashed curves are guides to the eye.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 062401 (2020); doi: 10.1063/1.5131689 116, 062401-4 Published under license by AIP Publishingimplementation of nano-magnonic devices utilizing spin-wave focus- ing, which is critically important for their real-world applications. We acknowledge the support from Deutsche Forschungsgemeinschaft (Project No. 423113162) and the French ANR Maestro (No.18-CE24-0021). REFERENCES 1Y. Sun, Y. Y. Song, H. Chang, M. Kabatek, M. Jantz, W. Schneider, M. Wu, H. Schultheiss, and A. Hoffmann, Appl. Phys. Lett. 101, 152405 (2012). 2O. d’Allivy Kelly, A. Anane, R. 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Rep. 6, 22367 (2016); H. S. K €orner, J. Stigloher, and C. H. Back, Phys. Rev. B 96, 100401(R) (2017). 31B. A. Kalinikos, IEE Proc. H 127, 4 (1980). 32A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, AIP Adv. 4, 107133 (2014). 33The calculations were performed at f ¼3.8 GHz. The static magnetic ﬁeld was increased to H ¼2900 Oe to obtain the same spin-wave wavelength of 0.6 lm, as in the case of the in-plane magnetized ﬁlm. 34V. E. Demidov, S. O. Demokritov, D. Birt, B. O’Gorman, M. Tsoi, and X. Li, Phys. Rev. B 80, 014429 (2009).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 062401 (2020); doi: 10.1063/1.5131689 116, 062401-5 Published under license by AIP Publishing
4.0000070.pdf	Struct. Dyn. 8, 024101 (2021); https://doi.org/10.1063/4.0000070 8, 024101 © 2021 Author(s).An assessment of different electronic structure approaches for modeling time- resolved x-ray absorption spectroscopy Cite as: Struct. Dyn. 8, 024101 (2021); https://doi.org/10.1063/4.0000070 Submitted: 14 December 2020 . Accepted: 11 February 2021 . Published Online: 12 March 2021 Shota Tsuru , Marta L. Vidal , Mátyás Pápai , Anna I. Krylov , Klaus B. Møller , and Sonia Coriani COLLECTIONS Paper published as part of the special topic on Theory of Ultrafast X-ray and Electron Phenomena ARTICLES YOU MAY BE INTERESTED IN Origin of core-to-core x-ray emission spectroscopy sensitivity to structural dynamics Structural Dynamics 7, 044102 (2020); https://doi.org/10.1063/4.0000022 Electronic circular dichroism spectra using the algebraic diagrammatic construction schemes of the polarization propagator up to third order The Journal of Chemical Physics 154, 064107 (2021); https://doi.org/10.1063/5.0038315 From orbitals to observables and back The Journal of Chemical Physics 153, 080901 (2020); https://doi.org/10.1063/5.0018597An assessment of different electronic structure approaches for modeling time-resolved x-ray absorption spectroscopy Cite as: Struct. Dyn. 8, 024101 (2021); doi: 10.1063/4.0000070 Submitted: 14 December 2020 .Accepted: 11 February 2021 . Published Online: 12 March 2021 Shota Tsuru,1,a) Marta L. Vidal,1 M/C19aty/C19asP/C19apai,1,b) Anna I. Krylov,2 Klaus B. Møller,1 and Sonia Coriani1,c) AFFILIATIONS 1DTU Chemistry, Technical University of Denmark, Kemitorvet Building 207, DK-2800 Kgs. Lyngby, Denmark 2Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA Note: This paper is part of the special issue on Theory of Ultrafast X-ray and Electron Phenomena. a)Present address: Arbeitsgruppe Quantenchemie, Ruhr-Universit €at Bochum, D-44780 Bochum, Germany. Electronic mail: Shota.Tsuru@ruhr-uni-bochum.de b)Present address: Wigner Research Center for Physics, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary. c)Author to whom correspondence should be addressed: soco@kemi.dtu.dk ABSTRACT We assess the performance of different protocols for simulating excited-state x-ray absorption spectra. We consider three different protocols based on equation-of-motion coupled-cluster singles and doubles, two of them combined with the maximum overlap method. The three pro- tocols differ in the choice of a reference conﬁguration used to compute target states. Maximum-overlap-method time-dependent density functional theory is also considered. The performance of the different approaches is illustrated using uracil, thymine, and acetylacetone asbenchmark systems. The results provide guidance for selecting an electronic structure method for modeling time-resolved x-ray absorptionspectroscopy. VC2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http:// creativecommons.org/licenses/by/4.0/ ).https://doi.org/10.1063/4.0000070 I. INTRODUCTION Since the pioneering study by Zewail’s group in the mid-1980s,1 ultrafast dynamics has been an active area of experimental research. Advances in light sources provide new means for probing dynamicsby utilizing core-level transitions. X-ray free electron lasers (XFELs)and instruments based on high-harmonic generation (HHG) enablespectroscopic measurements on the femtosecond 2–4and attosecond5–8 time scales. Methods for investigating femtosecond dynamics can beclassiﬁed into two categories: ( i) methods that track the electronic structure as parametrically dependent on the nuclear dynamics, suchas time-resolved photoelectron spectroscopy (TR-PES) 9–12and ( ii) methods that directly visualize nuclear dynamics, such as ultrafast x-ray scattering13–16and ultrafast electron diffraction.12,17Time- resolved x-ray absorption spectroscopy (TR-XAS) belongs to the for-mer category. Similar to x-ray photoelectron spectroscopy (XPS), XASis also element and chemical-state speciﬁc 18but is able to resolve the underlying electronic states better than TR-XPS. On the other hand,TR-XPS affords photoelectron detection from all the involved elec- tronic states with higher yield. XAS has been used to probe the localstructure of bulk-solvated systems, such as in most chemical reactionsystems in the lab and in cytoplasm. TR-XAS has been employed totrack photo-induced dynamics in organic molecules 19–22and transi- tion metal complexes.3,23–25With the aid of simulations,26nuclear dynamics can be extracted from experimental TR-XAS spectra. Similar to other time-resolved experimental methods from cate- gory ( i), interpretation of TR-XAS relies on computational methods for simulating electronic structure and nuclear wave-packet dynamics. In this context, electronic structure calculations should be able to pro- vide the following: (1) XAS of the ground states; (2) a description of the valence-excited states involved in the dynamics; and (3) XAS of the valence-excited states. Quantum chemistry has made major progress in simulations of XAS spectra of ground states.27,28Among currently available methods, the transition-potential density functional theory (TP-DFT) with the Struct. Dyn. 8, 024101 (2021); doi: 10.1063/4.0000070 8, 024101-1 VCAuthor(s) 2021Structural Dynamics ARTICLE scitation.org/journal/sdyhalf core-hole approximation29,30is widely used to interpret the XAS spectra of ground states.31,32Ehlert et al. extended the TP-DFT method to core excitations from valence-excited states33and imple- mented it in PSIXAS,34a plugin to the Psi4 code. TP-DFT is capable of simulating (TR-)XAS spectra of large molecules with reasonable accuracy, as long as the core-excited states can be described by a single electronic conﬁguration. Other extensions of Kohn–Sham DFT, suit- able for calculating the XAS spectra of molecules in their ground states, also exist.35Linear-response (LR) time-dependent (TD) DFT, a widely used method for excited states,36–39has been extended to the calculation of core-excited states40,41by means of the core-valence sep- aration (CVS) scheme,42a variant of truncated single excitation space (TRNSS) approach.43In the CVS scheme, conﬁgurations that do not involve core orbitals are excluded from the excitation space; this is jus- tiﬁed because the respective matrix elements are small, owing to the localized nature of the core orbitals and the large energetic gap between the core and the valence orbitals. Core-excitation energies calculated using TDDFT show errors up to/C2520 eV when standard exchange-correlation (xc) functionals such as B3LYP44are used. The errors can be reduced by using specially designed xc-functionals, such as those reviewed in Sec. 3.4.4. of Ref. 27.H a i ta n d Head-Gordon recently developed a square gradient minimum (SGM) algorithm for excited-state orbital optimization to obtain spin-pure restricted open-shell Kohn–Sham (ROKS) energies of core-excited states; they reported sub-eV errors in XAS transition energies.45 T h em a x i m u mo v e r l a pm e t h o d( M O M )46provides access to excited-state self-consistent ﬁeld (SCF) solutions and, therefore, can be used to directly compute core-level states. More importantly, MOM can be also combined with TDDFT to compute core excitations from a valence-excited state.20,22,47MOM-TDDFT is an attractive method for simulating TR-XAS spectra because it is computationally cheap and may provide excitation energies consistent with the TDDFT potential energy surfaces, which are often used in the nuclear dynam- ics simulations. However, in MOM calculations the initial valence- excited states are independently optimized and thus not orthogonal to each other. This non-orthogonality may lead to changes in the ener-getic order of the states. Moreover, open-shell Slater determinants pro- vide a spin-incomplete description of excited states (the initial state in an excited-state XAS calculation), which results in severe spin contam- ination of all states and may affect the quality of the computed spectra. Hait and Head-Gordon have presented SGM as an alternative general excited-state orbital-optimization method 48and applied it to compute XAS spectra of radicals.49 Applications of methods containing some empirical component, such as TDDFT, require benchmarking against the spectra computed with a reliable wave-function method, whose accuracy can be system- atically assessed. Among various post-HF methods, coupled-cluster (CC) theory yields a hierarchy of size-consistent ansatz for the ground state, with the CC singles and doubles (CCSD) method being the most practical.50CC theory has been extended to excited states via linear response51–53and equation-of-motion for excited states (EOM- EE)54–57formalisms. Both approaches have been adapted to treat core-excited states by using the CVS scheme,58including calculations of transition dipole moments and other properties.59–65The bench- marks illustrate that the CVS-enabled EOM-CC methods describe well the relaxation effects caused by the core hole as well as differential correlation effects. Given their robustness and reliability, the CC-basedmethods provide high-quality XAS spectra, which can be used to benchmark other methods. Aside from several CCSD investiga- tions,21,58–60,65–74core excitation and ionization energies have also been reported at the CC2 (coupled cluster singles and approximate dou- bles),66–68,73,75CC3 (coupled cluster singles, doubles and approximate tri- ples),21,76–78CCSDT (coupled cluster singles, doubles and triples),68,76,79 CCSDR(3),66,73,79and EOM-CCSD/C379levels of theory. XAS spectra have also been simulated with a linear-response (LR-)density cumulant theory (DCT),80which is closely related to the LR-CC methods. The algebraic diagrammatic construction (ADC) approach81,82 has also been used to model inner-shell spectroscopy. The second- order variant ADC(2)83yields valence-excitation energies with an accuracy and a computational cost [ OðN5Þ]s i m i l a rt oC C 2 ,84but within the Hermitian formalism. ADC(2) was extended to core excita- tions by the CVS scheme.85,86Because ADC(2) is inexpensive and is capable of accounting for dynamic correlation when calculating poten-tial energy surfaces, 87it promises to deliver reasonably accurate time- resolved XAS spectra at a low cost at each step of nuclear dynamic simulations. Neville et al. simulated TR-XAS spectra with ADC(2)88–90using multireference ﬁrst-order conﬁguration interaction (MR-FOCI) in their nuclear dynamics simulations. Neville and Schuurman also reported an approach to simulate XAS spectra using electronic wave packet autocorrelation functions based on TD- ADC(2).91Anad hoc extension of ADC(2), ADC(2)-x,92is known to give ground-state XAS spectra with relatively high accuracy [better than ADC(2)] employing small basis sets such as 6–31 þG,93but the improvement comes with a higher computational cost ½OðN6Þ/C138.L i s t et al. have recently used ADC(2)-x, along with restricted active-space second-order perturbation theory (RASPT2), to study competingrelaxation pathways in malonaldehyde by TR-XAS simulations. 94 An important limitation of the single-reference methods (at least those only including singles and double excitations) is that they can reliably treat only singly excited states. While transitions to the singly occupied molecular orbitals (SOMO) result in target states that are for- mally singly excited from the ground-state reference state, other ﬁnal states accessible by core excitation from valence-excited states can bedominated by conﬁgurations of double or higher excitation character relative to the ground-state reference. Consequently, these states are not well described by conventional response methods such as TDDFT, LR/EOM-CCSD, or ADC(2) (see Fig. 2 in II A). 60,94This is the main rational for using MOM within TDDFT. To overcome this problem while retaining a low computational cost, Seidu et al.95suggested to combine DFT and multireference conﬁguration interaction (MRCI) with the CVS scheme, which led to the CVS-DFT/MRCI method. The authors demonstrated that the semi-empirical Hamiltonian adjusted to describe the Coulomb and exchange interactions of the valence- excited states96works well for the core-excited states too. In the context of excited-state nuclear dynamics simulations based on complete active-space SCF (CASSCF) or CAS second-order perturbation theory (CASPT2), popular choices for computing core excitations from a given valence-excited state are restricted active- space SCF (RASSCF)97,98or RASPT2.99Delcey et al. have clearly sum- marized how to apply RASSCF for core excitations.100XAS spectra of valence-excited states computed by RASSCF/RASPT2 have been pre- sented by various authors.47,101,102RASSCF/RASPT2 schemes are suf- ﬁciently ﬂexible and even work in the vicinity of conical intersections; they also can tackle different types of excitations, including, forStructural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 024101 (2021); doi: 10.1063/4.0000070 8, 024101-2 VCAuthor(s) 2021example, those with multiply excited character.103However, the accu- racy of these methods depends strongly on an appropriate selection ofthe active space, which makes their application system speciﬁc. In addition, RASSCF simulations might suffer from insufﬁcient descrip- tion of dynamic correlation, whereas the applicability of RASPT2 maybe limited by its computational cost. Many of the methods mentioned above are available in standard quantum chemistry packages. Hence, the assessment of their perfor- mance would help for computational chemists who want to use thesemethods to analyze the experimental TR-XAS spectra. Since experimen- tal TR-XAS spectra are still relatively scarce, we set out assessing the per- formance of four selected single-reference methods from the perspectiveof the three requirements stated above. That is, they should be able to accurately describe the core and valence excitations from the ground state (GS), to give the transition strengths between the core-excited andvalence-excited states, and yield the XAS spectra of the valence-excited states over the entire pre-edge region, i.e., describe the spectral features due to the transitions of higher excitation character. More speciﬁcally, weextend the use of the MOM approach to the CCSD framework and eval- uate its accuracy relative to standard fc-CVS-EOM-EE-CCSD and to MOM-TDDFT. We note that MOM has been used in combination withCCSD to calculate double core excitations. 104For selected ground-state XAS simulations, we also consider ADC(2) results. We use the following systems to benchmark the methodology: uracil, thymine, and acetylacetone ( Fig. 1 ). Experimental TR-XAS spectra have not been recorded for uracil yet, but its planar symmetryat the Franck–Condon (FC) geometry and its similarities with thymine make it a computationally attractive model system. Experimental TR- XAS data are available at the O K-edge of thymine and at the C K-edge of acetylacetone. The paper is organized as follows: First, we describe the method- ology and computational details. We then compare the results obtained with the CVS-ADC(2), CVS-EOM-CCSD, and TDDFTmethods against the experimental ground-state XAS spectra. 20–22,105 We also compare the computed valence-excitation energies with UVabsorption and electron energy loss spectroscopy (EELS, often calledelectron impact spectroscopy when it is applied to gas-phase mole- cules). 106We then present the XAS spectra of the valence-excited states obtained with different CCSD-based protocols and comparethem with experimental TR-XAS spectra when available. 20–22Finally, we evaluate the performance of MOM-TDDFT. II. METHODOLOGY A. Protocols for computing XAS We calculated the energies and oscillator strengths for core and valence excitations from the ground states by standard LR/EOM methods: ADC(2),81,82,92EOM-EE-CCSD,50,54–57,107,108and TDDFT.In the ADC(2) and CCSD calculations of the valence-excited states, we employ the frozen core (fc) approximation. CVS58,59,86was applied to obtain the core-excited states within all methods. Within the fc-CVS-EOM-EE-CCSD framework,59we explored three different strategies to obtain the excitation energies and oscillator strengths forselected core-valence transitions, as summarized in Fig. 2 . In the ﬁrst one, referred to as standard CVS-EOM-CCSD, we assume that theﬁnal core-excited states belong to the set of excited states that can bereached by core excitation from the ground states (see Fig. 2 ,t o p panel). Accordingly, we use the HF Slater determinant, representing the ground state ( jU 0i) as the reference ( jUrefi) for the CCSD calcula- tion; the (initial) valence-excited and (ﬁnal) core-excited states arethen computed with EOM-EE-CCSD and fc-CVS-EOM-EE-CCSD,respectively. The transition energies for core-valence excitations aresubsequently computed as the energy differences between the ﬁnal core states and the initial valence state. The oscillator strengths for the transitions between the two excited states are obtained from the transi-tion moments between the EOM states according to the EOM-CC the-ory. 50,54,59In this approach, both the initial and the ﬁnal states are spin-pure states. However, the ﬁnal core-hole states that have multipleexcitation character with respect to the ground state are either not FIG. 1. Structures of (a) uracil, (b) thymine, and (c) acetylacetone.FIG. 2. Schematics of the standard CVS-EOM-CCSD, LSOR-CCSD, and HSOR- CCSD protocols. The crossed conﬁgurations are formally doubly excited withrespect to the ground-state reference.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 024101 (2021); doi: 10.1063/4.0000070 8, 024101-3 VCAuthor(s) 2021accessed or described poorly by this approach (the respective conﬁgu- rations are crossed in Fig. 2 ). In the second approach, named high-spin open-shell reference (HSOR) CCSD, we use as a reference for the CCSD calculations ahigh-spin open-shell HF Slater determinant that has the same elec-tronic conﬁguration as the initial singlet valence-excited state to beprobed in the XAS step. 60,64,109This approach is based on the assump- tion that the exchange interactions, which are responsible for theenergy gap between singlets and triplets, cancel out in calculations of the transition energies and oscillator strengths. An attractive feature of this approach is that the reference is spin complete (as opposed to alow-spin open-shell determinant of the same occupation) and that theconvergence of the SCF procedure is usually robust. A drawback ofthis approach is the inability to distinguish between the singlet andtriplet states with the same electronic conﬁgurations. In the third approach, we use low-spin (M s¼0) MOM references for singlet excited states and high-spin (M s¼1) MOM references for triplet excited states. We refer to this approach as low-spin open-shell reference (LSOR) CCSD. In both HSOR-CCSD and LSOR-CCSD, the calculation begins with an SCF optimization targeting the dominant conﬁguration of theinitial valence-excited state by means of the MOM algorithm, and theresulting Slater determinant is then used as the reference in the subse- quent CCSD calculation. Core-excitation energies and oscillator strengths from the high-spin and the low-spin references are com-puted with standard CVS-EOM-EE-CCSD. Such MOM-based CCSDcalculations can describe all target core-hole states, provided that theyhave singly excited character with respect to the chosen reference.Furthermore, in principle, initial valence-excited states of differentspin symmetries can be selected. However, in calculations using low-spin open-shell references (LSOR-CCSD states), variational collapse might occur. Moreover, the LSOR-CCSD treatment of singlet excited states suffers from spin contamination as the underlying open-shellreference is not spin complete (the well known issue of spin-completeness in calculations using open-shell references is discussedin detail in recent review articles. 110,111). We note that the HSOR-CCSD ansatz for a spin-singlet excited state is identical to the LSOR-CCSD ansatz of a (M s¼1) spin-triplet state having the same electronic conﬁguration as the spin-singletexcited state (see Fig. 2 ). In addition to the three CCSD-based protocols described above, we also considered MOM-TDDFT, which is often used for simulationof the time-resolved near-edge x-ray absorption ﬁne structure (TR-NEXAFS) spectra. 20,22,47We employed the B3LYP xc-functional,44as in Refs. 20,22,a n d 47. B. Computational details The equilibrium geometry of uracil was optimized at the MP2/ cc-pVTZ level. The equilibrium geometries of thymine and acetylace-tone were taken from the literature; 21,61they were optimized at the CCSD(T)/aug-cc-pVDZ and CCSD/aug-cc-pVDZ level, respectively.These structures represent the molecules at the FC points. The struc-tures of the T 1(pp/C3) and S 1(np/C3) states of acetylacetone, and of the S1(np/C3) state of thymine were optimized at the EOM-EE-CCSD/aug- cc-pVDZ level.61 We calculated near-edge x-ray absorption ﬁne structure (NEXAFS) of the ground state of all three molecules using CVS-ADC(2), CVS-EOM-CCSD, and TDDFT/B3LYP. The excitation ener- gies of the valence-excited states were calculated with ADC(2), EOM-EE-CCSD, and TDDFT/B3LYP. The XAS spectra of theT 1(pp/C3), T 2(np/C3), S1(np/C3), and S 2(pp/C3) states of uracil were calculated at the FC geometry. We used the FC geometry for all states in order to make a coherent comparison of the MOM-based CCSD methods withthe standard CCSD method and to ensure that the ﬁnal core-excited states are the same in the ground state XAS and transient state XAS calculations using standard CCSD. The spectra of thymine in theS 1(np/C3) state were calculated at the potential energy minimum of the S1(np/C3) state. The spectra of acetylacetone in the T 1(pp/C3) and S 2(pp/C3) states were calculated at the potential energy minima of the T 1(pp/C3) and S 1(np/C3) states, respectively. Our choice of geometries for acetyla- cetone is based on the fact that the S 2(pp/C3)-state spectra were mea- sured during wave packet propagation from the S 2(pp/C3) minimum (planar) toward the S 1(np/C3) minimum (distorted), and the ensemble was in equilibrium when the T 1(pp/C3)-state spectra were measured.22 The XAS spectra of the valence-excited states were computed with CVS-EOM-CCSD, HSOR-CCSD, and LSOR-CCSD. Pople’s6–311 þþG /C3/C3basis set was used throughout. In each spectrum, the oscillator strengths were convoluted with a Lorentzian function (empirically chosen FWHM ¼0.4 eV,60unless otherwise speciﬁed). We used the natural transition orbitals (NTOs)37,112–119to determine the character of the excited states. All calculations were carried out with the Q-Chem 5.3 electronic structure package.120The initial guesses [HOMO( b)]1[LUMO( a)]1 and [HOMO( a)]1[LUMO( a)]1were used in MOM-SCF for the spin- singlet and triplet states dominated by (HOMO)1(LUMO)1conﬁgura- tion, respectively. The SOMOs of the initial guess in a MOM-SCF pro- cedure are the canonical orbitals (or the Kohn–Sham orbitals) which resemble the hole and particle NTO of the transition from the groundstate to the valence-excited state. One should pay attention to the order of the orbitals obtained in the ground-state SCF, especially when the basis set has diffuse functions. In LSOR-CCSD calculations, the SCFconvergence threshold had to be set to 10 /C09Hartree. To ensure con- vergence to the dominant electronic conﬁguration of the desired elec- tronic state, we used the initial MOM (IMOM) algorithm121instead of regular MOM; this is important for cases when the desired state belongs to the same irreducible representation as the ground state. III. RESULTS AND DISCUSSION A. Ground-state NEXAFS Figure 3 shows the O K-edge NEXAFS spectra of uracil in the ground state computed by CVS-EOM-CCSD, CVS-ADC(2), andTDDFT/B3LYP. Table I shows NTOs of the core-excited states calcu- lated at the CVS-EOM-CCSD/6–311 þþG /C3/C3level, where rKare the singular values for a given NTO pair (their renormalized squares givethe weights of the respective conﬁgurations in the transition). 37,112–119 The NTOs for the other two methods are collected in the supplemen- tary material . Panel (d) of Fig. 3 shows the experimental spectrum (digitized from Ref. 105). The experimental spectrum has two main peaks at 531.3 and 532.2 eV, assigned to core excitations to the p/C3 orbitals from O4 and O2, respectively. Beyond these peaks, the inten- sity remains low up to 534.4 eV. The next notable spectral feature, attributed to Rydberg excitations, emerges at around 535.7 eV, just before the ﬁrst core-ionization onset (indicated as IE). The separationof/C240.9 eV between the two main peaks is reproduced at all threeStructural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 024101 (2021); doi: 10.1063/4.0000070 8, 024101-4 VCAuthor(s) 2021levels of theory. The NTO analysis at the CCSD level (cf. Table I )c o n - ﬁrms that the excitation to the 6A00state has Rydberg character and, after the uniform shift, the peak assigned to this excitation falls in the Rydberg region of the experimental spectrum. ADC(2) also yields a6A 00transition of Rydberg character, but it is signiﬁcantly red-shifted relative to the experiment. No Rydberg transitions are found at theTDDFT level. Only CVS-EOM-CCSD reproduces the separation between the 1A 00and the 6A00peaks with reasonable accuracy, 4.91 eV vs 4.4 eV in the experimental spectrum. The shoulder structure of theexperimental spectrum in the region between 532.2 and 534.4 eV isattributed to vibrational excitations or shakeup transitions. 18,122Figure 4 shows the ground-state NEXAFS spectra of thymine at the O K-edge. For construction of the theoretical absorption spectra, we used FWHM of 0.6 eV for the Lorentzian convolution function. Panel (d) shows the experimental spectrum (digitized from Ref. 21). Both the experimental and calculated spectra exhibit ﬁne structures, similar to those of uracil. Indeed, the ﬁrst and second peaks at 531.4 and 532.2 eV of the experimental spectrum were assigned to O 1s-hole states having the same electronic conﬁguration characters as the two lowest-lying O 1s-hole states of uracil. The NTOs of thymine can be found in the supplementary material .A g a i n ,o n l yC V S - E O M - C C S D reproduces reasonably well the Rydberg region after 534 eV. The sepa- ration of the two main peaks is well reproduced at all three levels oftheory. Figure 5 shows the C K-edge ground-state NEXAFS spectra of acetylacetone; the NTOs of the core excitations obtained at the CVS- EOM-CCSD/6–311 þþG /C3/C3level are collected in Table II . The experi- mental spectrum, plotted in panel (d) of Fig. 5 , was digitized from Ref. 22.Table II shows that the ﬁrst three core excitations are dominated by the transitions to the LUMO from the 1 sorbitals of the carbon atoms C2, C3, and C4. Transition from the central carbon atom, C3, appears as the ﬁrst relatively weak peak at 284.4 eV. We note that ace-tylacetone may exhibit keto–enol tautomerism. In the keto form, atoms C2 and C4 are equivalent. Therefore, transitions from these car- bon atoms appear as quasi-degenerate main peaks at /C25286.6 eV. The region around 288.2 eV is attributed to Rydberg transitions. The /C242 eV separation between the ﬁrst peak and the main peak due to the two quasi-degenerate transitions is well reproduced by ADC(2) and TDDFT/B3LYP, and slightly underestimated by CVS-EOM-CCSD (1.6 eV). On the other hand, the separation of /C241.6 eV between the main peak and the Rydberg resonance region is well reproduced only by CVS-EOM-CCSD. The results for the three considered molecules illustrate that CVS-EOM-CCSD describes well the entire pre-edge region of theFIG. 3. Uracil. Ground-state NEXAFS at the oxygen K-edge calculated with (a) ADC(2); (b) CVS-EOM-CCSD; (c) TDDFT/B3LYP. The calculated IEs are 539.68and 539.86 eV (fc-CVS-EOM-IP-CCSD/6-311 þþG/C3/C3). In panel (d), the computed spectrum of (b) is shifted by /C01.8 eV and superposed with the experimental spec- trum105(black curve). Basis set: 6-311 þþG/C3/C3. TABLE I. Uracil. CVS-EOM-CCSD/6-311 þþG/C3/C3energies, strengths, and NTOs of the O 1score excitations from the ground state at the FC geometry (NTO isosurface is 0.04 for the Rydberg transition and 0.05 for the rest). Final state Eex(eV) Osc. strength Hole r2 K Particle 1A00533.17 0.036 7 0.78 2A00534.13 0.034 3 0.79 3A00537.55 0.000 3 0.76 4A00537.66 0.000 4 0.78 6A00538.08 0.002 2 0.82 FIG. 4. Thymine. Ground-state oxygen K-edge NEXAFS calculated with (a) ADC(2), (b) CVS-EOM-CCSD, (c) TDDFT/B3LYP. The computed ionization ener- gies (IEs) are 539.67 and 539.73 eV (fc-CVS-EOM-IP-CCSD). In panel (d), the CVS-EOM-CCSD spectrum of (b) is shifted by /C01.7 eV and superposed with the experimental one21(black curve). Basis set: 6-311 þþG/C3/C3. FWHM of the Lorentzian convolution function is 0.6 eV.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 024101 (2021); doi: 10.1063/4.0000070 8, 024101-5 VCAuthor(s) 2021NEXAFS spectrum. CVS-ADC(2) and TDDFT/B3LYP describe well the core excitations to the LUMO and LUMO þ1 (apart from a sys- tematic shift), but generally fail to describe the transitions at higherexcitation energies.B. Valence-excited states Table III shows the excitation energies of the two lowest triplet states, the three lowest singlet states, plus the S 5(pp/C3) state of uracil, calculated at the FC geometry, along with the values derived from the EELS123and UV absorption experiments.124The EOM-EE-CCSD/ 6–311 þþG/C3/C3NTOs are collected in Table IV , and the NTOs for other methods are given in the supplementary material . We refer to Ref. 125 for an extensive benchmark study of the one-photon absorption and excited-state absorption of uracil. In EELS, the excited states are probed by measuring the kinetic energy change of a beam of electrons after inelastic collision with theprobed molecular sample. 106In the limit of high incident energy or small scattering angle, the transition amplitude takes a dipole form and the selection rules are same as those of UV-Vis absorption. Otherwise, the selection rules are different and optically dark statescan be detected. Furthermore, spin–orbit coupling enables excitationinto triplet states. Assignment of the EELS spectral signatures is based on theoretical calculations. Note that excitation energies obtained withFIG. 5. Acetylacetone. Ground-state NEXAFS at carbon K-edge calculated with (a) ADC(2); (b) CVS-EOM-CCSD; (c) TDDFT/B3LYP . The ionization energies (IEs) are291.12, 291.88, 292.11, 294.10, and 294.56 eV (fc-CVS-EOM-IP-CCSD). In panel(d), the computational result of (b) is shifted by /C00.9 eV and superposed with the experimental spectrum 22(black curve). Basis set: 6-311 þþG/C3/C3. TABLE II. Acetylacetone. CVS-EOM-CCSD/6-311 þþG/C3/C3NTOs of the C 1score excitations from the ground state at the FC geometry (NTO isosurface is 0.03 for the Rydberg transition and 0.05 for the rest). Final state Eex(eV) Osc. strength Hole r2 K Particle 1A 285.88 0.013 3 0.76 2A 287.36 0.067 1 0.82 3A 287.53 0.067 3 0.81 9A 288.63 0.021 3 0.79 11A 289.13 0.020 2 0.82 13A 289.27 0.020 5 0.83 14A 289.28 0.017 5 0.82 15A 289.30 0.017 4 0.81 TABLE III. Uracil. Excitation energies (eV) at the FC geometry and comparison with experimental values from EELS123and UV absorption spectroscopy.124 ADC(2) ADC(2)-x EOM-CCSD TDDFT EELS UV T1(pp/C3) 3.91 3.36 3.84 3.43 3.75 T2(np/C3) 4.47 3.79 4.88 4.27 4.76 S1(np/C3) 4.68 3.93 5.15 4.65 5.2 S2(pp/C3) 5.40 4.70 5.68 5.19 5.5 5.08 S3(pRyd) 5.97 5.39 6.07 5.70 … S5(pp/C3) 6.26 5.32 6.74 5.90 6.54 6.02 TABLE IV. Uracil. EOM-EE-CCSD/6-311 þþG/C3/C3NTOs for the transitions from the ground state to the lowest valence-excited states at the FC geometry (NTO isosur-face is 0.05). Final state Eex(eV) Osc. strength Hole r2 K Particle T1(A0;pp/C3) 3.84 … 0.82 T2(A00,np/C3) 4.88 … 0.82 S1(A00,np/C3) 5.15 0.000 0 0.81 S2(A0,pp/C3) 5.68 0.238 6 0.75 S3(A00;pRyd) 6.07 0.0027 0.85 S5(A0,pp/C3) 6.74 0.0573 0.73 Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 024101 (2021); doi: 10.1063/4.0000070 8, 024101-6 VCAuthor(s) 2021EELS may be blue-shifted compared to those from UV-Vis absorption due to momentum transfer between the probing electrons and theprobed molecule. EOM-EE-CCSD excitation energies for all valence states of uracil agree well with the experimental values from EELS. Both the EOM-EE-CCSD and EELS values slightly overestimate the UV-Vis results.For the two triplet states and the S 1(A00,np/C3) and S 2(A0;pp/C3)s t a t e s , ADC(2) also gives fairly accurate excitation energies. ADC(2)-x, onthe other hand, seems unbalanced for the valence excitations (regard-less of the basis set). The TDDFT/B3LYP excitation energies are red-shifted with respect to the EELS values, but the energy differencesbetween the T 1(A0;pp/C3), T 2(A00,np/C3), S 1(A00,np/C3), and S 2(A0;pp/C3) states are in reasonable agreement with the corresponding experimen-tally derived values. Table V shows the excitation energies of the ﬁve lowest triplet and singlet states of thymine, along with the experimental valuesobtained by EELS. 126We did not ﬁnd literature data for the UV absorption of thymine in the gas phase. The energetic order is basedon EOM-EE-CCSD. Here, we reassign the peaks of the EELS spec-tra 126on the basis of the following considerations: ( i) optically bright transitions also exhibit strong peaks in the EELS spectra; ( ii) the excita- tion energy of a triplet state is lower than the excitation energy of thesinglet state with the same electronic conﬁguration; ( iii) the strengths of the transitions to triplet states are smaller than the strengths ofthe transitions to singlet states; ( iv) among the excitations enabled by spin–orbit coupling, p!p /C3transitions have relatively large transition moments. Except for T 1(pp/C3), the ADC(2) excitation energies are red- shifted relative to EOM-CCSD. Hence, the ADC(2) excitation energiesof the states considered here are closest, in absolute values, to theexperimental values from Table V . However, the energy differences between the singlet states (S 1,S2,S4,a n dS 5) are much better repro- duced by EOM-CCSD. TDDFT/B3LYP accurately reproduces theexcitation energies of the T 2(np/C3), S1(np/C3), and S 2(pp/C3)s t a t e s . Table VI shows the excitation energies of the two lowest triplet and singlet states, and the lowest Rydberg states of acetylacetone, alongwith the experimental values obtained from EELS 127and UV absorption128(the exact state ordering of states in the singlet Rydberg manifold is unknown). Table VII shows the NTOs obtained at theEOM-EE-CCSD/6–311 þþG/C3/C3level. Remarkably, for this molecule the excitation energies from EELS agree well with those from UV absorption. Note that the EELS spectra of acetylacetone were recordedwith incident electron energies of 25 and 100 eV, 127whereas those for uracil123were obtained with 0–8.0 eV. The higher incident electron energies reduce the effective acceptance angle of the electrons, whichmay hinder the detection of electrons that have undergone momen-tum transfer. The transitions to the T 1(pp/C3)a n dT 2(np/C3) states appeared only with the 25 eV incident electron energy and a scatteringangle of 90 /C14(see Fig. 3 of Ref. 127). The peaks were broad and, fur- thermore, an order of magnitude less intense than the S 0!S2(pp/C3) transition. Consequently, it is difﬁcult to resolve the excitation energiesof T 1(pp/C3) and T 2(np/C3). ADC(2) yields the best match with the exper- imental results for acetylacetone. These results indicate that the excitation energies of the valence- excited states computed by EOM-EE-CCSD, ADC(2), and TDDFT/ TABLE V. Thymine. Excitation energies (eV) at the FC geometry compared with the experimental values from EELS.126The oscillator strengths are from EOM-EE-CCSD and used for the re-assignment. ADC(2) EOM-CCSD TDDFT EELS Osc. strength T1(pp/C3) 3.70 3.63 3.19 3.66 … T2(np/C3) 4.39 4.81 4.25 4.20 … S1(np/C3) 4.60 5.08 4.64 4.61 0.000 0 S2(pp/C3) 5.18 5.48 4.90 4.96 0.228 9 T3(pp/C3) 5.27 5.32 4.61 5.41 …. T4(pRyd) 5.66 5.76 5.39 … … S3(pRyd) 5.71 5.82 5.46 … 0.000 5 T5(pp/C3) 5.87 5.91 5.10 5.75 … S4(np/C3) 5.95 6.45 5.72 5.96 0.000 0 S5(pp/C3) 6.15 6.63 5.87 6.17 0.067 9TABLE VI. Acetylacetone. Excitation energies (eV) at the FC geometry compared with the values obtained in EELS127and UV absorption spectroscopy.128 ADC(2) ADC(2)-x EOM-CCSD TDDFT EELS UV T1(pp/C3) 3.76 3.16 3.69 3.23 3.57? … T2(np/C3) 3.79 3.13 4.11 3.75 ? … S1(np/C3) 4.03 3.29 4.39 4.18 4.04 4.2 S2(pp/C3) 4.96 4.28 5.24 5.08 4.70 4.72 T3ðpRydÞ 5.91 5.45 6.02 5.66 5.52 … S3?ðpRydÞ5.98 5.53 6.13 5.72 5.84 5.85 S5?ðpRydÞ6.87 6.30 7.06 6.64 6.52 6.61 TABLE VII. Acetylacetone. EOM-EE-CCSD/6-311 þþG/C3/C3NTOs of the excitations from the ground state to the lowest-lying valence-excited states at the FC geometry(NTO isosurface is 0.03 for the Rydberg transitions and 0.05 for the rest). Final state Eex(eV) Osc. strength Hole r2 K Particle T1(A0,pp/C3) 3.69 … 0.82 T2(A00,np/C3) 4.11 … 0.82 S1(A00,np/C3) 4.39 0.000 6 0.81 S2(A0,pp/C3) 5.24 0.329 9 0.77 T3½pRydðsÞ/C138 6.02 … 0.86 S3?½pRydðsÞ/C138 6.13 0.007 2 0.86 S5?½pRydðpÞ/C138 7.06 0.057 1 0.85 Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 024101 (2021); doi: 10.1063/4.0000070 8, 024101-7 VCAuthor(s) 2021B3LYP are equally (in)accurate. Which method yields the best match with experiment depends on the molecule. C. Core excitations from the valence-excited states In Secs. III A andIII B, we analyzed two of our three desiderata for a good electronic structure method for TR-XAS—that is, the abilityto yield accurate results for ground-state XAS as well as for thevalence-excited states involved in the dynamics. In this subsection, wefocus on the remaining item, i.e., the ability to yield accurate XAS of valence-excited states. For uracil, we conﬁrmed that EOM-CCSD and CVS-EOM- CCSD yield fairly accurate results for the valence-excited T 1(pp/C3), T2(np/C3), S 1(np/C3), and S 2(pp/C3) states and for the (ﬁnal) singlet (O 1s) core-excited states at the FC geometry, respectively. It is thus reason-able to consider the oxygen K-edge XAS spectra of the S 1(np/C3)a n d S2(pp/C3) states of uracil obtained from CVS-EOM-CCSD as our refer- ence, even though CVS-EOM-CCSD only yields the peaks of the core-to-SOMO transitions. Figure 6 shows the oxygen K-edge XAS of uracil in the (a) S 1(np/C3), (b) S 2(pp/C3), (c) T 2(np/C3), and (d) T 1(pp/C3) states, calculated using CVS-EOM-CCSD (blue curve) and LSOR-CCSD (red curve) atthe FC geometry. Note that the HSOR-CCSD spectra of S 1(np/C3)a n d S2(pp/C3) are identical to the LSOR-CCSD spectra for the T 2(np/C3)a n d T1(pp/C3) states, respectively, because their orbital electronic conﬁgura- tion are the same, see Table IV . The ground-state spectrum (green curve) is included in all panels for comparison. The LSOR-CCSDNTOs of the transitions underlying the peaks in the S 1(np/C3), S2(pp/C3) and T 1(pp/C3) spectra are given in Tables VIII–X ,r e s p e c t i v e l y . The CVS-EOM-CCSD spectrum of S 1(np/C3) exhibits a relatively intense peak at 528.02 eV, and tiny peaks at 532.40 and 532.52 eV. Theintense peak is due to transition from the 1 sorbital of O4 to SOMO, which is a lone-pair-type orbital localized on O4. The tiny peak at532.40 eV is assigned to the transition to SOMO from the 1 sorbital of O2, whereas the peak at 532.52 eV is assigned to a transition with mul-tiply excited character. The LSOR-CCSD spectrum exhibits the strongcore-to-SOMO transition peak at 526.39 eV, which is red-shifted from the corresponding CVS-EOM-CCSD one by 1.63 eV. As Table VIII shows that the peak at 534.26 eV is due to transition from the 1 s orbital of O2 to a p /C3orbital, and it corresponds to the second peak in the ground-state spectrum. In the S 1(np/C3) XAS spectrum, there is no peak corresponding to the ﬁrst band in the ground-state spectrum,there assigned to the O4 1 s!p /C3transition. This suggests that thisTABLE VIII. Uracil. LSOR-CCSD/6-311 þþG/C3/C3NTOs of the O 1score excitations from the S 1(np/C3) state at the FC geometry (NTO isosurface value is 0.05). Eex(eV) Osc. strength Spin Hole r2 K Particle 526.39 0.045 1 a 0.86 534.26 0.032 3 a 0.56 b 0.23 TABLE IX. Uracil. LSOR-CCSD/6-311 þþG/C3/C3NTOs of the O 1score excitations from the S 2(pp/C3) state at the FC geometry (NTO isosurface value is 0.05). Eex(eV) Osc. strength Spin Hole r2 K Particle 530.16 0.010 2 a 0.68 530.54 0.013 1 a 0.67 532.96 0.018 6 b 0.74 534.74 0.015 5 b 0.80 535.70 0.007 6 a 0.77 535.88 0.008 5 a 0.76 FIG. 6. Uracil. Oxygen K-edge NEXAFS of the four lowest-lying valence states: (a) S1(np/C3); (b) S 2(pp/C3); (c) T 2(np/C3); and (d) T 1(pp/C3)]. The blue and red curves corre- spond to the CVS-EOM-CCSD and LSOR-CCSD results, respectively. Note that the HSOR spectra for S 1and S 2are identical to the LSOR-CCSD spectra for T 2 and T 1. Basis set: 6-311 þþG/C3/C3. FC geometry. The ground state XAS (green curve) is included for comparison.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 024101 (2021); doi: 10.1063/4.0000070 8, 024101-8 VCAuthor(s) 2021transition is suppressed by the positive charge localized on O4 in the S1(np/C3) state. The S 1(np/C3) state from LSOR-CCSD is spin-contaminated, with hS2i¼1:033. The spectra of S 1(np/C3) yielded by LSOR-CCSD [panel (a)] and by HSOR-CCSD [panel (c)] are almost identical. This is not too surprising, as the spectra of S 1(np/C3) and T 2(np/C3) from CVS-EOM- CCSD are also almost identical. This is probably a consequence ofsmall exchange interactions in the two states (the singlet and the trip- let) due to negligible spatial overlap between the lone pair (n) and p /C3 orbitals. In the CVS-EOM-CCSD spectrum of S 2(pp/C3), see panel (b), the peaks due to the core-to-SOMO ( p) transitions from O4 and O2 occur at 527.50 and 531.87 eV, respectively. The additional peak at 531.99 eV is assigned to a transition with multiple electronic excitation. In theLSOR-CCSD spectrum, the core-to-SOMO peaks appear at 530.16and 530.54 eV, respectively. As shown in Table IX ,w ea s s i g nt h ep e a k sa t5 3 2 . 9 6a n d 534.74 eV in the LSOR-CCSD spectrum to transitions from the 1 s orbitals of the two oxygens to the p /C3orbital, which is half occupied in S2(pp/C3). The NTO analysis reveals that they correspond to the ﬁrst and second peak of the ground-state spectrum. Note that hS2i¼1.326 for the S 2(pp/C3) state obtained from LSOR-CCSD. In the HSOR-CCSD spectrum of the S 2(pp/C3)s t a t e[ w h i c hi s equal to the LSOR-CCSD spectrum of the T 1(pp/C3) state in panel (d)], t h ep e a k so ft h ec o r e - t o - S O M O( p) transitions from O4 and O2 appear at 529.81 and 532.39 eV, respectively (see Table X ). They arefollowed by transitions to the half-occupied p/C3orbital at 534.15 and 535.09 eV, respectively. In contrast to what we observed in the S 1(np/C3) spectra, the LSOR-CCSD and HSOR-CCSD spectra of the S 2(pp/C3) state are qualitatively different. This can be explained, again, in terms of importance of the exchange interactions in the initial and ﬁnal states. On one hand, there is a stabilization of the T 1(pp/C3) (initial) state over the S 2(pp/C3) state by exchange interaction as the overlap between thepandp/C3orbitals is not negligible. The exchange interaction between the strongly localized core-hole orbital and the half-occupiedvalence/virtual orbital in the ﬁnal core-excited state, on the other hand, is expected to be small. To evaluate the accuracy of the excited-state XAS spectra from CVS-EOM-CCSD and LSOR-CCSD, we also calculated the XAS spec-tra of the S 1(np/C3) state of thymine at the potential energy minimum of S1(np/C3), see panel (a) of Fig. 7 . For construction of the surface cut of the theoretical absorption spectra, we chose FWHM of 0.6 eV for theLorentzian convolution function. Panel (b) shows the spectra of S 1(np/C3) multiplied by 0.2 and added to the ground-state spectrum multiplied by 0.8. These factors 0.2 and 0.8 were chosen for the best ﬁtwith the experimental spectrum. A surface cut of the experimental TR-NEXAFS spectrum at the delay time of 2 ps (Ref. 21)i sa l s os h o w n in panel (b) of Fig. 7 . The reconstructed computational spectra are shifted by /C01.7 eV. In the experimental spectrum, the core-to-SOMO transition peak occurs at 526.4 eV. In the reconstructed theoretical spectrum, the core-to-SOMO transition peaks appear at 526.62 and524.70 eV, for CVS-EOM-CCSD and LSOR-CCSD, respectively. Thus, the CVS-EOM-CCSD superposed spectrum agrees slightly better with experiment than the LSOR-CCSD spectrum. Nonetheless, the accu-racy of the LSOR-CCSD spectrum is quite reasonable, as compared with the experimental spectrum. Due to the lack of experimental data, not much can be said about the accuracy of CVS-EOM-CCSD and LSOR-CCSD/HSOR-CCSD for core excitations from a triplet excited state in uracil and thymine. Furthermore, we are unable to unambiguously clarify, using uracil andthymine as model system, which of the two methods, LSOR-CCSD or HSOR-CCSD, should be considered more reliable when they give qualitatively different spectra for the singlet excited states. Therefore, we turn our attention to the carbon K-edge spectra of acetylacetone and show, in Fig. 8 , the spectra obtained using CVS- EOM-CCSD (blue), LSOR-CCSD (red), and HSOR-CCSD (magenta) FIG. 7. Thymine. (a) Oxygen K-edge NEXAFS in the S 1(np/C3) state at its potential energy minimum. Blue: CVS-EOM-CCSD. Red: LSOR-CCSD. Thin green line:ground-state spectrum at the FC geometry. (b) Thick black: Experimental spectrumat the delay time of 2 ps. 21Blue: computational spectrum made from the blue and green curves of (a), shifted by /C01.7 eV. Red: computational spectrum made from the red and green curves of (a), shifted by /C01.7 eV. The blue and red curves from (a) were scaled by 0.2 in (b). The ground-state spectrum from (a) was scaled by0.8 in (b). FWHM of the Lorentzian convolution function is 0.6 eV.TABLE X. Uracil. LSOR-CCSD/6-311 þþG/C3/C3NTOs of the O 1score excitations from the T 1(pp/C3) state at the FC geometry (NTO isosurface is 0.05). Eex(eV) Osc. strength Spin Hole r2 K Particle 529.81 0.021 2 b 0.79 532.39 0.011 5 b 0.78 534.15 0.018 7 a 0.76 535.09 0.010 0 a 0.73 535.58 0.006 2 b 0.77 535.61 0.008 1 b 0.72 Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 024101 (2021); doi: 10.1063/4.0000070 8, 024101-9 VCAuthor(s) 2021for the T 1(pp/C3)[ p a n e l( a ) ]a n dS 2(pp/C3) [panel (b)] states. The T 1(pp/C3) spectra were obtained at the potential energy minimum of T 1(pp/C3). The spectra of S 2(pp/C3) were calculated at the potential energy mini- mum of the S 1(np/C3) state. In doing so, we assume that the nuclear wave packet propagates on the S 2(pp/C3) surface toward the potential energy minimum of the S 1(np/C3) surface. Note that CVS-EOM-CCSD does not describe all the core excitations from a valence-excited state(seeFig. 2 ). In panels (c) and (d), the LSOR-CCSD spectra were multi- plied by 0.75 and subtracted from the ground-state spectrum, scaled by 0.25, and superposed to the surface cuts of the experimentaltransient-absorption NEXAFS at delay times 7–10 ps and 120–200 fs,respectively. The calculated transient-absorption spectra were shifted by/C00.9 eV, i.e., by the same amount as the spectrum of the ground s t a t e[ s e ep a n e l( b )o f Fig. 5 ]. For construction of the surface cut of the theoretical transient-absorption spectra, we used FWHM of 0.6 eV for the Lorentzian convolution function. The scaling factors values 0.75 and 0.25 were chosen to yield the best ﬁt with the experimental spec-tra. The NTOs of the core excitations from T 1(pp/C3)a n dS 2(pp/C3)a r e shown in Tables XI andXII, respectively. In the experimental study,22 it was concluded that S 2(pp/C3) is populated at the shorter timescale, whereas at the longer timescale it is T 1(pp/C3) that becomes populated. The surface cut of the experimental transient-absorption spectra at longer times (7–10 ps) features two peaks at 281.4 and 283.8 eV. In panel (a) of Fig. 8 , the CVS-EOM-CCSD spectrum of T 1(pp/C3)s h o w s the core-to-SOMO transition peaks at 282.69 and 284.04 eV, whereas the LSOR-CCSD ones appear at 281.76 and 283.94 eV. The LSOR- CCSD spectrum also shows a peak corresponding to a transition fromC4 to the half-occupied p /C3orbital at 286.96 eV (see Table XI ). The separation of 2.4 eV between the two core-to-SOMO peaks in theexperiment is well reproduced by LSOR-CCSD. Spin contamination is small, hS2i¼2.004 for the T 1(pp/C3) state obtained using LSOR-CCSD. Therefore, it is safe to say, that LSOR-CCSD accurately describes coreexcitations from the low-lying triplet states.FIG. 8. Acetylacetone. Carbon K-edge NEXAFS from the T 1(pp/C3) (a) and S 2(pp/C3) (b) states. The spectra of T 1(pp/C3) were computed at the potential energy minimum of T 1(pp/C3). The spectra of S 2(pp/C3) were computed at the potential energy mini- mum of S 1(np/C3). Blue: CVS-EOM-CCSD. Red: LSOR-CCSD. Magenta: HSOR- CCSD. Green: Ground-state spectrum at the FC geometry. (c), (d) Black:Experimental transient absorption spectra at the delay times of 7–10 ps and 120–200 fs,22respectively. Red: computational transient absorption spectra made from the red and the green curves of (a) and (b), respectively, shifted by /C00.9 eV as the spectrum of the ground state [see panel (b) of Fig. 5 ]. The red curves of pan- els (a) and (b) were scaled by 0.75 and from these, the green ground-state spec- trum, scaled by 0.25, was subtracted. FWHM of the Lorentzian convolution function is 0.4 eV for panels (a) and (b), 0.6 eV for panels (c) and (d), respectively. Basisset: 6-311 þþG/C3/C3.TABLE XI. Acetylacetone. LSOR-CCSD/6-311 þþG/C3/C3NTOs of the C 1score excita- tions from the T 1state at the potential energy minimum (NTO isosurface is 0.05). Eex(eV) Osc. strength Spin Hole r2 K Particle 281.76 0.034 7 b 0.86 283.94 0.031 8 b 0.84 285.69 0.003 6 b 0.72 286.96 0.033 4 a 0.65 b 0.14 TABLE XII. Acetylacetone. LSOR-CCSD/6-311 þþG/C3/C3NTOs of the C 1score excita- tions from the S 2state at the potential energy minimum of S 1(NTO isosurface is 0.05). Eex(eV) Osc. strength Spin Hole r2 K Particle 281.30 0.022 8 a 0.77 283.69 0.008 5 a 0.71 285.43 0.026 9 b 0.76 286.07 0.038 1 b 0.76 287.39 0.005 7 b 0.64 Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 024101 (2021); doi: 10.1063/4.0000070 8, 024101-10 VCAuthor(s) 2021The surface cut of the transient-absorption spectra at shorter times, 120–240 fs, features relatively strong peaks at 284.7, 285.9 and a ground-state bleach at 286.6 eV. The CVS-EOM-CCSD spectrum of the S 2(pp/C3) state shows the core-to-SOMO peak at 280.77. The LSOR- CCSD spectrum (red) has core-to-SOMO transition peaks at 281.30 and 283.69 eV, plus the peaks due to the transitions from the core of C2, C4, and C3 to the half-occupied p/C3orbital at 285.43, 286.07 and 287.39 eV, respectively (see Table XII ). Note that the peaks at 285.43 and 286.07 eV correspond to the main degenerate peaks of the ground-state spectrum, as revealed by inspection of the NTOs. The HSOR-CCSD spectrum (magenta) exhibits the core-to-SOMO transi- tion peaks at 281.99 and 283.17 eV, followed by only one of the quasi- degenerate peaks corresponding to transitions to the half-occupied p/C3 orbital, at 287.95 eV. Since the experimental surface-cut spectrum does not clearly show the core-to-SOMO transition peaks, it is difﬁcult to assess the accuracy of these peaks as obtained in the calculations. When it comes to the experimental peaks at 284.7 and 285.9 eV, only LSOR-CCSD reproduces them with reasonable accuracy. The experi- mental peak at 288.4 eV is not reproduced. In the case of acetylace- tone, the HSOR-CCSD approximation fails to correctly mimic the spectrum of S 2(pp/C3), since it does not give the peaks at 284.7 and 285.9 eV. The differences between LSOR-CCSD and HSOR-CCSD spectra for S 2(pp/C3) can be rationalized as done for uracil. We emphasize that the assignment of the transient absorption signal at shorter time to S 2(pp/C3) is based on peaks assigned to transi- tions to the p/C3orbitals (almost degenerate in the ground state), which cannot be described by CVS-EOM-CCSD (see Fig. 2 in Sec. II A). On the basis of the above analysis, we conclude that, despite spin contamination, LSOR-CCSD describes the XAS of singlet valence- excited states with reasonable accuracy. LSOR-CCSD could even be used as benchmark for other levels of theory, especially when experi- mental TR-XAS spectra are not available. We conclude this section by analyzing the MOM-TDDFT results for the transient absorption. As seen in Secs. III A andIII B, ADC(2) and TDDFT/B3LYP yield reasonable results for the lowest-lying core- excited states and for the valence-excited states of interest in the n u c l e a rd y n a m i c s .T h en e x tq u e s t i o ni st h u sw h e t h e rM O M - T D D F T / B3LYP can reproduce the main peaks of the time-resolved spectra with reasonable accuracy. We attempt to answer this question by com- paring the MOM-TDDFT/B3LYP spectra of thymine and acetylace- tone with the surface cuts of the experimental spectra. The MOM-TDDFT/B3LYP O K-edge NEXAFS spectrum of thy- mine in the S 1(np/C3) state is shown in Fig. 9 , panel (a). For construction of the surface cut of the theoretical absorption spectra, we used FWHM of 0.6 eV for the Lorentzian convolution function. A theoreti- cal surface cut spectrum was constructed as sum of the MOM-TDDFT spectrum and the standard TDDFT spectrum of the ground state, scaled by 0.2 and 0.8, respectively. This is shown in panel (b), together with the experimental surface cut spectrum at 2 ps delay. 21 The MOM-TDDFT/B3LYP peaks due to the core transitions from O4and O2 to SOMO (n) are found at 511.82 and 513.50 eV, respectively. The peak corresponding to the ﬁrst main peak of the ground-state spectrum is missing, and the one corresponding to the second main peak in the ground state appears at 517.71 eV. These features are equivalent to what we observed in the LSOR-CCSD case (see Fig. 7 ). Thus, the separation between the core-to-SOMO peak and the ground-state main peaks is accurately reproduced.Next, we consider the carbon K-edge spectra of acetylacetone in the T 1(pp/C3) [at the minimum of T 1(pp/C3)] and S 2(pp/C3)[ a tt h em i n i - mum of S 1(np/C3)] states, as obtained from MOM-TDDFT. They are plotted in panels (a) and (b) of Fig. 10 , respectively. Surface cuts of the transient-absorption NEXAFS spectra were constructed by subtracting the TDDFT spectrum, scaled by 0.25, with the MOM-TDDFT spectra scaled by 0.75. For this construction, we convoluted the oscillator strengths with a Lorentzian function (FWHM ¼0.6 eV) and chose the factors 0.75 and 0.25 for the best ﬁt with the experimental spectra. They are superposed with those from experiment at delay times of 7–10 ps and 120–200 fs in Fig. 10 ,p a n e l s( c )a n d( d ) .T h eM O M - TDDFT spectrum of T 1(pp/C3) exhibits the core-to-SOMO transition peaks at 270.88 and 272.41 eV. A peak due to the transition to the half-occupied p/C3orbital occurs at 274.16 eV. All peaks observed in the LSOR-CCSD spectrum were also obtained by MOM-TDDFT. TheFIG. 9. (a) Red: Oxygen K-edge NEXAFS for thymine in the S 1(np/C3) state calcu- lated at the MOM-TDDFT/B3LYP/6-311 þþG/C3/C3level at the potential energy mini- mum. Green: Ground-state spectrum. (b) Black: Experimental spectrum at the delay time of 2 ps,21Red: computational spectrum made from the red and the green curves of (a), shifted by þ14.8 eV. The red curve of (a) was scaled by 0.2. The green curve of (a) was scaled by 0.8. FWHM of the Lorentzian convolution function is 0.6 eV. FIG. 10. (a) and (b) Carbon K-edge NEXAFS for acetylacetone in the T 1(pp/C3) and S2(pp/C3) states calculated at the MOM-TDDFT/B3LYP/6-311 þþG/C3/C3level at the potential energy minima of T 1(pp/C3) and S 1(np/C3), respectively. The green curve is the ground-state spectrum. In panels (c) and (d), the experimental transient absorp-tion spectra at delay times of 7–10 ps and 120–200 fs are reported with blacklines. 22In red are the computational transient absorption spectra reconstructed from the red and green curves of panels (a) and (b), respectively, shifted by þ10.9 eV. The red curves of (a) and (b) were scaled by 0.75, and subtracted from the green curves, which were scaled by 0.25. FWHM of the Lorentzian convolutionfunction is 0.4 eV for panels (a) and (b), 0.6 eV for panels (c) and (d), respectively.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 024101 (2021); doi: 10.1063/4.0000070 8, 024101-11 VCAuthor(s) 2021ﬁne structure of the surface-cut transient absorption spectrum is quali- tatively reproduced. The MOM-TDDFT spectrum of S 2(pp/C3) exhibits the core-to- SOMO( p/C3) transition peaks at 269.94 and 271.73 eV. The peaks due to the transitions to the half-occupied p/C3orbital appear at 274.17 and 274.98 eV. The reconstructed transient-absorption spectrum agrees well with the experimental surface-cut spectrum. IV. SUMMARY AND CONCLUSIONS We have analyzed the performance of different single-reference electronic structure methods for excited-state XAS calculations. Theanalysis was carried out in three steps. First, we compared the results for the ground-state XAS spectra of uracil, thymine, and acetylacetone computed using CVS-ADC(2), CVS-EOM-CCSD, and TDDFT/B3LYP, and with the experimental spectra. Second, we computed the excitation energies of the valence-excited states presumably involved in the dynamics at ADC(2), EOM-EE-CCSD, and TDDFT/B3LYP lev-els, and compared them with the experimental data from EELS and UV absorption. Third, we analyzed different protocols for the XAS spectra of the lowest-lying valence-excited states based on the CCSDansatz, namely, regular CVS-EOM-CCSD for transitions betweenexcited states, and EOM-CCSD applied on the excited-state reference state optimized imposing the MOM constraint. The results for thy- mine and acetylacetone were evaluated by comparison with the experi-mental time-resolved spectra. Finally, the performance of MOM- TDDFT/B3LYP for TR-XAS was evaluated, again on thymine and ace- tylacetone, by comparison with the LSOR-CCSD and the experimentalspectra. In the ﬁrst step, we found that CVS-EOM-CCSD reproduces well the entire pre-edge region of the ground-state XAS spectra. On theother hand, CVS-ADC(2) and TDDFT/B3LYP only describe thelowest-lying core excitations with reasonable accuracy, while the Rydberg region is not captured. In the second step, we observed that EOM-EE-CCSD, ADC(2), and TDDFT/B3LYP treat the valence-excited states with a comparable accuracy. Among the methods analyzed in the third step, only LSOR- CCSD and MOM-TDDFT can reproduce the entire pre-bleachingregion of the excited-state XAS spectra for thymine and acetylacetone, despite spin contamination of the singlet excited states. LSOR-CCSD could be used as the reference when evaluating the performance ofother electronic structure methods for excited-state XAS, especially ifno experimental spectra are available. For the spectra of the spin- singlet states, CVS-EOM-CCSD yields slightly better core !SOMO positions. We note that the same procedure can be used to assess the performance of other xc-functional or post-HF methods for TR- XAS calculations. We also note that description of an initial statewith the MOM algorithm is reasonably accurate only when the initial state has a single conﬁgurational wave-function character. The low computational scaling and reasonable accuracy ofMOM-TDDFT makes it rather attractive for the on-the-ﬂy calcu-lation of TR-XAS spectra in the excited-state nuclear dynamics simulations. SUPPLEMENTARY MATERIAL See the supplementary material for the NTOs of all core and valence excitations.ACKNOWLEDGMENTS The research leading to the presented results has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement Nos. 713683 (COFUNDfellowsDTU) and 765739 (COSINE, COmputational Spectroscopy In Natural sciences and Engineering), from DTU Chemistry, from the Danish Council for Independent Research (now Independent Research Fund Denmark), Grant Nos. 7014-00258B, 4002-00272, 014-00258B, and 8021-00347B, and from the Hungarian National Research,Development and Innovation Fund, Grant No. NKFIH PD 134976. A.I.K. was supported by the U.S. National Science Foundation (No. CHE-1856342). A.I.K. is president and part-owner of Q-Chem, Inc. DATA AVAILABILITY The data that support the ﬁndings of this study are available within the article and its supplementary material . REFERENCES 1N. F. Scherer, J. L. 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1.4941417.pdf	A magnetoplasmonic electrical-to-optical clock multiplier C. J. Firby and A. Y. Elezzabi Citation: Appl. Phys. Lett. 108, 051111 (2016); doi: 10.1063/1.4941417 View online: http://dx.doi.org/10.1063/1.4941417 View Table of Contents: http://aip.scitation.org/toc/apl/108/5 Published by the American Institute of Physics Articles you may be interested in Magnetoplasmonic RF mixing and nonlinear frequency generation Applied Physics Letters 109, 011101 (2016); 10.1063/1.4955455A magnetoplasmonic electrical-to-optical clock multiplier C. J. Firbya)and A. Y . Elezzabi Ultrafast Optics and Nanophotonics Laboratory, Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta T6G 1H9, Canada (Received 1 December 2015; accepted 25 January 2016; published online 5 February 2016) We propose and investigate an electrical-to-optical clock multiplier, based on a bismuth-substituted yttrium iron garnet (Bi:YIG) magnetoplasmonic Mach-Zehnder interferometer (MZI). Transient magnetic ﬁelds induce a precession of the magnetization vector of the Bi:YIG, which in turnmodulates the nonreciprocal phase shift in the MZI arms, and hence the intensity at the output port. We show that the device is capable of modulation depth of 16.26 dB and has a tunable output frequency between 279.9 MHz and 5.6 GHz. Correspondingly, the input electrical modulationfrequency can be multiplied by factors of up to 2 :1/C210 3in the optical signal. Such a device is envisioned as a critical component in the development of hybrid electrical-optical circuitry. VC2016 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4941417 ] Nanoplasmonics provides a convenient platform for hybrid- izing complimentary-metal- oxide-semiconductor (CMOS) and optical technology on a single chip. Signiﬁcant researchefforts have been devoted to developing optical analogues of various electronic components. Coherent, integrable optical sources, in the form of nanoscale plasmonic lasers, can pro-vide on-chip optical signal generation. 1For encoding data onto the lightwave, optical switches and modulators can beimplemented with either electrical stimulus, utilizing elec-tro-optic 2or thermo-optic3effects, or optical stimulus, employing free-carrier-generation4or the Kerr nonlinearity.5 Additionally, nanoplasmonic logic gates, whose electricalanalogues form the foundation of CMOS logical circuits, have been demonstrated. 6These components mirror complex CMOS logic designs in use today. One key device that has yet to manifest in the optical re- gime is the clock multiplier (CM). The role of a CM is to scaleup the clock signal frequency at different locations within asystem. A nanoplasmonic CM would transfer an electrical sig- nal to the optical domain, upconvert the modulation fre- quency, and synchronize the electrical and optical logicsystems operating at different speeds. Thus, one requires anoptical material that facilitates a mechanism of generating reg-ular optical modulation between stimulating electrical pulses,in the absence of excitation. Previously explored materialproperties such as electro-optic, thermo-optic, and nonlineareffects are unable to display this characteristic, as they do not exhibit long-lasting, resonantly driven oscillations that can be mapped into the phase of the optical wave. A prime material candidate is a resonant magnetic sys- tem, such as bismuth-substituted yttrium iron garnet(Bi:YIG). In the optical regime, Bi:YIG possesses a largemagneto-optical ﬁgure of merit, exhibits a transparency win-dow around the communications wavelength of k¼1550 nm, 7and is well established for its high-speed mod- ulation capabilities via magnetization switching.8 When light propagates through a Bi:YIG waveguide in a transverse magnetic ﬁeld, the magnetization breaks the timereversal symmetry and imparts a nonreciprocal phase shift (NRPS) onto the optical mode. As such, the phase gained by forward and backward propagating modes will differ. A per- pendicular transient magnetic ﬁeld can exert a torque on the magnetization vector, changing the magnetization state.9 Upon conclusion of the transient pulse, the magnetization will precess around the static ﬁeld vector as it relaxes back to its initial state. This precession manifests as an oscillatory response of the perpendicular magnetization components, ini- tiating dynamic actuation of the NRPS. Integrating this wave- guide into a Mach-Zehnder interferometer (MZI) converts thedynamic NRPS into a sequence of optical clock pulses. In this letter, we present the design of an electrical-to- optical CM utilizing the NRPS in magnetoplasmonic wave- guides. Employing the transient magnetic ﬁeld from a nearby pulsed current clock signal, precession of the magnetization vector in a Bi:YIG magnetoplasmonic MZI can be excited, and consequently, the intensity at the output of the MZI ismodulated at a higher frequency. Our results show a tunable output frequency between 279.9MHz and 5.6 GHz and corre- sponding frequency multiplication factors between 1 :1/C2 10 3and 2 :1/C2103. The MZI under consideration is illustrated in Fig. 1(a). This device is composed of long-range dielectric-loaded mag- netoplasmonic waveguide (LRDLMPW) arms and photonicY-junctions for the input and output. The centers of the arms are 10 lm apart, while the Y-junctions connect the arms to input/output guides over 55 lm. One branch of each Y-junction is slightly longer, introducing a static phase bias of p=2r e l a - tive to the other arm. As such, the arms are then designed tothe length required to generate a p=2N R P S ,o r L p=2. The waveguides are built on top of a SiO 2substrate (nSiO 2¼1:444),10capped with a dAl2O3¼175 nm thick layer of Al 2O3(nAl2O3¼1:746)10and a dSi3N4¼175 nm thick layer of Si 3N4(nSi3N4¼1:977).11The gyrotropic waveguide cores are constructed from Bi:YIG and have dimensions of wYIG¼320 nm and dYIG¼400 nm. This material exhibits a refractive index of nYIG¼2.3, a saturation magnetization of l0MS¼9 mT, a speciﬁc Faraday rotation of hF¼0:25/C14/lm atk¼1550 nm,12and a Gilbert damping parameter ofa)Electronic mail: ﬁrby@ualberta.ca 0003-6951/2016/108(5)/051111/4/$30.00 VC2016 AIP Publishing LLC 108, 051111-1APPLIED PHYSICS LETTERS 108, 051111 (2016) a¼10/C04.13By incorporating an Ag layer ( nAg¼0:145 þ11:438i)14having dimensions wAg¼160 nm and dAg¼15 nm at the bottom of the Bi:YIG ridge, the structure becomes a LRDLMPW, having a characteristic propagation length of Lprop¼3.0 mm. The jEzj2proﬁles of the photonic and plasmonic modes are shown in Figs. 1(b) and1(c). The underlying layers can be fabricated with standard depositionand lithographic processes, while the Bi:YIG can be depos-ited through pulsed laser deposition, and etched to form the waveguide cores with a focussed ion beam. The static biasing ﬁeld is generated along the y-axis by an external magnet over the magnetoplasmonic arms, i.e.,H static¼h0;þHy;0i. Transient magnetic ﬁelds are applied to the arms as a result of current pulses passing through two Ag microwires (having a 2 lm/C22lm cross sectional area, length ofLp=2, and a separation of 16 lm), as shown in Fig. 1(a).T h e current pulses comprising the clock signal, I(t), resemble practi- cal square wave pulses with ﬁnite rise and fall times and a super-Gaussian form. As suc h, ﬁelds of opposite polarity, hðtÞ¼h 0;0;6hzðtÞi, are generated over the MZI arms. The NRPS is calculated through ﬁnite-difference-time- domain simulations. The gyrotropic effects of the Bi:YIG are modeled with an asymmetric permittivity tensor7 erMðÞ ¼n2 YIG iknYIG pMz MShF/C0iknYIG pMy MShF /C0iknYIG pMz MShF n2 YIG iknYIG pMx MShF iknYIG pMy MShF/C0iknYIG pMx MShF n2 YIG2 66666643 7777775: (1) Since the primary electric ﬁeld component of the plasmonic mode is E z, a NRPS will only occur when the magnetization vector, M,is oriented along the 6x-axis.As such, in calculating the NRPS of the waveguide, we consider M¼h þ MS;0;0i. The NRPS is found to be Db¼/C01:77 rad/mm. The NRPS is dependent on Mx, and thus, introducing a temporal variation in this component modulates Dband the interference condition accordingly. The temporal evolutionofMcan be modeled through the Landau-Lifshitz-Gilbert equation, which is given by 15 dM dt¼/C0l0c0 1þa2M/C2HstaticþhtðÞ ðÞ ½/C138 /C0l0c0a MS1þa2 ðÞM/C2M/C2HstaticþhtðÞ ðÞ ½/C138 ; (2) where c0is the gyromagnetic ratio. For the operation of a tunable CM, the magnitude of the static magnetic ﬁeld is crucial in the device operation as it deﬁnes initial/ﬁnal states of M, dictates the ferromagnetic resonance (FMR) frequency, and determines the precessionaxis of M. In this case, H static ¼h0;þHy;0iinitially satu- rates Malong the y-axis, M¼h0;þMS;0i. Short, transient magnetic ﬁeld pulses, h(t), tip Maway from this axis, and it precesses around the y-axis as it relaxes. This precession manifests as a decaying oscillatory response in MxandMz with a characteristic FMR at the Larmor frequency, /C23¼c0l0Hy=ð2pÞ.15 Figure 2depicts the Mx=MSamplitude of the preces- sional oscillations that can be excited as functions of thestatic magnetic ﬁeld, H y, and peak transient magnetic ﬁeld, hz;pk, for pulse widths, sp¼100 ps (Fig. 2(a)) and sp¼500 ps (Fig. 2(b)). When hðtÞ6¼0,Mis subject to an effective magnetic ﬁeld due to the superposition of h(t) and Hstatic. As such, Mbegins to precess around this resultant vector. When the transient ﬁeld is turned off (i.e., hðtÞ¼0), FIG. 1. (a) Illustration of the electrical-to-optical CM. (b) jEzj2proﬁle and geometry of the photonic waveguide. (c) jEzj2proﬁle and geometry of the plasmonic waveguide. FIG. 2. Plots of the excited Mx=MSprecession oscillation amplitude as func- tions of Hyand hz;pkfor (a) sp¼100 ps and (b) sp¼500 ps. The black dashed lines depict the contour where the Mx=MSoscillation amplitude is 1 (maximum), while the red dashed line marks the hz;pk¼Hyboundary. (c)–(e) The plane of precession around the effective magnetic ﬁeld when (c) hz;pk¼Hy, (d) hz;pk<Hy, and (e) hz;pk>Hy.051111-2 C. J. Firby and A. Y . Elezzabi Appl. Phys. Lett. 108, 051111 (2016)Mis subject only to Hstatic, and it precesses around the y-axis with a frequency, /C23. The magnitude of Mxdepends on the orientation of Mash(t) turns off. For maximum amplitude oscillations of Mx, the precession must begin when My¼0. This condition can be satisﬁed, as shown in Figs. 2(a) and 2(b), with an appropriate choice of hz;pk,Hy, and sp. Since the magnitude of Mis a constant value ( MS), the tip of the Mvector moves on the surface of a sphere of radius MS(denoted as the magnetization sphere). The initial deﬂec- tion of Mis set by its orientation and the direction of the effective magnetic ﬁeld vector. As such, the tip of Mtraverses around the magnetization sphere within the plane whose nor-mal is the effective ﬁeld vector, and which passes through the initial state (denoted as the plane of precession, or POP). For the condition M y¼0t ob es a t i s ﬁ e d , Mmust lie within the x-z plane. When hz;pk¼Hy, the effective ﬁeld vector makes an angle of 45/C14with both the y-a n d z-axes, and thus, the POP intersects the z-axis at one point (point Cin Fig. 2(c))o nt h e magnetization sphere. When hz;pk<Hy,t h ee f f e c t i v eﬁ e l d vector lies closer to the y-axis, and thus, the POP does not intersect the x-zplane at any point on the sphere (Fig. 2(d)). Interestingly, for hz;pk>Hy, the effective ﬁeld vector lies closer to the z-axis, and the POP intersects the x-zplane on the magnetization sphere, as shown in Fig. 2(e). Therefore, maxi- mum amplitude oscillations in Mxcan be excited. The arch patterns within Figs. 2(a)and2(b) reﬂect such dynamics. When hz;pk<Hy, the amplitude of the Mxoscilla- tions is <1, since the My¼0 condition can never be satisﬁed. However, when hz;pk>Hy, the trajectory of Msatisﬁes the My¼0 condition at two points (within the x-zplane) per revo- lution (points AandBin Fig. 2(e)). If the ﬁelds are chosen such that Mends at one of these states, Mxcan be made to oscillate between 61. At a ﬁxed magnitude of Hy, the corre- sponding hz;pkcan be applied such that Mis deﬂected from its initial state to point A. Increasing hz;pkfurther increases the effective precession frequency, allowing Mto overshoot point Aand reach point Bby the time h(t) concludes. These two points manifest as the two branches of the arch patterns in Figs. 2(a)and2(b). At high values of Hy, the effective ﬁeld vector shifts more towards the y-axis, and thus, the points A and Bcoalesce into a single point (point C)a n dt h et w o branches of the arch merge into one. Notably, points AandB should merge into point Cwhen hz;pk¼Hy. However, devia- tion from this is observed, especially in Fig. 2(b) for sp¼500 ps pulses. This deviation is the result of the nonzero rise and fall time of h(t), where Mis subject to a time varying effective ﬁeld vector. Here, Mis deﬂected as the effective ﬁeld vector moves, and when it reaches its peak, Mno longer resides at the initial state. Thus, the POP shifts and the condi- tion required to maximize the Mxamplitude changes. A ss h o w ni nF i g . 2(b), a long spallows Mto complete several revolutions around the effective ﬁeld vector, which is characterized by multiple arch es and coalescing points in the Hyvshz;pkplot, resulting in multiple frequencies in the output optical train. Clearly, such behaviour is undesirable for single frequency CM operation. Since the CM frequency is propor- tional to Hy, higher precession frequencies are more readily attainable when utilizing electri cal pulses of shorter durations. To map the long-lasting Mxoscillations into optical in- tensity modulation, the MZI arm length must be Lp=2. With aNRPS of Db¼/C01:77 rad/mm, Lp=2¼886:1lm. Such an interaction length provides the maximum NRPS and inten-sity modulation. To limit the current pulses ( I(t)) to a practi- cal range, we consider peak currents of no more than 1 A,which in the described transmission lines, generate peakmagnetic ﬁelds of 636 mT at the waveguides. Applying higher static ﬁelds, and accessing higher FMR frequencies, requires compensation for the reduction in M xoscillation amplitude by increasing Lp=2. An exemplary parameter set for which maximum optical modulation can be attained includes sp¼500 ps, l0hz;pk ¼19 mT (i.e., a peak current of 0.53 A), and l0Hy¼10 mT (corresponding to a CM output frequency of /C23CM¼279:9 MHz). Since higher /C23CMrequires shorter sp, we consider the scenario where sp¼100 ps. As shown in Fig. 2(a),f o ra maximum attainable ﬁeld of l0hz;pk¼36 mT, the CM fre- quencies between /C23CM¼279:9 MHz and /C23CM¼5:6 GHz are attainable for static ﬁeld biases of l0Hy¼10 mT and l0Hy¼200 mT, respectively. These ﬁeld conﬁgurations cor- respond to reduced Mx=MSoscillation amplitudes of 0.59 and 0.34 (or NRPS of Db¼/C01:04 rad/mm and Db¼/C00:60 rad/mm), respectively. Therefore, the MZI arm length mustbe increased to L p=2¼1510 :7lm and Lp=2¼2607 :4lm. Note that in each case, Lpropis longer than Lp=2, and thus, a complete p=2 NRPS can always be obtained. Figures 3(a)–3(c) display the MZI transmission as a func- tion of time. For sp¼500 ps (Fig. 3(a)),l0hz;pk¼19 mT and l0Hy¼10 mT produce an optical pulse train at /C23CM¼279:9 MHz, having transmission between /C01.85 dB and/C018.11 dB. Thus, a total modulation depth of 16.26 dB is attainable when the NRPS is oscillating maximally. Figure3(b) depicts the transmission for L p=2¼886:1lma n d Lp=2¼1510 :7lmi nr e s p o n s et oa sp¼100 ps pulse at l0hz;pk¼36 mT and l0Hy¼10 mT. At Lp=2¼886:1lm, the 279.9 MHz output oscillates between /C02.30 dB and /C010.96 dB, resulting in a modulation depth of 8.65 dB. Optimizing Lp=2to 1510.7 lm improves the modulation depth to 16.26 dB. Similarly, Fig. 3(c)shows the transmission for a FIG. 3. MZI transmission versus time for (a) sp¼500 ps, l0hz;pk¼19 mT, and l0Hy¼10 mT; (b) sp¼100 ps, l0hz;pk¼36 mT, and l0Hy¼10 mT; and (c) sp¼100 ps, l0hz;pk¼36 mT, and l0Hy¼200 mT. Note that the red curve represents the triggering pulse (not shown to scale).051111-3 C. J. Firby and A. Y . Elezzabi Appl. Phys. Lett. 108, 051111 (2016)sp¼100 ps pulse with l0hz;pk¼36 mT and l0Hy¼200 mT. These parameters produce a /C23CM¼5:6 GHz signal, modu- lated between /C03.04 dB and /C07.65 dB (a modulation depth of 4.61 dB) for Lp=2¼886:1lm. At the optimized Lp=2 ¼2607 :4lm, the modulation depth is improved to 16.26 dB. Similar to electronic CMs, the MZI must be able to respond to a continuous train of electrical pulses comprising the triggering clock signal. As the Mxresponse decays over the relaxation time of the Bi:YIG, one must set a minimum threshold for the magnetization (and NRPS) decay to initiate the next triggering cycle. As a design ﬁgure of merit, weassign such a threshold to the time when the M xamplitude drops to 50% of its initial value. The clock repetition fre- quency, frep, can then be adjusted over a wide range. For syn- chronous clock operation, one must account for a slight difference between the transit time for the initial tipping of M at the onset of the pulse train, and the tipping required to re-establish the original oscillation after decaying to 50%. As such, the perturbation between the ﬁrst two triggering pulses and the remaining train is on the order of 0.001% of the trig-gering clock frequency in the examples presented here. The three optimized exemplary cases discussed above are depicted under stimulus from a regular clock signal in Fig. 4. The s p¼500 ps electrical pulse train with l0hz;pk¼19 mTcan be applied at frep¼132.0 kHz (Fig. 4(a)). At /C23CM¼279:9 MHz, the CM exhibits a multiplication factor of 2:1/C2103. Similarly, Fig. 4(b) depicts the output due to an electrical pulse train of sp¼100 ps ( l0hz;pk¼36 mT) pulses with frep¼228.0 kHz, where the input frequency is multiplied b yaf a c t o ro f1 :2/C2103to/C23CM¼279:9M H z . N o t e t h a t d e - spite the same FMR frequency, th is case has a lower multiplier due to the threshold condition and the nonlinear relaxation of the precession. Since the decay p rocess is nonlinear, it takes less time for these oscillations to decay to 50% of their initial value compared to the case in Fig. 4(a), and as such, frepis greater and the multiplication factor is reduced. Figure 4(c) shows the output for the higher FMR, /C23CM¼5:6G H z .At r a i n ofsp¼100 ps pulses can be applied at frep¼4.9 MHz, result- i n gi na1 :1/C2103multiplication factor. Clearly, the MZI is ca- pable of large multiplication factors, in excess of 103,a n di s tunable over a wide range of frequencies. The repetition rate can be tailored at the cycle level, and hence, relaxing the setthreshold condition and employing detectors with higher sensi- tivity can lead to much larger multiplication factors. Notably, due to the low duty cycle of the electrical clock, the averagepower dissipation is less than 5 mW. In summary, we have presented the design for a magne- toplasmonic electrical-to-optical CM. Exploiting magneticprecession and the NRPS in an active Bi:YIG magnetoplas- monic MZI, we proposed a device capable of providing 16.26 dB modulation at frequencies ranging from 279.9 MHzto 5.6 GHz, and correspondingly upconverting the input sig- nal repetition rate by factors up to 2 :1/C210 3. Such a device is envisioned to satisfy crucial applications in the develop-ment of hybrid electrical-optical circuitry. This work was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC). 1R.-M. Ma, R. F. Oulton, V. J. Sorger, and X. Zhang, Laser Photonics Rev. 7, 1 (2013). 2C. Haffner, W. Heni, Y. Fedoryshyn, J. Niegemann, A. Melikyan, D. L. Elder, B. Baeuerle, Y. Salamin, A. Josten, U. Koch, C. Hoessbacher, F. Ducry, L. Juchli, A. Emboras, D. Hillerkuss, M. Kohl, L. R. Dalton, C.Hafner, and J. Leuthold, Nat. Photonics 9, 525 (2015). 3J. Gosciniak, S. I. Bozhevolnyi, T. B. Andersen, V. S. Volkov, J. Kjelstrup- Hansen, L. Markey, and A. Dereux, Opt. Express 18, 1207 (2010). 4S. Sederberg, D. Driedger, M. Nielsen, and A. Y. Elezzabi, Opt. Express 19, 23494 (2011). 5H. Lu, X. Liu, L. Wang, Y. Gong, and D. Mao, Opt. Express 19, 2910 (2011). 6Y. Fu, X. Hu, C. Lu, S. Yue, H. Yang, and Q. Gong, Nano Lett. 12, 5784 (2012). 7A. K. Zvezdin and V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials (IOP Publishing, Bristol, 1997). 8A. Y. Elezzabi and M. R. Freeman, Appl. Phys. Lett. 68, 3546 (1996). 9C. J. Firby and A. Y. Elezzabi, Optica 2, 598 (2015). 10E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, San Diego, 1998). 11A. Arbabi and L. L. Goddard, Opt. Lett. 38, 3878 (2013). 12S. E. Irvine and A. Y. Elezzabi, IEEE J. Quantum Electron. 38, 1428 (2002). 13H. Kurebayashi, O. Dzyapko, V. E. Demidov, D. Fang, A. J. Ferguson,and S. O. Demokritov, Nat. Mater. 10, 660 (2011). 14P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972). 15R. F. Soohoo, Microwave Magnetics (Harper & Row, New York, 1985).FIG. 4. Transmission response to a train of pulses with the following proper- ties: (a) sp¼500 ps, l0hz;pk¼19 mT, l0Hy¼10 mT, and frep¼132.0 kHz; (b)sp¼100 ps, l0hz;pk¼36 mT, l0Hy¼10 mT, and frep¼228.0 kHz; and (c) sp¼100 ps, l0hz;pk¼36 mT, l0Hy¼200 mT, and frep¼4.9 MHz. These plots depict the overall envelope of the transmission, and on this time scale, the individual optical pulses are indistinguishable. The optical pulsetrain occurs in the blue band, as shown in the insets. Note that the red curve represents the triggering pulses (not shown to scale).051111-4 C. J. Firby and A. Y . Elezzabi Appl. Phys. Lett. 108, 051111 (2016)
5.0006002.pdf	J. Chem. Phys. 152, 184108 (2020); https://doi.org/10.1063/5.0006002 152, 184108 © 2020 Author(s).Psi4 1.4: Open-source software for high- throughput quantum chemistry Cite as: J. Chem. Phys. 152, 184108 (2020); https://doi.org/10.1063/5.0006002 Submitted: 26 February 2020 . Accepted: 12 April 2020 . Published Online: 13 May 2020 Daniel G. A. Smith , Lori A. Burns , Andrew C. Simmonett , Robert M. Parrish , Matthew C. Schieber , Raimondas Galvelis , Peter Kraus , Holger Kruse , Roberto Di Remigio , Asem Alenaizan , Andrew M. James , Susi Lehtola , Jonathon P. Misiewicz , Maximilian Scheurer , Robert A. Shaw , Jeffrey B. Schriber , Yi Xie , Zachary L. Glick , Dominic A. Sirianni , Joseph Senan O’Brien , Jonathan M. Waldrop , Ashutosh Kumar , Edward G. Hohenstein , Benjamin P. Pritchard , Bernard R. Brooks , Henry F. Schaefer , Alexander Yu. Sokolov , Konrad Patkowski , A. Eugene DePrince , Uğur Bozkaya , Rollin A. King , Francesco A. Evangelista , Justin M. Turney , T. Daniel Crawford , and C. David Sherrill COLLECTIONS Paper published as part of the special topic on Electronic Structure Software Note: This article is part of the JCP Special Topic on Electronic Structure Software. ARTICLES YOU MAY BE INTERESTED IN NWChem: Past, present, and future The Journal of Chemical Physics 152, 184102 (2020); https://doi.org/10.1063/5.0004997 TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed- matter simulations The Journal of Chemical Physics 152, 184107 (2020); https://doi.org/10.1063/5.0004635 Essentials of relativistic quantum chemistry The Journal of Chemical Physics 152, 180901 (2020); https://doi.org/10.1063/5.0008432The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp PSI41.4: Open-source software for high-throughput quantum chemistry Cite as: J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 Submitted: 26 February 2020 •Accepted: 12 April 2020 • Published Online: 13 May 2020 Daniel G. A. Smith,1 Lori A. Burns,2 Andrew C. Simmonett,3 Robert M. Parrish,2 Matthew C. Schieber,2 Raimondas Galvelis,4 Peter Kraus,5 Holger Kruse,6 Roberto Di Remigio,7 Asem Alenaizan,2 Andrew M. James,8 Susi Lehtola,9 Jonathon P. Misiewicz,10 Maximilian Scheurer,11 Robert A. Shaw,12 Jeffrey B. Schriber,2 Yi Xie,2 Zachary L. Glick,2 Dominic A. Sirianni,2 Joseph Senan O’Brien,2 Jonathan M. Waldrop,13 Ashutosh Kumar,8 Edward G. Hohenstein,14 Benjamin P. Pritchard,1 Bernard R. Brooks,3Henry F. Schaefer III,10 Alexander Yu. Sokolov,15 Konrad Patkowski,13 A. Eugene DePrince III,16 U˘gur Bozkaya,17 Rollin A. King,18 Francesco A. Evangelista,19 Justin M. Turney,10 T. Daniel Crawford,1,8 and C. David Sherrill2,a) AFFILIATIONS 1Molecular Sciences Software Institute, Blacksburg, Virginia 24061, USA 2Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0400, USA 3National Institutes of Health – National Heart, Lung and Blood Institute, Laboratory of Computational Biology, Bethesda, Maryland 20892, USA 4Acellera Labs, C/Doctor Trueta 183, 08005 Barcelona, Spain 5School of Molecular and Life Sciences, Curtin University, Kent St., Bentley, Perth, Western Australia 6102, Australia 6Institute of Biophysics of the Czech Academy of Sciences, Královopolská 135, 612 65 Brno, Czech Republic 7Department of Chemistry, Centre for Theoretical and Computational Chemistry, UiT, The Arctic University of Norway, N-9037 Tromsø, Norway 8Department of Chemistry, Virginia Tech, Blacksburg, Virginia 24061, USA 9Department of Chemistry, University of Helsinki, P.O. Box 55 (A. I. Virtasen aukio 1), FI-00014 Helsinki, Finland 10Center for Computational Quantum Chemistry, University of Georgia, Athens, Georgia 30602, USA 11Interdisciplinary Center for Scientific Computing, Heidelberg University, D-69120 Heidelberg, Germany 12ARC Centre of Excellence in Exciton Science, School of Science, RMIT University, Melbourne, VIC 3000, Australia 13Department of Chemistry and Biochemistry, Auburn University, Auburn, Alabama 36849, USA 14SLAC National Accelerator Laboratory, Stanford PULSE Institute, Menlo Park, California 94025, USA 15Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, USA 16Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390, USA 17Department of Chemistry, Hacettepe University, Ankara 06800, Turkey 18Department of Chemistry, Bethel University, St. Paul, Minnesota 55112, USA 19Department of Chemistry, Emory University, Atlanta, Georgia 30322, USA Note: This article is part of the JCP Special Topic on Electronic Structure Software. a)Author to whom correspondence should be addressed: sherrill@gatech.edu ABSTRACT PSI4is a free and open-source ab initio electronic structure program providing implementations of Hartree–Fock, density functional theory, many-body perturbation theory, configuration interaction, density cumulant theory, symmetry-adapted perturbation theory, and coupled- cluster theory. Most of the methods are quite efficient, thanks to density fitting and multi-core parallelism. The program is a hybrid of C++ and J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Python, and calculations may be run with very simple text files or using the Python API, facilitating post-processing and complex workflows; method developers also have access to most of PSI4’s core functionalities via Python. Job specification may be passed using The Molecular Sciences Software Institute (MolSSI) QCSCHEMA data format, facilitating interoperability. A rewrite of our top-level computation driver, and concomitant adoption of the MolSSI QCARCHIVE INFRASTRUCTURE project, makes the latest version of PSI4well suited to distributed computation of large numbers of independent tasks. The project has fostered the development of independent software components that may be reused in other quantum chemistry programs. Published under license by AIP Publishing. https://doi.org/10.1063/5.0006002 .,s I. INTRODUCTION The PSIseries of programs for quantum chemistry (QC) has undergone several major rewrites throughout its history. This is also true of the present version, PSI4,1which bears little resemblance to its predecessor, PSI3. While PSI3is a research code aimed at pro- viding a handful of high-accuracy methods for small molecules, PSI4aims to be a user-friendly, general-purpose code suitable for fast, automated computations on molecules with up to hundreds of atoms. In particular, PSI4has seen the introduction of efficient multi-core, density-fitted (DF) algorithms for Hartree–Fock (HF), density functional theory (DFT), symmetry-adapted perturbation theory (SAPT),2,3second- and third-order many-body perturba- tion theory (MP2, MP3), and coupled-cluster (CC) theory through perturbative triples [CCSD(T)].4While PSI3is a stand-alone pro- gram that carries the assumption that QC computations were the final desired results and so offered few capabilities to interface with other program packages, PSI4is designed to be part of a soft- ware ecosystem in which quantum results may only be interme- diates in a more complex workflow. In PSI4, independent compo- nents accomplishing well-defined tasks are easily connected, and accessibility of key results through a Python interface has been emphasized. Although the PSIproject was first known as the BERKELEY pack- age in the late 1970s, it was later renamed to reflect its geographi- cal recentering alongside Henry F. Schaefer III to the University of Georgia. The code was ported to hardware-independent program- ming languages (Fortran and C) and UNIX in 1987 for PSI2; rewritten in an object-oriented language (C++), converted to free-format user input and flexible formatting of scratch files, and released under an open-source GPL-2.0 license in 1999 for PSI3;5reorganized around a programmer-friendly library for easy access to molecular integrals and related quantities and then unified into a single executable com- bining C++ for efficient QC kernels with Python for input parsing and for the driver code in 2009 for PSI4;6and, most recently, con- verted into a true Python module calling core C++ libraries, reor- ganized into an ecosystem with narrow data connections to external projects, opened to public development and open-source best prac- tices, and relicensed as LGPL-3.0 to facilitate use with a greater vari- ety of computational molecular sciences (CMS) software in 2017 for PSI4v1.1.1 These rewrites have addressed challenges particular to quan- tum chemistry programs, including the following: (i) users want a fully featured program that can perform computations with the latest techniques; however, (ii) QC methods are generally complex and difficult to implement; even more challenging is that (iii) QC methods have a steep computational cost and therefore must beimplemented as efficiently as possible; yet this is a moving target as (iv) hardware is widely varied (e.g., from laptops to supercomputers) and frequently changing. We also note an emerging challenge: (v) thermochemical,7machine learning,8force-field fitting,9etc. appli- cations can demand large numbers (105–108) of QC computations that may form part of complex workflows. PSI4has been designed with these challenges in mind. For (i)– (iii), we have created a core set of libraries that are easy to program with and that provide some of the key functionalities required for modern QC techniques. These include the LIBMINTS library that pro- vides simple interfaces to compute one- and two-electron integrals, the DFHELPER library to facilitate the computation and transforma- tion of three-index integrals for DF methods, and a library to build Coulomb and exchange (J and K) matrices in both the conventional and generalized forms that are needed in HF, DFT, SAPT, and other methods (see Refs. 1 and 6 and Sec. V B for more details). These libraries are also intended to address challenge (iv) above, as they have been written in a modular fashion so that alternative algo- rithms may be swapped in and out. For example, the LIBMINTS library actually wraps lower-level integrals codes, and alternative integrals engines may be used as described in more detail in Sec. V G. Sim- ilarly, the object-oriented JK library is written to allow algorithms adapted for graphics processing units (GPUs) or distributed-parallel computing. Challenge (v) is tackled by allowing computations via a direct application programming interface (API) and by encouraging machine-readable input and output. The PSI4NUMPY project10further simplifies challenge (ii), the implementation of new QC methods in PSI4. By making the core PSI4libraries accessible through Python, it is now considerably eas- ier to create pilot or reference implementations of new methods, since Python as a high-level language is easier to write, understand, and maintain than the C++ code. Indeed, because the libraries them- selves are written in an efficient C++ code, a Python implementation of a new method is often sufficient as the final implementation as well, except in the cases that require manipulations of three- or four- index quantities that are not already handled by the efficient core PSI4 libraries. For reasons of readability, maintainability, and flexibility, the entire codebase is migrated toward more top-level functions in Python. Although the library design makes it easier for developers to add new methods into PSI4, we believe an even more powerful approach is to create a software ecosystem that facilitates the use of external software components. Our build system, driver, and dis- tribution system have been rewritten specifically with this goal in mind, as discussed in Ref. 1 and Sec. VIII. The Python interface to PSI4and the recently introduced ability to communicate via QCSCHEMA further enhance this interoperability. Our recent moves to the more J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp permissive LGPL-3.0 license and to fully open development on a public GitHub site (https://github.com/psi4/psi4) are also meant to foster this ecosystem. Our recent infrastructure work since Ref. 1 is mainly focused on challenge (v), so that QC calculations can be routinely under- taken in bulk for use in various data analysis pipelines. As discussed in Sec. IV, PSI4has reworked its driver layout to simplify nested post-processing calls and greatly promote parallelism and archiving. Python within PSI4’s driver sets keywords according to the molec- ular system and method requested, allowing straightforward input files. Additionally, PSI4as a Python module (since v1.1, one can import psi4 ) means that codes may easily call PSI4from Python to perform computations and receive the desired quantities directly via Python, either through the application programming interface (PSIAPI ) or through JavaScript Object Notation (JSON) structured data. Below, we present an overview of the capabilities of PSI4(Sec. II). We then discuss the performance improvements in PSI4’s core QC libraries (Sec. V), the expanding ecosystem of software components that can use or be used by PSI4(Secs. VI and VII), and how the software driver has been rewritten to collect key quantities into a standard data format and to allow for parallel computation of independent tasks (Sec. IV). II. CAPABILITIES PSI4provides a wide variety of electronic structure methods, either directly or through interfaces to external community libraries and plugins. Most of the code is threaded using OPENMP to run effi- ciently on multiple cores within a node. The developers regularly use nodes with about six to eight cores, so performance is good up to that number; diminishing returns may be seen for larger numbers of cores. Hartree–Fock and Kohn–Sham DFT . Conventional, integral- direct, Cholesky, and DF algorithms are implemented for self- consistent field (SCF) theory. Thanks to the interface with the LIBXC library (see Sec. V A), nearly all popular functionals are available. The DF algorithms are particularly efficient, and computations on hundreds of atoms are routine. Energies and gradients are avail- able for restricted and unrestricted Hartree–Fock and Kohn–Sham (RHF, RKS, UHF, UKS), and restricted open-shell Hartree–Fock (ROHF). RHF and UHF Hessians are available for both conventional and DF algorithms. Perturbation theory .PSI4features Møller–Plesset perturbation theory up to the fourth order. Both conventional and DF imple- mentations are available for MP2, MP3, and MP2.5,11including gra- dients.1,12,13For very small molecules, the full configuration inter- action (CI) code can be used14,15to generate arbitrary-order MP n and Z-averaged perturbation theory (ZAPT n)16results. Electron affinities and ionization potentials can now be computed through second-order electron-propagator theory (EP2)17and the extended Koopmans’s theorem (EKT).18–20 Coupled-cluster theory .PSI4 supports conventional CC ener- gies up to singles and doubles (CCSD) plus perturbative triples [i.e., CCSD(T)]4for any single determinant reference (including RHF, UHF, and ROHF) and analytic gradients for RHF and UHF references.5For the DF, energies and analytic gradients up toCCSD(T) are available for RHF references.21–23Cholesky decom- position CCSD and CCSD(T) energies21and conventional CC224 and CC325energies are also available. To lower the computational cost of CC computations, PSI4supports26approximations based on frozen natural orbitals (FNOs)27–30that may be used to truncate the virtual space. Excited-state properties in PSI4are supported with equation-of-motion CCSD31,32and the CC2 and CC3 approxima- tions.33Linear-response properties, such as optical rotation,34are also available. PSI4also supports additional CC methods through interfaces to the CCT3 (see Sec. VI C 6) and MRCC programs.35 Orbital-optimized correlation methods . CC and Møller–Plesset perturbation methods are generally derived and implemented using the (pseudo)canonical Hartree–Fock orbitals. Choosing to instead use orbitals that minimize the energy of the targeted post-HF wavefunction has numerous advantages, including simpler analytic gradient expressions and improved accuracy in some cases. PSI4 supports a range of orbital-optimized methods, including MP2,36 MP3,37MP2.5,38and linearized coupled-cluster doubles (LCCD).39 DF energies and analytic gradients are available for all of these methods.40–43 Symmetry-adapted perturbation theory . PSI4 features wavefunction-based SAPT through the third-order to compute intermolecular interaction energies (IEs) and leverages efficient, modern DF algorithms.44–48PSI4also offers the ability to compute the zeroth-order SAPT (SAPT0) IEs between open-shell molecules with either UHF or ROHF reference wavefunctions.49–51In addition to conventional SAPT truncations, PSI4also features the atomic52and functional-group53partitions of SAPT0 (ASAPT0 and F-SAPT0, respectively), which partition SAPT0 IEs and components into con- tributions from pairwise atomic or functional group contacts. Fur- thermore, PSI4also offers the intramolecular formulation of SAPT0 (ISAPT0),54which can quantify the interaction between fragments of the same molecule as opposed to only separate molecules. The extensive use of core library functions for DF Coulomb and exchange matrix builds and integral transformations (see Sec. V B) has greatly accelerated the entire SAPT module in PSI4, with all SAPT0-level methods routinely deployable to systems of nearly 300 atoms (∼3500 basis functions); see also Secs. V C–V F for a new SAPT functionality. Configuration interaction .PSI4provides configuration interac- tion singles and doubles (CISD), quadratic CISD (QCISD),55and QCISD with perturbative triples [QCISD(T)]55for RHF references. It also provides an implementation56of full configuration interac- tion (FCI) and the restricted active space configuration interaction (RASCI) approach.57 Multi-reference methods .PSI4 provides conventional and DF implementations of the complete-active-space SCF (CASSCF)58,59 and restricted-active-space SCF (RASSCF).60Through the CHEMPS2 code, the density-matrix renormalization group (DMRG)61,62based CASSCF63and CASSCF plus second-order perturbation theory (CASPT2)64are available. The state-specific multireference CC method of Mukherjee and co-workers (Mk-MRCC) is implemented in PSI4with singles, doubles, and perturbative triples.65A comple- mentary second-order perturbation theory based on the same for- malism (Mk-MRPT2) also exists.66PSI4can perform multireference CC calculations through an interface to the MRCC program of Kállay and co-workers,35,67where high-order excitations (up to sex- tuples) as well as perturbative methods are supported. Additional J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp methods for strong correlation are available through the FORTE68–70 and V2RDM_CASSCF71(see Sec. VI C 5) plugins. Density cumulant theory .PSI4offers the reference implemen- tation of Density Cumulant Theory (DCT), which describes elec- tron correlation using the cumulant of the two-electron reduced density matrix (RDM) instead of a many-electron wavefunction.72 PSI4includes an implementation73of the original DCT formula- tion,72a version with an improved description of the one-particle density matrix (DC-12),74its orbital-optimized variants (ODC- 06 and ODC-12),75and more sophisticated versions that include N-representability conditions and three-particle correlation effects [ODC-13 and ODC-13( λ3)].76In particular, ODC-12 maintains CCSD scaling but is much more tolerant of open-shell character and mild static correlation.77,78Analytic gradients are available for DC-06, ODC-06, ODC-12, and ODC-13 methods.75,76,79 Relativistic corrections .PSI4can perform electronic structure computations with scalar relativistic corrections either by calling the external DKH library for up to fourth-order Douglas–Kroll– Hess (DKH)80,81or by utilizing the exact-two-component (X2C)82–92 approach to supplement the one-electron Hamiltonian of a non- relativistic theory for relativistic effects. At present, only the point nuclear model is supported. Composite and many-body computations .PSI4provides a sim- ple and powerful user interface to automate multi-component computations, including focal-point93–95approximations, complete- basis-set (CBS) extrapolation, basis-set superposition corrections [counterpoise (CP), no-counterpoise (noCP), and Valiron–Mayer functional counterpoise (VMFC)],96–98and many-body expansion (MBE) treatments of molecular clusters. These capabilities can all be combined to obtain energies, gradients, or Hessians, as discussed below in Sec. IV. For example, one can perform an optimization of a molecular cluster using focal-point gradients combining MP2/CBS estimates with CCSD(T) corrections computed in a smaller basis set, with counterpoise corrections. The MBE code allows for differ- ent levels of theory for different terms in the expansion (monomers, dimers, trimers, etc.) and also supports electrostatic embedding with point charges.III. PSIAPI Introduced in v1.1,1the PSI4API ( PSIAPI ) enables deployment within custom Python workflows for a variety of applications, including quantum computing and machine learning, by making PSI4 a Python module (i.e., import psi4 ). Using PSI4in this manner is no more difficult than writing a standard PSI4input file, as shown in the middle and left panels of Fig. 1, respectively. The true power of the PSIAPI lies in the user’s access to PSI4’s core C++ libraries and data structures directly within the Python layer. The PSIAPI thereby can be used to, e.g., combine highly optimized computational kernels for constructing Coulomb and exchange matrices from HF theory with syntactically intuitive and verbose Python array manipulation and linear algebra libraries such as NUMPY .99An example of the PSIAPI for rapid prototyping is given in Sec. V I 1. A. Psi4NumPy Among the most well-developed examples of the advantages afforded by the direct Python-based PSIAPI is the PSI4NUMPY project,10 whose goal is to provide three services to the CMS community at large: (i) to furnish reference implementations of computational chemistry methods for the purpose of validation and reproducibil- ity, (ii) to lower the barrier between theory and implementation by offering a framework for rapid prototyping where new methods could be easily developed, and (iii) to provide educational materials that introduce new practitioners to the myriad of practical considera- tions relevant to the implementation of quantum chemical methods. PSI4NUMPY accomplishes these goals through its publicly available and open-source GitHub repository,100containing both reference imple- mentations and interactive tutorials for many of the most common quantum chemical methods, such as HF, Møller–Plesset perturba- tion theory, CC, CI, and SAPT. Furthermore, since its publication in 2018, 17 separate projects to date have leveraged the PSI4NUMPY framework to facilitate their development of novel quantum chem- ical methods.101–117Finally, PSI4NUMPY is a thoroughly community- driven project; interested readers are highly encouraged to visit the repository100for the latest version of PSI4NUMPY and to participate in FIG. 1 . Input modes for PSI4. A coupled-cluster calculation is run equivalently through its preprocessed text input language (PSIthon; left), through the Python API ( PSIAPI ; middle), and through structured JSON input ( QCSCHEMA ; right). J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the “pull request” code review, issue tracking, or contributing the code to the project itself. B. Jupyter notebooks Inspired by notebook interfaces to proprietary computer alge- bra systems (e.g., Mathematica and Maple), a JUPYTER notebook is an open-source web application that allows users to create and share documents containing an executable code, equations, visualizations, and text.118JUPYTER notebooks are designed to support all stages of scientific computing, from the exploration of data to the cre- ation of a detailed record for publishing. Leveraging PSI4within this interface, therefore, provides interactive access to PSI4’s data struc- tures and functionalities. Visualization and analysis of properties such as geometry and orbitals can be facilitated with tools avail- able within The Molecular Sciences Software Institute’s119(MolSSI) QCARCHIVE120,121project. Additionally, the ability to combine exe- cutable code cells, equations, and text makes JUPYTER notebooks the perfect environment for the development and deployment of interactive educational materials, as illustrated by the PSI4NUMPY and PSI4EDUCATION122projects, or for living supplementary material that allows readers to reproduce the data analysis.123,124 IV. TASK-BASED DISTRIBUTED DRIVER The recursive driver introduced in 2016 for PSI4v1.0 to reor- ganize the outermost user-facing functions into a declarative inter- face has been refactored for PSI4v1.4 into the distributed driver that emphasizes high-throughput readiness and discretized com- munication through schema. In the earlier approach, the user employed one of a few driver functions [ energy() ,gradient() , optimize() ,hessian() ,frequency() ], and everything else was handled either by the driver behind the scenes [e.g., selecting ana- lytic or finite-difference (FD) derivatives] or through keywords (e.g., “mp2/cc-pv[t,q]z ,” “bsse_type=cp ,”dertype= “energy ”). When a user requested a composite computation that requires many individual computations (for example, a gradient calculation of a basis-set extrapolated method on a dimer with counterpoise cor- rection), internal logic directed this into a handler function (one each for many-body expansion, finite-difference derivatives, and composite methods such as basis-set extrapolations and focal-point approximations) which broke the calculation into parts; then, each part re-entered the original function, where it could be directed to the next applicable handler (hence a “recursive driver”). At last, the handlers called the function on an analytic task on a single chemi- cal system, at which point the actual QC code would be launched. However, the code to implement this functionality was complex and not easily extendable to the nested parallelism (among many-body, finite-difference, and composite) to which these computations are naturally suited. Because of these limitations, the internal structure of the driver has been reorganized so that all necessary QC input representations are formed before any calculations are run. The motivation for the driver refactorization has been the shift toward task-based computing and particularly integration with the MolSSI QCARCHIVE120,121project to run, store, and analyze QC compu- tations at scale. The QCARCHIVE software stack, collectively QCARCHIVE INFRASTRUCTURE , consists of several building blocks: QCSCHEMA125for JSON representations of QC objects, job input, and job output;QCELEMENTAL126for Python models (constructors and helper functions) for QCSCHEMA as well as fundamental physical constants and periodic table data; QCENGINE127for compute configuration (e.g., memory, nodes) and QCSCHEMA adaptors for QC programs; and QCFRACTAL128for batch compute setup, compute management, stor- age, and query. PSI4 v1.1 introduced a psi4 --json input mode that took in a data structure of molecular coordinates, drivers, methods, and keyword strings and returned a JSON structure with the requested driver quantity (energy, gradient, or Hessian), a success boolean, QCVariables (a map of tightly defined strings such as CCSD CORRELATION ENERGY orHF DIPOLE to float or array quantities), and string output. Since then, QC community input under MolSSI guidance has reshaped that early JSON into the current QCSCHEMA AtomicInput model capable of representing most non-composite computations. (“Atomic” here refers not to an atom vs a molecule but to a single energy/derivative on a single molecule vs multistage computations.) PSI4v1.4 is fully capable of being directed by and emitting MolSSI QCSCHEMA v1 (see Fig. 1, right) via psi4 --schema input orpsi4.run_qcschema(input) , where input is a Python dictionary, JSON text, or binary MESSAGEPACK ed structure of NUMPY arrays and other fields. Since PSI4speaks QCSCHEMA natively, its adap- tor in QCENGINE is light, consisting mostly of adaptations for older versions of PSI4and of schema hotfixes. Several other QC packages without QCSCHEMA input/ouput (I/O) have more extensive QCENGINE adaptors that construct input files from AtomicInput and parse output files into AtomicResult (discussed below). The distributed driver is designed to communicate through QCSCHEMA and QCENGINE so that the driver is independent of the community adoption of QCSCHEMA . TheAtomicInput data structure includes a molecule, driver function name, method and basis set (together “model”), and key- word dictionary, while the output data structure AtomicResult additionally includes the primary return scalar or array, any appli- cable of a fixed set of QCSCHEMA properties, as well as PSI4 spe- cialties such as QCVariables. Importantly, the customary output file is included in the returned schema from a PSI4 computa- tion. The driver has been revamped to use the AtomicInput and AtomicResult structures as the communication unit. In order for the above-mentioned handler procedures (now “Computer” objects) of the PSI4driver to communicate, specialized schemas that are supersets of AtomicResult have been developed. New fields have been introduced, including bsse_type andmax_nbody for ManyBodyComputer; stencil_size (the number of points in the finite-difference approximation) and displacement_space for FiniteDifferenceComputer ;scheme andstage forComposite Computer ; anddegeneracy andtheta_vib for the vibrational procedure. These contents are being optimized for practical use in PSI4and have been or will be submitted to MolSSI QCSCHEMA and QCELE- MENTAL for community input and review. A recently official schema already implemented in PSI4is for wavefunction data and encodes orbital coefficients, occupations, and other information in the stan- dard common component architecture (CCA) format.129This new schema is supported by the native PSI4infrastructure to permit seri- alization and deserialization of PSI4’s internal Wavefunction class that contains more fields than the schema stores. Although not yet used for communication, PSI4can also emit the BasisSet schema. The layered procedures of the distributed driver involve sums of J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp potentially up to thousands of schema-encoded results and are thus susceptible to numerical noise that a pure-binary data exchange would avoid. Nominally, JSON does not serialize NUMPY arrays or binary floats. However, the QCELEMENTAL /QCSCHEMA models support extended serialization through MESSAGEPACK130so that NUMPY arrays99 can be transparently and losslessly moved through the distributed driver. The task-oriented strategy for the distributed driver is illus- trated in Fig. 2. The user interface with the customary driver func- tions, Fig. 2(a), remains unchanged. If a single analytic computation is requested, it proceeds directly into the core QC code of PSI4(left- most arrow), but if any of the handlers are requested, the driver diverts into successively running the “planning” function of each prescribed procedure [Fig. 2(b) with details in Fig. 2(z)] until a pool of analytic single-method, single-molecule jobs in the QCSCHEMA AtomicInput format is accumulated. Although these could be run internally through the API counterpart of psi4 --schema [Fig. 2(c.i)], PSI4executes through QCENGINE so that other programs can be executed in place of PSI4if desired [Fig. 2(c–ii)]. An additional strategic benefit of running through QCENGINE is that the job pool can be run through QCFRACTAL [Fig. 2(c–iv)], allowing for simultane- ous execution of all jobs on a cluster or taking advantage of milder parallelism on a laptop, just by turning on the interface ( ∼5 addi- tional Python lines). The database storage and QCSCHEMA indexing inherent to QCFRACTAL means that individual jobs are accessible after completion; if execution is interrupted and restarted, completedtasks are recognized, resulting in effectively free coarse-grained checkpointing. Alternatively, for the mild boost of single-node par- allelism without the need to launch a QCFRACTAL database, one can run in the “snowflake” mode [Fig. 2(c.iii)], which employs all of QCFRACTAL ’s task orchestration, indexing, and querying technology, except the internal database vanishes in the end. The use of these modes in input is shown in Fig. 3. When all jobs in the pool are complete (all QCSCHEMAAtomicResult are present), the “assemble” functions of each procedure are run in a reverse order of invocation [Fig. 2(d) with details in Fig. 2(z)]. That is, model chemistry ener- gies are combined into composite energies by the CompositeCom- puter class, then composite energies at different displacements are combined into a gradient by the FiniteDifferenceComputer class, then gradients for different molecular subsystems and basis sets are combined into a counterpoise-corrected gradient by the ManyBodyComputer class, and finally, the desired energy, gra- dient, or Hessian is returned, Fig. 2(e). The schema returned by driver execution has the same apparent (outermost) struc- ture as a simple AtomicResult with a molecule, return result, properties, and provenance, so it is ready to use by other soft- ware expecting a gradient (like a geometry optimizer). However, each procedure layer returns its own metadata and the con- tributing QC jobs in a specialized schema, which is presently informal, so that the final returned JSON document is self- contained. Ensuring maintainability by merging code routes was given high priority in the distributed driver redesign: parallel and FIG. 2 . Structure of the distributed driver: see the final paragraph in Sec. IV for details. In brief, a user request (a) for a multi-molecule, multi-model-chemistry, or non-analytic derivative passes into planning functions (b) defined in procedure tiles (z) that generate a pool of QCSCHEMA for single-molecule, single-model-chemistry, analytic derivative inputs. These can run in several modes (c), depending on desired parallelism and recoverability. Completed QCSCHEMA passes through assembly functions (d) defined in procedure tiles (z) and denoted “ASM” that reconstitute (e) into the requested energy (“E”), gradient (“G”), or Hessian (“H”). J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Input file illustrating a CBS and many-body gradient run through the distributed driver in the continuous mode [white-background lines; Fig. 2(c.ii)], distributed mode with FractalSnowflake [Fig. 2(c.iii); additional blue-background lines], and distributed mode with the full storage and queuing power of QCFRAC- TAL[Fig. 2(c.iv); additional red-background lines]. The lower example is “free” when using QCFRACTAL since the components required for BSSE corrections have already been computed during the upper VMFC. While this example exposes the returned QCSCHEMAAtomicResult , the traditional syntax of grad = psi4.gradient(“HF/cc-pV[DT]Z,” bsse_type=“vmfc”) runs in the mode as in Fig. 2(c.ii) and is identical to the upper example. serial executions take the same routes, intra-project (API) and inter- project communications use the same QCSCHEMA medium, and (in a future revision) a generic QC driver calling PSI4can proceed through QCSCHEMA . V. NEW FEATURES AND PERFORMANCE IMPROVEMENTS A. DFT The DFT module now uses LIBXC131to evaluate the exchange- correlation terms. PSI4thus has access to 400+ functionals, of which ∼100 are routinely tested against other implementations. Modern functionals, such as ωB97M-V132and the SCAN family,133are nowavailable. Support for hybrid LDA functionals such as LDA0, pend- ing their release in a stable version of LIBXC , is also implemented. The new functional interface is Python-dictionary-based and uses LIBXC -provided parameters where possible. Additional capabilities for dispersion-inclusive, tuned range-separated, and double-hybrid functionals are defined atop LIBXC fundamentals. The interface also allows users to easily specify custom functionals, with tests and examples provided in the documentation. The DFT module in PSI4v1.4 is significantly faster than the one in PSI4v1.1, in both single-threaded and multi-threaded use cases. Recent versions are compared in Fig. 4, showing the speed improve- ments for the adenine ⋅thymine (A ⋅T) stacked dimer from the S22 database.135WithωB97X-D/def2-SVPD (Fig. 4, upper), this test case corresponds to 234 and 240 basis functions for each monomer and 474 for the dimer, while the problem size is approximately doubled in B3LYP-D3(BJ)/def2-TZVPD (Fig. 4, lower). Much of the speed improvement is due to improved handling of the DFT grids. Collocation matrices between basis functions and the DFT grid are now formed by an optimized library ( GAU2GRID ; Sec. VI B 3) and are automatically cached if sufficient memory is available, thus removing the need for their re-computation in every iteration. The whole module, including the generation of quadra- ture grids and collocation matrices, is now efficiently parallelized. The overall speedup between v1.1 and v1.4 is 1.9 ×on a single core. Notable speedups are obtained for range-separated functionals (e.g., theωB97X-D functional, see Fig. 4, upper), as the MemDFJK FIG. 4 . Wall-time comparison for the interaction energy of the adenine ⋅thymine stacked dimer from the S22 database with various versions of PSI4using 1 (darker green) to 16 (brown) threads, in multiples of two.134PSI4v1.4 data are obtained with the robust grid pruning algorithm. J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp algorithm is now implemented for this class of methods (see Sec. V B). As of PSI4v1.4, grid screening based on exchange-correlation weights is applied with a conservative default cutoff of 10−15. Grid pruning schemes are also implemented, the default robust scheme removing ∼30% of the grid points. Grid pruning on its own is responsible for a 1.3 ×single-core speedup in the case of the A ⋅T dimer with B3LYP-D3(BJ)/def2-TZVPD. However, a loss of accu- racy can be expected in the pruning of smaller grids ( <0.1 kcal mol−1 for IEs in the A24 database136). B. MemDFJK algorithm The SCF Coulomb (J) and exchange (K) builds are the corner- stone of all SCF-level operations in PSI4, such as SCF iterations, MP2 gradients, SAPT induction terms, SCF response, time-dependent DFT (TDDFT), and more. Over the past decade, the ability to per- form raw floating point operations per second (FLOPS) of modern central processing units (CPUs) has grown much faster than the speed of memory I/O, which can lead to memory I/O rather than raw FLOPS limiting operations. A large data copy quickly became the bottleneck of the PSI4v1.1 JK algorithm, especially when running on many concurrent cores. Examining the canonical K equations with the DF shows the following (using the Einstein summation convention): Dλσ=CiσCiλ, (1) ζPνi=(P∣νλ)Ciλ, (2) K[Dλσ]μν=ζPμiζPνi, (3) where iis an occupied index, Pis the index of the auxiliary basis function, and μ,ν,λ, andσare atomic orbital (AO) indices. The C,D,K, and ( P\|νλ) tensors are the SCF orbitals matrix, density matrix, exchange matrix, and three-index integral tensor (includ- ing auxiliary basis Coulomb metric term), respectively. Holding the (P\|νλ) quantity in a tensor TPνλoffers the benefit of a straightfor- ward optimized matrix–matrix operation in Eq. (2). However, this neglects the symmetricity and sparsity of the three-index integrals (P\|νλ). Accounting for both of these properties leads to the previ- ously stored form of TPνλνwhere theλindex was represented sparsely for each Pνpair by removing all duplicate or zero values; the spar- sity of the index λdepends on the value of νand hence the nota- tionλν. This form provides a highly compact representation of the (P\|νλ) tensor; however, the matrix–matrix operation to form ζPνiin Eq. (2) requires unpacking to a dense form, causing the previously mentioned data bottleneck. To overcome this issue, the new J and K builds in PSI4hold the (P\|νλ) quantity in a TνPλνrepresentation, where there is a unique mapping for the Pλindices for each νindex. While full sparsity can also be represented in this form, the symmetry of the AOs is lost, leading to this quantity being twice as large in the memory or disk. This form requires the Ciλνmatrix to be packed for every νindex for optimal matrix–matrix operations in Eq. (2). While both the TPνλν and TνPλνforms require packing or unpacking of tensors, the former requires QN2operations, while the latter requires N2ooperations, where Qis the size of the auxiliary index, Nis the number of basis functions, and ois the size of the occupied index. In practice, o≪Q,often resulting in 15 ×less data movement, and generally all but removing the bottleneck. This small data organization change combined with vector- ization and parallelization improvements has led to performance increases, especially for a high number of cores and when the system is very sparse, with the drawback of doubling the memory footprint. For a system of two stacked benzene molecules in the cc-pVDZ basis set (228 basis functions), the new JK algorithm is 2.6, 3.6, 3.7, and 4.3×faster than the old algorithm for 1, 8, 16, and 32 threads, respec- tively. For a more extensive system of 20 stacked benzene molecules with cc-pVDZ (2280 basis functions), the respective speedups are 1.5, 1.7, 2.1, and 2.2 ×.PSI4automatically detects which algorithm to use based on the amount of available memory. C. Additive dispersion models PSI4 specializes in providing convenient access to methods with additive dispersion corrections. Several have long been avail- able, such as Grimme’s three-component corrections to mean- field methods, HF-3c137and PBE-3c138(external via DFTD3139and GCP140executables), and the simpler pairwise additive schemes - D2141(internal code) and -D3142,143(external via a DFTD3 executable). Now also available are a similar correction to perturbation the- ory, MP2-D144(external via an MP2D145executable), and a non- local correction to DFT through the VV10 functional, DFT-NL146 (internal code). These are simply called gradient(“mp2-d”) or energy(“b3lyp-nl”) . See Table I for details of external software. PSI4v1.4 uses the -D3 correction in a new method, SAPT0-D. While SAPT0 has long been applicable to systems with upward of 300 non-hydrogen atoms by leveraging optimized DF routines for both JK builds and MP2-like E(20) dispandE(20) exch-dispterms, it is limited by theO(N5)scaling of the second-order dispersion ( Nproportional to the system size). By refitting the -D3 damping parameters against a large training set of CCSD(T)/CBS IEs and using the result in place of the analytic SAPT0 dispersion component, SAPT0-D at O(N4) scaling achieves a 2.5 ×speedup for systems with about 300 atoms (increasing for larger systems).147 The SAPT0-D approach is also applicable to the func- tional group partition of SAPT.53The resulting F-SAPT0-D has been applied to understand the differential binding of the β1- adrenoreceptor ( β1AR) (Fig. 5) in its active (G-protein coupled) vs inactive (uncoupled) forms to the partial agonist salbutamol. While experimentally determined ΔΔGbind was previously justified with respect to changes in the binding site geometry upon β1AR activa- tion,148F-SAPT0-D quantifies the contribution of each functional group contact, revealing that differential binding is due in large part to cooperativity of distant amino acid residues and peptide bonds, rather than only local contacts. D. SAPT(DFT) PSI4now provides SAPT(DFT),149also called DFT-SAPT,150 which approximately accounts for the intramolecular electron corre- lation effects that are missed in SAPT0 by including correlation-like effects found in DFT. The Hartree–Fock orbitals are replaced with Kohn–Sham orbitals,151and induction terms are solved using the coupled-perturbed Kohn–Sham equations. The long-range behav- ior that is important for dispersion interactions is known to be J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Quantum chemistry software that PSI4can use (upstream interaction). SoftwareaGroup Added License Language Comm.bCitecCapability Upstream required C-link LIBINT1 Valeev v1.0dLGPL-3.0 C C API 163 ... Two-electron and properties integrals LIBINT2 Valeev v1.4 LGPL-3.0 C++C++API 164 ... Two-electron and properties integrals LIBXC Marques v1.2 MPL-2.0 C C API 179 131 Definitions, compositions of density functionals GAU2GRID Smith v1.2 BSD-3-Cl C/Py C API 180 ... Gaussian collocation grids for DFT Upstream required Py-link QCELEMENTAL MolSSI v1.3 BSD-3-Cl Py Py API 126 121 Physical constants and molecule parsing QCENGINE MolSSI v1.4 BSD-3-Cl Py Py API 127 121 QC schema runner with dispersion and opt engines Upstream optional C-link DKH Reiher v1.0 LGPL-3.0 Fortran C API 181 80 and 81 Relativistic corrections LIBEFP Slipchenko v1.0eBSD-2-Cl C C API 182 183 Fragment potentials GDMA Stone v1.0 GPL-2.0 Fortran C API 184 185 Multipole analysis CHEMPS2 Wouters v1.0 GPL-2.0 C++C++API 186 187 and 188 DMRG and multiref. PT2 methods PCMSOLVER Frediani v1.0 LGPL-3.0 C++/Fortran C++API 189 190 Polarizable continuum/implicit solvent modeling ERD QTP v1.0dGPL-2.0 Fortran C API 191 192 Two-electron integrals SIMINT Chow v1.1 BSD-3-Cl C C API 193 165 Vectorized two-electron integrals AMBIT Schaefer v1.2 LGPL-3.0 C++/Py C++API 194 ... Tensor manipulations Upstream optional Py-link or exe DFTD3 Grimme v1.0 GPL-1.0 Fortran QCSCHEMA 139 142 and 143 Empirical dispersion for HF and DFT MRCC Kallay v1.0 pty C++/Fortran Text file ... 35 Arbitrary order CC and CI GCP Grimme v1.1 GPL-1.0 Fortran Py intf./CLI 140 137 and 138 Small-basis corrections PYLIBEFP Sherrill v1.3 BSD-3-Cl C++/Py Py API 195 ... Python API for libefp MP2D Beran v1.4 MIT C++QCSCHEMA 145 144 Empirical dispersion for MP2 CPPE Dreuw v1.4 LGPL-3.0 C++/Py Py API 196 197 Polarizable embedding/explicit solvent modeling ADCC Dreuw v1.4 GPL-3.0+pty C++/Py Py API 198 113 Algebraic-diagrammatic construction methods aBinary distributions available from Anaconda Cloud for all projects except for MRCC . For the channel in conda install <project >-c<channel >, usepsi4 except for ADCC fromadcc and GAU2GRID ,QCELEMENTAL , and QCENGINE fromconda-forge , the community packaging channel. bMeans by which PSI4 communicates with the project. cThe first reference is a software repository. The second is theory or software in the literature. dNo longer used. LIBINT1 last supported before v1.4. ERD last supported before v1.2. eSince v1.3, LIBEFP called through PYLIBEFP . problematic for generalized gradient approximation (GGA) func- tionals, and in DFT-SAPT, this is corrected by gradient-regulated asymptotic correction (GRAC)152in obtaining the Kohn–Sham orbitals. Dispersion energies are obtained by solving for the TDDFT propagator of each monomer and integrating the product of the propagators over the frequency domain.153,154In PSI41.4, we have improved the TDDFT dispersion capabilities to allow hybrid ker- nels in the TDDFT equations,155which can significantly improve accuracy when hybrid functionals are used to determine the orbitals.150,156E. SAPT0 without the single-exchange approximation The SAPT module in PSI4now has an option to compute the second-order SAPT0 exchange corrections E(20) exch-ind, respand E(20) exch-disp without the use of the common S2approximation, that is, using the complete antisymmetrizer in the expressions instead of its approximation by intermolecular exchanges of a single electron pair. The working equations for the non-approximate second-order cor- rections were derived and implemented for the first time in Refs. 157 and 158 in the molecular-orbital (MO) form prevalent in the classic J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . F-SAPT0-DD3M(0)/jun-cc-pVDZ analysis of 459 atoms (5163 orbitals and 22 961 auxiliary basis functions) from the β1AR–salbutamol co-crystal (PDB: 6H7M). (Left) Geometry of ligands (wide sticks) and residues (thin sticks) within 7 Å. (Right) Order-2 F-SAPT difference analysis of an active vs an inactive complex, with functional groups colored by contribution to ΔΔEint(red: more attractive in the activated state; blue: more attractive in the inactive state; color saturation at ±10 kcal mol−1). SAPT developments. We have recast the nonapproximate formu- las for E(20) exch-ind, respand E(20) exch-dispin Refs. 157 and 158 into the AO form and implemented them efficiently in PSI4with DF. As these AO- based expressions have not been published before, we present them together with an outline of their derivation in the supplementary material. Thanks to this new development, the entire SAPT0 level of theory (but not higher levels such as the second-order, SAPT2) is now available in PSI4without the single-exchange approxima- tion. Preliminary numerical tests show157–159that the replacement ofE(20) exch-disp(S2)with its nonapproximated counterpart introduces inconsequential changes to the SAPT0 interaction potentials at short intermolecular separations. In contrast, the full E(20) exch-ind, respvalues often deviate significantly from E(20) exch-ind, resp(S2)at short ranges, especially for interactions involving ions.160At the usual SAPT0 level (as defined, e.g., in Ref. 161), this difference between E(20) exch-ind, resp and E(20) exch-ind, resp(S2)cancels out when the δE(2) HFterm that approx- imates the higher-order induction and exchange induction effects from a supermolecular HF calculation is taken into account. How- ever, the removal of the S2approximation from second-order SAPT0 will significantly affect SAPT results computed without the δE(2) HF correction. F. SF-SAPT An open-shell SAPT feature that is currently unique to PSI4is the ability to compute the leading exchange term, E(10) exch(S2), for an arbitrary spin state of the interacting complex, not just its highest spin state. This spin-flip SAPT (SF-SAPT ) method was introduced in Ref. 162 and so far applies to the interaction between two open- shell systems described by their ROHF determinants. Such an inter- action leads to a bundle of asymptotically degenerate states of the interacting complex, characterized by different values of the spin quantum number S. These states share the same values of all electro- static, induction, and dispersion energies, and the splitting between them arises entirely out of electron exchange. In such a case, the SF-SAPT approach implemented in PSI4can provide an inexpensive [cost is similar to the standard E(10) exch(S2)] and qualitatively correct first-order estimate of the splittings between different spin states of the complex. In addition, all terms can be computed usingstandard SCF JK quantities and have been implemented within PSI4 in a PSI4NUMPY formalism, as the best performance can be achieved without any additional compiled code. G. Libint2 and Simint The LIBINT package163has been the default engine for two- electron integrals since the development of PSI3two decades ago. Allowing arbitrary levels of angular momentum and numerous inte- gral kernels, LIBINT has proven to be a reliable tool for generating the integrals that are central to QC. However, modern CPUs increas- ingly derive their power from a combination of multi-core and single instruction, multiple data (SIMD) technologies, rather than the reg- ular strides in clock speed that were realized around the time of PSI3’s development. While PSI4has exploited multi-core technologies for some time via OPENMP , its SIMD capabilities were previously limited to the linear algebra libraries used to power SCF and post-HF meth- ods. In PSI4v1.4, the LIBINT package has been superseded by LIBINT2 ,164 which partially exploits SIMD capabilities by vectorizing the work needed for a given shell quartet, making it better suited for mod- ern computer architectures. LIBINT2 permits additional integral ker- nels, including the Yukawa- and Slater-type geminal factors, which expand the range of DFT and explicitly correlated methods that may be implemented. LIBINT2 is also preferable from a software sustain- ability perspective as it is actively maintained and developed, unlike the original LIBINT . Although LIBINT2 is now the default integrals engine, PSI4is writ- ten to allow the use of alternative integrals packages, and an interface toSIMINT165,166is also provided. SIMINT was designed from the begin- ning with SIMD parallelism in mind. By reordering shell pairs to be grouped by common angular momentum classes, SIMINT achieves a compelling level of vectorization on the latest AVX512 chipsets. The PSI4integrals interface has been generalized to allow the shell pairs to be given in an arbitrary order and to account for the possibility of batching among them, thus allowing SIMINT to take full advantage of its approach to vectorization. H. SCF guesses The reliability of the atomic solver used for the superposi- tion of atomic densities167,168(SAD) initial guess has been greatly improved in PSI4, and the SAD guess has been made the default J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp also for open-shell and restricted open-shell calculations, result- ing in significantly faster convergence, especially for systems con- taining heavy atoms such as transition metal complexes. Although powerful in many cases, the SAD guess does not yield molecu- lar orbitals, and it may thereby be harder to build a guess with the wanted symmetry. The traditional alternatives to SAD that do yield molecular orbitals, the core orbital guess or the generalized Wolfsberg–Helmholz169modification thereof, fail to account for electronic screening effects whose importance increases rapidly with the increasing nuclear charge, resulting in horrible performance.170 However, guesses that both account for electronic screening and yield guess orbitals have recently been described in Ref. 170 and are now implemented in PSI4: an extended Hückel guess employing the atomic orbitals and orbital energies from the SAD solver, the SAD natural orbitals (SADNO) guess, and the superposition of atomic potentials (SAP) guess that constructs a guess Fock matrix from a sum of atomic effective potentials computed at the complete-basis- set limit.171,172With the improvements to SAD and the introduction of the three novel guesses, PSI4can be applied even to more challeng- ing open-shell and transition metal systems. Calculations are now possible even in overcomplete basis sets, as redundant basis func- tions are removed automatically by default in PSI4via the pivoted Cholesky decomposition procedure.173,174 I. TDDFT We have recently added time-dependent DFT capabilities using either the full TDDFT equations [also known as the random- phase approximation (RPA)] or the Tamm–Dancoff approximation (TDA).175The former yields a generalized eigenvalue problem, and our solver leverages the Hamiltonian structure of the equations to ensure robust convergence.176The latter corresponds to a Hermi- tian eigenvalue problem, and we employ a Davidson solver.177Theexcitation energies and vectors are obtained from the following gen- eralized eigenvalue problem, also known as the response eigenvalue problem : (A B B∗A∗)(Xn Yn)=ωn(1 0 0−1)(Xn Yn). (4) The excitation eigenvectors, (Xn,Yn)T, provide information on the nature of the transitions and can be used to form spectroscopic observables, such as oscillator and rotatory strengths. The AandB matrices appearing on the left-hand side are the blocks of the molec- ular electronic Hessian178whose dimensionality is ( ov)2, with oand vbeing the number of occupied and virtual MOs, respectively. Due to this large dimensionality, rather than forming AandBexplicitly, one instead uses subspace iteration methods to extract the first few roots. In such methods, the solutions are expanded in a subspace of trial vectors bi, and the most compute- and memory-intensive oper- ations are the formation and storage of the matrix–vector products (A+B)biand ( A−B)bi. These matrix–vector products are equiv- alent to building generalized Fock matrices; the efficient JK-build infrastructure of PSI4(Sec. V B) can thus be immediately put to use also for the calculation of TDDFT excitation energies. In fact, con- struction of these product vectors is the only part written in C++. All other components, including the subspace iteration techniques, are written in Python for easy readability and maintainability. Follow- ing our design philosophy, we have written the required subspace solvers for the response eigenvalue problems in a generic way, so that they may be reused for future features. 1. Example of rapid prototyping To illustrate the use of PSI4and PSI4NUMPY to rapidly implement new features, Fig. 6 shows an easy oscillator strength implemen- tation at the Python layer. Excitations are obtained by calling the FIG. 6 . Example Python implementation of TDDFT oscillator strengths. J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . UV-Vis spectrum of rhodamine 6G at the PBE0/aug-pcseg-2 level of theory. The spectra computed using full TDDFT (RPA) and the Tamm–Dancoff approximation (TDA) are reported in blue and orange, respectively. tdscf_excitations() function, and dipole moment integrals are calculated trivially in four lines of code by accessing the occu- pied and virtual parts of the SCF coefficient matrix and the dipole moment integrals from LIBMINTS . The oscillator strengths are then computed from the MO basis electric dipole moment integrals ⟨ϕa∣ˆμ∣ϕi⟩and the right excitation vectors Xn+Ynas follows: f=2 3ωn∑ u=x,y,zocc ∑ ivir ∑ a∣(Xn+Yn)ia⟨ϕa∣ˆμu∣ϕi⟩∣2. (5) Figure 7 shows an example UV-Vis spectrum using these oscillator strengths, as fitted by applying a Gaussian-shaped broadening to the computed excitation energies. We are also working on the imple- mentation of gauge-including atomic orbitals (London orbitals) to enable magnetic response evaluations needed to calculate properties such as optical rotation and electronic circular dichroism. VI. SOFTWARE ECOSYSTEM Like all QC packages, PSI4strives to continuously expand its capabilities to advance research in both method development and applications. New methods are introduced frequently in electronic structure theory, and it would be a challenge to implement all the latest advances. The PSI4team prefers to encourage the development of reusable libraries, so that new methods need to be implemented only once (by the experts) and can then be adopted by any QC code with merely a short, custom interface. This ecosystem-building approach has the advantages of (i) not binding a community library’s use to a single software package, (ii) encouraging smaller software projects that are more modular in function and ownership and more localized in (funding) credit, and (iii) facilitating the propagation of new features and bug fixes by using a generic interface rather than embedding a code snapshot. Since v1.1, PSI4has added new projects to its ecosystem, contributed back to existing projects, and disgorged some of its own code into projects that are more tightly defined. Dis- cussed below is a selection of illustrative or newly interfaced projects. The full ecosystem of external, connected software is collected intoTable I, code used by PSI4(upstream packages), and Table II, code that uses PSI4(downstream packages). A. Sustainability through community libraries The introduction of LIBINT2 and LIBXC not only provides new fea- tures (see Secs. V G and V A, respectively) but also results in substan- tial simplifications to the codebase. The previous version of LIBINT only provided the recursion routines, relying on the calling pro- gram to provide the fundamental s-type integrals used as the start- ing point. There were also restrictions on the angular momentum ordering among the four centers, requiring bookkeeping to apply permutations to the resulting integrals in the case where reorder- ings were necessary to satisfy these requirements. Furthermore, LIBINT1 provided only the minimal number of integral derivatives required by translational invariance,239,240requiring the calling code to compute the missing terms by application of the relationships. The combination of applying permutations and translational invari- ance amounted to over 3000 lines of code in previous PSI4versions, primarily due to the complexity introduced by second derivative integrals. In LIBINT2 , the fundamental integrals are provided and the translational invariance is applied automatically for derivatives, and the shells can be fed in any order of the angular momenta. With these tasks outsourced to LIBINT2 , the latest PSI4codebase is significantly cleaner and more maintainable. With the transition to the LIBXC131library for DFT calculations, in accordance with the modular development model, PSI4gains con- tinuous fixes and new features, which is especially important as none of the primary PSI4development groups specialize in DFT. Thanks to LIBXC ,PSI4now supports over 400 functionals of various rungs. Final DFT compositions suitable for energy() are now defined by LIBXC and are directly subsumed into PSI4’s functional list, making for a more maintainable code. In cooperation with the LIBXC upstream, the PSI4authors have contributed an alternate CMAKE build system and a Python API, PYLIBXC , to LIBXC , and also provided help in porting to Windows. B. Launching community libraries 1. QCElemental When the needs of ongoing research projects outgrew LIB- MINTS ’s C++ parsing of molecule specification strings, a redesign was implemented in Python and transferred to QCELEMENTAL to serve as the backend to QCSCHEMA Molecule validation. The resulting code is easily extensible, mirrors the schema (though with additional fields to accomodate PSI4’s Z-matrix and deferred geometry finaliza- tion features), and accepts and returns dictionary-, schema-, array- , or string-based representations. Additionally, it performs strong physics-based validation and defaulting for masses, mass numbers, total and fragment charges and multiplicities, and basis function ghosting, saving considerable validation code in PSI4as a QCELEMENTAL client. QCELEMENTAL additionally provides a light Python interface over NIST CODATA and periodic table data and other “look-up” quanti- ties such as van der Waals and covalent radii. By switching to QCELE- MENTAL API calls in PSI4’s Python code and using its header-writing utilities for the C++ code, readability has improved, and datasets are easier to update. J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcpTABLE II . Chemistry software that can use PSI4(downstream interaction). SoftwareaGroup V.bLicense Language Comm.cCitedPSI4provides Downstream optional C-link, plugins V2RDM_CASSCF DePrince v1.0 GPL-2.0 C++/Fortran C++API 71 199 Backend for variational 2-RDM-driven CASSCF FORTE Evangelista v1.0 LGPL-3.0 C++/Py C++API 70 68 and 69 Backend for multiref. many-body mtds and sel. CI CCT3 Piecuch v1.1 LGPL-3.0 Fortran C++API 200 201 and 202 Backend for actv-sp CCSDt, CC(t;3), CR-CC(2,3) GPU_DFCC DePrince v1.2 GPL-2.0 C++/Cuda C++API 203 204 Backend for GPU-accelerated DF-CCSD and (T) Downstream optional Py-link or exe WEBMO Polik v1.0 pty Java/Perl PSIthon ... 205 QC engine for GUI/web server MOLDEN Schaftenaar v1.0 pty Fortran Molden file 206 207 Orbitals for orbital/density visualization JANPA Bulavin v1.0 BSD-4-Cl Java Molden file 208 209 Orbitals for natural population analysis (NPA) PSI4NUMPY Smith v1.1 BSD-3-Cl Py PsiAPI 100 10 QC essentials for rapid prototyping and QC edu. PSI4EDUCATION McDonald v1.1 BSD-3-Cl Py PsiAPI 210 122 QC engine for instructional labs PSIOMM Sherrill v1.1 BSD-3-Cl Py PsiAPI 211 ... Self for interface between PSI4and OPENMM HTMD/PARAMETERIZE Acellera v1.1 pty Py PSIthon 212 213 and 214 QC engine for force-field parametrization for MD GPUGRID De Fabritiis v1.1 pty Py PSIthon 215 216 QC torsion scans for MD-at-home PYREX Derricotte v1.1 BSD-3-Cl Py PsiAPI 217 ... Engine for reaction coordinate analysis SNS-MP2 D. E. Shaw v1.2 BSD-2-Cl Py PsiAPI 218 219 Backend for spin-network-scaled MP2 method RESP Sherrill v1.2 BSD-3-Cl Py PsiAPI 220 115 ESP for restrained ESP (RESP) fitting QCENGINE MolSSI v1.2 BSD-3-Cl Py QCSCHEMA 127 121 QC engine for QC schema runner QISKIT-AQUA IBM v1.2 Apache-2.0 Py PSIthon 221 ... Engine for quantum computing algorithms MS QUANTUM Microsoft v1.2 MIT C#/Q# PsiAPI 222 ... Engine for quantum computing algorithms ORION OpenEye v1.2 pty Go/Py PsiAPI ... ... QC engine for drug-design workflow CRYSTALATTE Sherrill v1.2 LGPL-3.0 Py PSIthon 223 224 QC and MBE engine for crystal lattice energies OPENFERMION Google v1.3 Apache-2.0 Py PSIthon 225 226 Engine for quantum computing algorithms OPENFERMION-PSI4 Google v1.3 LGPL-3.0 Py PSIthon 227 226 Self for interface between PSI4and OpenFermion QCDB Sherrill v1.3 BSD-3-Cl Py QCSCHEMA 228 ... Engine for QC common driver OPTKING King v1.3 BSD-3-Cl Py QCSCHEMA 229 ... Gradients for geometry optimizer PSIXAS Gryn’ova v1.3 GPL-3.0 Py PsiAPI 230 ... Backend for x-ray absorption spectra FOCKCI Mayhall v1.3 BSD-3-Cl Py PsiAPI 231 116 CAS engine for Fock-space CI ASE ASE v1.4 LGPL-2.1 Py PsiAPI 232 233 QC engine for CMS code runner I-PI Ceriotti v1.4 GPL-3.0 Fortran/Py PsiAPI 234 235 QC gradients for MD runner MDI MolSSI v1.4 BSD-3-Cl C PsiAPI 236 ... QC engine for standardized CMS API GEOMETRIC Wang v1.4eBSD-3-Cl Py QCSCHEMA 237 238 QC gradients for geometry optimizer QCFRACTAL MolSSI v1.4 BSD-3-Cl Py QCSCHEMA 128 121 QC engine for database and compute manager aBinary distributions available from Anaconda Cloud for some projects. For the channel in conda install <PROJECT >-C<CHANNEL >, use psi4 for V2RDM_CASSCF ,GPU_DFCC ,SNS-MP2 ,RESP ,OPENFERMION , and OPENFERMION -PSI4; ACELLERA forHTMD/PARAMETERIZE ; andconda-forge , the community packaging channel, for QCENGINE ,ASE,MDI,GEOMETRIC , and QCFRACTAL . bEarliest version of PSI4 with which software works. cApart from compiled plugins that interact directly with PSI4’s C++ layer, downstream projects use established file formats such as Molden or one of the three input modes of Fig. 1. dThe first reference is a software repository. The second is theory or software in the literature. eGeomeTRIC has called PSI4 through PSIthon since v1.0. QCENGINE has driven geomeTRIC to drive PSI4 through QCSCHEMA since v1.3. PSI4 can itself call geomeTRIC through QCSCHEMA since v1.4. J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 2. QCEngine PSI4has long supplemented its internal empirical dispersion capabilities (Sec. V C) with external projects, namely, DFTD3 and MP2D executables. These were run via a Python interface that addition- ally stores fitting and damping parameters at the functional level, so that the programs are used solely for compute and not for inter- nal parameters. Since operation is independent of PSI4, the Python interfaces have been adapted to QCSCHEMA and moved to the QCENGINE repository where they can be of broader use. 3. Gau2Grid Improvements to the PSI4DFT code highlighted a bottleneck at the computation of the collocation matrix between basis functions and the DFT grid. It was found that the simple loops existing in PSI4 did not vectorize well and exhibited poor cache performance. Much in the same way that modern two-electron libraries work, GAU2GRID180 begins with a template engine to assist in writing unrolled C loops for arbitrary angular momentum and up to third-order derivatives. This template engine also allows multiple performance strategies to be employed and adjusted during code generation, depending on the angular momentum, the derivative level of the requested matrix, and the hardware targeted. GAU2GRID also has a Python interface to allow usage in Python programs that need fast collocation matrices. 4. PylibEFP In the course of shifting control of SCF iterations from C++ to Python, it became clear that the effective fragment potential241,242 (EFP) capabilities through Kaliman and Slipchenko’s LIBEFP library183 would be convenient in Python. Since LIBEFP provides a C interface, a separate project of essentially two files, PYLIBEFP ,195wraps it into an importable Python module. PYLIBEFP includes a full test suite, conve- nient EFP input parsing, and an interface amenable to schema com- munication (a QCENGINE adaptor is in progress). PSI4employs PYLIBEFP for EFP/EFP energies and gradients and EFP/SCF energies. C. Selected new features from community libraries 1. adcc ADC-connect ( ADCC ),113a hybrid Python/C++ toolkit for excited-state calculations based on the algebraic-diagrammatic con- struction scheme for the polarization propagator (ADC),243–245 equips PSI4with ADC methods (in-memory only) up to the third order in perturbation theory. Expensive tensor operations use an efficient C++ code, while the entire workflow is controlled by Python. PSI4and ADCC can connect in two ways. First, PSI4can be the main driver; here, method keywords are given through the PSI4 input file and ADCC is called in the background. Second, the PSI4 Wavefunction object from a SCF calculation can be passed to ADCC directly in the user code; here, there is more flexibility for complex workflows or for usage in a JUPYTER notebook. 2. SNS-MP2 McGibbon and co-workers219applied a neural network trained on HF and MP2 IEs and SAPT0 terms to predict system-specific scaling factors for MP2 same- and opposite-spin correlation ener- gies to define the spin-network-scaled, SNS-MP2, method. This hasbeen made available in a PSI4pure-Python plugin218so that users can callenergy(“sns-mp2”) , which manages several QC calcula- tions and the model prediction in the background and then returns an IE likely significantly more accurate than conventional MP2.219 By using PSI4’s export of wavefunction-level arrays to Python, the developers were able to speed up calculations through custom den- sity matrix manipulations of basis projection, fragment stacking, and fragment ghosting. 3. CPPE PSI4now supports the polarizable embedding (PE) model246,247 through the CPPE library.197In the PE model, interactions with the environment are represented by a multi-center multipole expansion for electrostatics, and polarization is modeled through dipole polar- izabilities usually located at the expansion points. The interface to the CPPE library is entirely written in Python and supports a fully self- consistent description of polarization for all SCF methods inside PSI4. In the future, PE will also be integrated in a fully self-consistent man- ner for molecular property calculations and TDDFT. Integration of CPPE motivated efficiency improvements to the electric field integrals and multipole potential integrals, which also benefit the related EFP method. 4. GeomeTRIC Wang and Song237,238developed a robust geometry optimiza- tion procedure to explicitly handle multiple noncovalently bound fragments using a translation-rotation-internal coordinate (TRIC) system. Their standalone geometry optimizer, GEOMETRIC , supports multiple QC packages including PSI4through a command-line inter- face. QCENGINE offers a GEOMETRIC procedure, allowing PSI4and oth- ers to use the new optimizer with a Python interface. The latest PSI4release adds native GEOMETRIC support, allowing users to spec- ify the geometry optimizer within an input, e.g., optimize (..., engine= “geometric ”). 5. v2rdm_casscf PSI4can perform large-scale approximate CASSCF computa- tions through the v2rdm_casscf plugin,71which describes the elec- tronic structure of the active space using the variational two-electron RDM approach.199,248,249Version 0.9 of v2rdm_casscf can per- form approximate CASSCF calculations involving active spaces as large as 50 electrons in 50 orbitals199and is compatible with both conventional four-center electron repulsion integrals (ERIs) and DF/Cholesky decomposition approximations. Active-space specifi- cation inv2rdm_casscf is consistent with other active-space meth- ods in PSI4, and users can write RDMs to the disk in standard formats (e.g., FCIDUMP) for post-processing or for post-CASSCF meth- ods. Geometry optimizations using analytic energy gradients can also be performed (with four-center ERIs).250While most use cases ofv2rdm_casscf involve calls to PSI4’senergy() orgradient() functions, key components of the plugin such as RDMs are also accessible directly through Python. 6. CCT3 The CCT3 plugin200to PSI4is capable of executing a number of closed- and open-shell CC calculations with up to triply excited J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (T3) clusters. Among them are the active-space CC approach abbre- viated as CCSDt,251–254which approximates full CCSDT by select- ing the dominant T3amplitudes via active orbitals, and the CC(t;3) method, which corrects the CCSDt energies for the remaining, pre- dominantly dynamical, triple excitations that have not been cap- tured by CCSDt.201,202The CC(t;3) approach belongs to a larger family of methods that rely on the generalized form of biorthogonal moment expansions defining the CC( P;Q) formalism.201,202 The CCSDt method alone is already advantageous, since it can reproduce electronic energies of near-CCSDT quality at a small frac- tion of the computational cost while accurately describing select multireference situations, such as single bond breaking. CC(t;3) improves on the CCSDt energetics even further, being practically as accurate as full CCSDT for both relative and total electronic energies at essentially the same cost as CCSDt. CCSDt and CC(t;3) converge systematically toward CCSDT as the active space is increased. The CCT3 plugin can also be used to run CCSD and com- pletely renormalized (CR) CR-CC(2,3) calculations. This can be done by making the active orbital set (defined by the user in the input) empty, since in this case CCSDt = CCSD and CC(t;3) = CR-CC(2,3). We recall that CR-CC(2,3) is a completely renormal- ized triples correction to CCSD, which improves the results obtained with the conventional CCSD(T) approach without resorting to any multireference concepts and being at most twice as expensive as CCSD(T).255–257 VII. DOWNSTREAM ECOSYSTEM A. Computational molecular science drivers In addition to the closely associated ecosystem in Sec. VI, PSI4is robust and simple enough that projects can develop interfaces that use it as a “black box,” and such programs are considered part of the downstream ecosystem. Of these, the one exposing the most PSI4 capabilities is the QCARCHIVE INFRASTRUCTURE project QCENGINE , which can drive almost any single-command computation (e.g., gradient or complete-basis-set extrapolation, in contrast to a structure optimiza- tion followed by a frequency calculation) through the QCSCHEMA spec- ification. By launching PSI4through QCFRACTAL , the QCARCHIVE database has stored 18M computations over the past year and is growing rapidly. A recent addition is the interface to the Atomic Simulation Environment232,233(ASE) through which energies and gradients are accessible as a Calculator . All PSI4capabilities are available in the ASEby using the in-built psi4 module in the PSIAPI . Another MolSSI project, the MolSSI Driver Interface236(MDI), devised as a light communication layer to facilitate complex QM/MM and machine learning workflows, has a PSI4interface covering energies and gra- dients of HF and DFT methods. Finally, the I-PIuniversal force engine driver234,235has a PSI4interface covering gradients of most methods. B. Quantum computing PSI4is also used in several quantum computing packages to pro- vide orbitals, correlated densities, and molecular integrals. Its flexi- ble open-source license (LGPL) and Python API are factors that have favored its adoption in this area. For example, PSI4is interfaced to the open-source quantum computing electronic structure packageOPENFERMION225,226via the OPENFERMION-PSI4 plugin.227The QISKIT AQUA suite of algorithms for quantum computing developed by IBM221 is also interfaced to PSI4via an input file. The Microsoft Quantum Development Kit222is another open-source project that takes advan- tage of PSI4’s Python interface to generate molecular integrals and then transform them into the Broombridge format, a YAML-based quantum chemistry schema. C. Aiding force-field development for pharmaceutical infrastructure Many classical simulation methods have been developed with the aid of PSI4. As an illustrative example, torsion scans have been performed9using the OpenEye’s ORION platform to provide a first principles evaluation of conformational preferences in crystals, and the related methodology is used by the Open Force Field consor- tium258to parameterize force fields within the QCARCHIVE frame- work. PSI4has also found use in the development of nascent polariz- able, anisotropic force fields by providing the distributed multipoles and MP2 electrostatic potentials (ESPs) needed to parameterize the AMOEBA force field.259Moreover, the efficient SAPT code has been used in many recent developments in advanced force fields,260 including the emerging successors to AMOEBA.261,262In collabora- tion with Bristol Myers Squibb, we performed nearly 10 000 SAPT0 computations with PSI4to train a pilot machine-learning model of hydrogen-bonding interactions,8and a much larger number is being computed for a follow-up study. The restrained electostatic potential (RESP) model263is a pop- ular scheme for computing atomic charges for use in force field computations. A Python implementation was initially contributed to the PSI4NUMPY project, and later, an independent open-source pack- age was developed,115,220both of which employ PSI4for the quantum electrostatic potential. The package supports the standard two-stage fitting procedure and multi-conformational fitting and also allows easy specification of complex charge constraints. VIII. DEVELOPMENT AND DISTRIBUTION A choose-your-own-adventure guide to obtaining PSI4is avail- able at http://psicode.org/downloads. Here, users and developers can select their operating system (Linux, Windows, Mac) and Python version and then choose between downloading standalone installers for production-quality binaries, using the CONDA264package manager, and building the software from the source. While stan- dalone installers are only provided for stable releases, the source and CONDA installations also support the development branch. A new and substantial access improvement has been the porting of PSI4 to native Windows by one of the authors (R.G.) for the Acellera company (previously it was only available via Windows Subsys- tem for Linux, WSL) for GPUGRID , a distributed computing infras- tructure for biomedical research.215This involved separate ports of the required dependency projects and introduction of Windows continuous integration to conserve compatibility during the course of largely Linux-based development. The resulting uniform access to PSI4in a classroom setting has been especially valuable for the PSI4EDUCATION project. The cultivation of an ecosystem around PSI4led to changes in the build system (Sec. 3 of Ref. 1), notably the maintain-in-pieces J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-15 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp build-as-a-whole scheme where upstream and downstream depen- dencies remain in their own development repositories and are connected to PSI4through a single-file footprint in the CMAKE build system. Through a “superbuild” setup, PSI4and ecosystem projects can be flexibly built together upon a single command and use either pre-built packages or build dependencies from the source. For dis- tribution, we rely upon Anaconda Python (and its associated pack- age manager, CONDA ), which specializes in cross-platform building and management of Python/C++/Fortran software for the scien- tific community. Conda packages for Linux and Mac of PSI4and its dependencies (such that conda install psi4 -c psi4 yields a working installation) were in place by v1.1, when 11 packages were built for the psi4 channel. Since the v1.1 era, PSI4developers have focused on modern- ization and compatibility. With the release of CONDA-BUILD265v3 in late 2017 supporting enhanced build recipe language and built-in sysroots, PSI4has upgraded to use the same compilers as the foun- dational Anaconda defaults and community conda-forge channels. A substantial improvement is that, with the widespread availabil- ity of the Intel Math Kernel Library (MKL) through CONDA ,PSI4 now uses the same libraries ( mkl_rt ) as those in packages such as NUMPY , rather than statically linking LAPACK, thereby elimi- nating a subtle source of import errors and numerical discrepan- cies. After these improvements, PSI4today may be installed with- out fuss or incompatibility with other complex packages such as JUPYTER ,OPENMM , and RDKIT . While maintaining compatibility with defaults and conda-forge channels, PSI4packages additionally build with Intel compilers and use flags that simultaneously generate an optimized code for several architectures so that the same binary can run on old instruction sets such as SSE2 but also run in an opti- mal fashion on AVX2 and AVX512. In keeping with our ecosys- tem philosophy, PSI4will help a project with CONDA distribution on their own channel or ours or the community channel, or leave them alone, whichever level of involvement the developers pre- fer. We presently manage 23 packages. Since distributing through CONDA ,PSI4has accumulated 68k package manager and 93k installer downloads. With a reliable distribution system for production-quality bina- ries to users, PSI4can allow fairly modern code standards for develop- ers, including C++14 syntax, Python 3.6+, and OPENMP 3+. By stream- lining the build, PSI4can be compiled and tested within time limits on Linux and Windows with multiple compilers. By performing this continuous integration testing on cloud services, developers receive quality control feedback on their proposed code changes. These include the following: through testing, rough assurance that changes do not break the existing functionality; through coverage analysis, confidence that changes are being tested and a notice of testing gaps; and through static analysis, alerts that changes have incorrect syntax, type mismatches, and more. The last reflects the advantages of using standard CMAKE build tools: the static analysis tool correctly guesses how to build the PSI4source purely by examining build-language files in the repository. IX. LIMITATIONS PSI4’s current focus on high-throughput quantum chemistry on conventional hardware has limited development of distributedparallel multi-node computing capabilities except for independent tasks managed by QCFRACTAL, as described in Sec. IV. GPU support is also limited beyond the GPU_DFCC module;203,204however, due to the plugin structure of PSI4, interfacing a GPU-based Coulomb (J) and exchange (K) code would enhance the majority of PSI4’s capa- bilities, and PSI4is in discussions to integrate such a plugin. Several other features have been requested by users such as advanced algo- rithms for transition state searching, implicit solvent gradients, and additional implicit solvent methods. Beyond the above capability weaknesses, a primary downside of the open-source code is that there is no dedicated user support. While help can be found through a user forum at http://forum.psicode.org , a Slack workspace, and GitHub Issues, this support always comes from volunteers, and while questions are answered in the majority of cases, this is not guaranteed. On the other hand, the open-source software model empowers do-it-yourself fixes and extensions for power users and developers. X. CONCLUSIONS PSI4is a freely available, open-source quantum chemistry (QC) project with a broad feature set and support for multi-core paral- lelism. The density-fitted MP2 and frozen natural orbital CCSD(T) codes are particularly efficient, even in comparison with commer- cial QC programs. PSI4provides a number of uncommon features, including orbital-optimized electron correlation methods, density cumulant theory, and numerous intermolecular interaction meth- ods in the symmetry-adapted perturbation theory family. With sev- eral input modes—text file, powerful Python application program- ming interface, and structured data—we can serve QC to traditional users, power users, developers, and database backends. The rewrite of our driver to work with task lists and integration with the MolSSI QCARCHIVE INFRASTRUCTURE project make PSI4uniquely positioned for high-throughput QC. Our development efforts and capabilities have been tremen- dously boosted by the “inversion” of PSI4into a Python module in v1.1. We are able to rely more heavily on Python for driver logic, simplifying export of structured data and transition to the new distributed driver. The hybrid C++/Python programming strategy has also greatly aided development in the multiconfigurational SCF (MCSCF) and SAPT modules. We are able to transparently con- vert between NUMPY and PSI4linear algebra structures and fully access performance-critical C++ classes, facilitating rapid prototyping of novel SAPT and orbital-optimized MP nmethods. We are able to load into Python scripts and connect easily with other CMS software such as OPENMM and ASE. Finally, we have fostered a QC software ecosystem meant to benefit the electronic structure software community beyond PSI4. Our adoption of the MolSSI QCSCHEMA should facilitate interoper- ability efforts, and our switch to a more permissive LGPL-3.0 license should aid developers and users who wish to deploy PSI4as part of a larger toolchain or in cloud computing environments. We sin- cerely hope that the uptick in reusable software elements will con- tinue in the future, so that new methods may be adopted quickly by many QC packages simply by interfacing a common implementa- tion that is continuously updated, rather than developing redundant implementations in every code. J. Chem. Phys. 152, 184108 (2020); doi: 10.1063/5.0006002 152, 184108-16 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp SUPPLEMENTARY MATERIAL See the supplementary material for working equations for second-order SAPT0 without the single-exchange ( S2) approxima- tion using an atomic orbital formulation with density fitting. ACKNOWLEDGMENTS The authors are grateful to the contributors of all earlier ver- sions of the PSIprogram, as well as to all the developers of external libraries, plugins, and interfacing projects. The authors thank Pro- fessor Piotr Piecuch for providing the text describing the CCT3 plugin. Several of the co-authors were supported in their devel- opment of PSI4 and affiliated projects by the U.S. National Sci- ence Foundation through Grant Nos. CHE-1351978, ACI-1449723, CHE-1566192, ACI-1609842, CHE-1661604, CHE-1554354, CHE- 1504217 ACI-1547580, and CHE-1900420; by the U.S. Department of Energy through Grant Nos. DE-SC0018412 and DE-SC0016004; by the Office of Basic Energy Sciences Computational Chemical Sciences (CCS) Research Program through Grant No. AL-18-380- 057; and by the Exascale Computing Project through Grant No. 17- SC-20-SC, a collaborative effort of the U.S. Department of Energy Office of Science and the National Nuclear Security Administra- tion. U.B. acknowledges support from the Scientific and Technolog- ical Research Council of Turkey (Grant Nos. TUBITAK-114Z786, TUBITAK-116Z506, and TUBITAK-118Z916) and the European Cooperation in Science and Technology (Grant No. CM1405). The work at the National Institutes of Health was supported by the intramural research program of the National Heart, Lung, and Blood Institute. T.D.C. and The Molecular Sciences Software Institute acknowledge the Advanced Research Computing at Vir- ginia Tech for providing computational resources and technical support. H.K. was supported by the SYMBIT project (Reg. No. CZ.02.1.01/0.0/0.0/15_003/0000477) financed by the ERDF. S.L. was supported by the Academy of Finland (Suomen Akatemia) through Project No. 311149. R.D.R. acknowledges partial support by the Research Council of Norway through its Centres of Excel- lence scheme, Project No. 262695, and through its Mobility Grant scheme, Project No. 261873. P.K. acknowledges support of the For- rest Research Foundation and the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia. D.G.A.S. also acknowledges the Open Force Field Consortium and Initiative for financial and scientific support. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1R. M. Parrish, L. A. Burns, D. G. A. Smith, A. C. Simmonett, A. E. DePrince III, E. G. Hohenstein, U. Bozkaya, A. Y. Sokolov, R. Di Remigio, R. M. Richard, J. F. Gonthier, A. M. James, H. R. McAlexander, A. Kumar, M. 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1.3035911.pdf	Oscillating Ponomarenko dynamo in the highly conducting limit Marine Peyrot, Andrew Gilbert, and Franck Plunian Citation: Physics of Plasmas (1994-present) 15, 122104 (2008); doi: 10.1063/1.3035911 View online: http://dx.doi.org/10.1063/1.3035911 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/15/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Sheath oscillation characteristics and effect on near-wall conduction in a krypton Hall thruster Phys. Plasmas 21, 113501 (2014); 10.1063/1.4900764 Large quasineutral electron velocity oscillations in radial expansion of an ionizing plasma Phys. Plasmas 19, 092118 (2012); 10.1063/1.4754865 Numerical investigation of edge plasma phenomena in an enhanced D-alpha discharge at Alcator C-Mod: Parallel heat flux and quasi-coherent edge oscillations Phys. Plasmas 19, 082311 (2012); 10.1063/1.4747503 High Frequency Oscillations In 2D Fluid Model Of Hall Effect Thrusters AIP Conf. 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Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 08:14:18Oscillating Ponomarenko dynamo in the highly conducting limit Marine Peyrot,1,a/H20850Andrew Gilbert,2,b/H20850and Franck Plunian1,c/H20850 1Laboratoire de Géophysique Interne et Tectonophysique, Université Joseph Fourier, CNRS, Maison des Géosciences, B.P . 53, 38041 Grenoble Cedex 9, France 2Mathematics Research Institute, School of Engineering, Computing and Mathematics, University of Exeter, Exeter, EX4 4QF , United Kingdom /H20849Received 15 July 2008; accepted 5 November 2008; published online 8 December 2008 /H20850 This paper considers dynamo action in smooth helical ﬂows in cylindrical geometry, otherwise known as Ponomarenko dynamos, with periodic time dependence. An asymptotic framework isdeveloped that gives growth rates and frequencies in the highly conducting limit of large magneticReynolds number, when modes tend to be localized on resonant stream surfaces. This theory isvalidated by means of numerical simulations. © 2008 American Institute of Physics . /H20851DOI: 10.1063/1.3035911 /H20852 I. INTRODUCTION A well-known kinematic dynamo model goes back to the work of Ponomarenko,1who found that magnetic modes could be ampliﬁed in a ﬂow ﬁeld in cylindrical geometry/H20849depending only on distance from the axis /H20850, which generally possesses helical streamlines. In recent studies this Ponomar-enko dynamo has been investigated when the helical ﬂow ismodulated in time, 2,3with a focus on the dynamo threshold. The aim here is to quantify the effect of simple time-periodicﬂuctuations on the mean ﬂow, and the effect of these on thethreshold for magnetic growth. The main conclusion is thatthe dynamo threshold is larger than the one obtained withoutﬂuctuations, suggesting that large scale ﬂuctuations are notdesirable when optimizing a dynamo experiment. In dynamoexperiments, such large scale ﬂuctuations have been avoidedsimply by adding inner walls 4,5or a ﬂow-stabilizing ring.6A further experiment in preparation7is based on a single helical ﬂow, again avoiding large scale ﬂuctuations. The above simulations2,3were purely numerical and in order to give some theoretical backing to the results it isnecessary to use an asymptotic limit where approximate re-sults can be obtained. The steady Ponomarenko dynamo 1has been studied for Rm /H112711 in the kinematic case8–10and equili- brated solutions have been found taking into account thenonlinear feedback on the ﬂow. 11In both cases the underly- ing ﬂow is steady, and our aim here is to extend the kine-matic results to the case of oscillatory ﬂow ﬁelds. We therefore adopt the limit of large magnetic Reynolds number /H20849Rm/H112711/H20850, generally much above the threshold. Al- though our aim is to derive asymptotic results which are valid for this general class of ﬂows, with an eye to experi-mental dynamos and numerical simulations, we note thatPonomarenko dynamos may also occur in bipolar jetlike out-ﬂows commonly observed in protostellar systems, in whichmagnetic ﬁeld probably plays an important role. Though it isargued that such magnetic ﬁelds are produced inside the pro-tostellar disk 12we cannot exclude the existence of aPonomarenko-type dynamo in such a helical jet in which strong time ﬂuctuations may also occur. Here we assume a time-periodic ﬂow depending only on radius, limit our investigation to the kinematic approxima-tion, and study the asymptotic limit of large Rm /H20849Sec. II /H20850. Our analysis will be compared to direct numerical simula-tions for three cases /H20849Sec. III /H20850: stationary ﬂow and oscillatory ﬂow with zero mean ﬂow /H20849ZM/H20850, or nonzero mean ﬂow /H20849NZM /H20850. Finally, in Sec. IV we generalize our analysis to the case of time-varying resonant radius. II. MODEL AND ASYMPTOTIC APPROXIMATION The time evolution of the magnetic ﬁeld is given by the dimensionless induction equation /H9255/H11509B /H11509t=/H11612/H11003/H20849U/H11003B/H20850+/H9255/H116122B, /H208491/H20850 with/H9255=Rm−1and where we have adopted a diffusion time scale for our time variable /H20849in contrast to Ref. 8, but in ac- cord with Ref. 3/H20850. We consider a time-dependent helical ﬂow expressed in cylindrical coordinates /H20849r,/H9258,z/H20850by U/H20849r,t/H20850=/H208510,r/H9024/H20849r/H20850,V/H20849r/H20850/H20852F/H20849t/H20850for r/H333551, /H208492/H20850 U= 0 for r/H110221, where /H9024andVare smooth functions of randFis a given function of time. In the stationary case F=1, and for the nonstationary case we will consider two functional forms forthe time-dependence, F/H20849t/H20850= cos /H9275t/H20849ZM/H20850,F/H20849t/H20850=1+/H9267cos/H9275t/H20849NZM /H20850, /H208493/H20850 the “zero-mean” and “nonzero mean” ﬂows, respectively. For the linear, kinematic dynamo problem we may con- sider a magnetic ﬁeld of the form B/H20849r,t/H20850=ei/H20851m/H9258+kz+/H9278/H20849t/H20850/H20852b/H20849r,t/H20850, /H208494/H20850 where mandkare the azimuthal and vertical wave numbers of the ﬁeld and /H9278/H20849t/H20850is a phase, put in for convenience, thata/H20850Electronic mail: Marine.Peyrot@ujf-grenoble.fr. b/H20850Electronic mail: A.D.Gilbert@ex.ac.uk. c/H20850Electronic mail: Franck.Plunian@ujf-grenoble.fr.PHYSICS OF PLASMAS 15, 122104 /H208492008 /H20850 1070-664X/2008/15 /H2084912/H20850/122104/8/$23.00 © 2008 American Institute of Physics 15, 122104-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 08:14:18we will choose shortly. The solenoidality of the ﬁeld, /H11612·B =0, expressed in cylindrical coordinates, /H11509rbr+r−1br+imr−1b/H9258+ikbz=0 , /H208495/H20850 shows that it is enough to solve the induction equation for br andb/H9258only. It has been shown8,9that for a stationary ﬂow the mag- netic ﬁeld is generated in a resonant layer located at r=r0, where the magnetic ﬁeld lines are aligned with the shear andthus minimize their diffusion. In Eq. /H208492/H20850we see that since there is the same time-dependent factor F/H20849t/H20850multiplying an- gular and axial velocities, this radius is independent of time, and so this surface is ﬁxed and given by m/H9024 /H11032/H20849r0/H20850+kV/H11032/H20849r0/H20850=0 . /H208496/H20850 For more complex time-dependence r0may vary with time, leading to a succession of growing and damping magneticﬁeld states. 3In Sec. III we will consider the velocity ﬁeld in Eq. /H208492/H20850with r0ﬁxed, and we assume that r0lies in the ﬂuid, with 0 /H11021r0/H110211/H20849otherwise modes are strongly damped /H20850. For the more general case of distinct time-dependence for angu-lar and axial ﬂows, where r 0does vary with time, the equa- tions are set out and discussed in Sec. IV. With a given resonant surface r=r0the leading action of the ﬂow on the ﬁeld is simply advection of the mode by theangular and axial velocities, on the fast advective time scalet=O/H20849/H9255/H20850. We take this out of consideration by deﬁning the phase /H9278/H20849t/H20850as /H9278/H20849t/H20850=−/H9255−1/H20851m/H9024/H20849r0/H20850+kV/H20849r0/H20850/H20852/H20885 0t F/H20849t/H11032/H20850dt/H11032 /H208497/H20850 to leave behind only evolution through dynamo action, dif- fusion and reconnection, on slower time scales. Introducing Eqs. /H208492/H20850,/H208494/H20850, and /H208497/H20850in Eq. /H208491/H20850,w eﬁ n d /H20853/H9255/H11509t+/H20851im/H9024/H20849r/H20850+ikV/H20849r/H20850−im/H9024/H20849r0/H20850−ikV/H20849r0/H20850/H20852F/H20849t/H20850/H20854br =/H9255/H20851/H20849L−r−2/H20850br−2imr−2b/H9258/H20852, /H208498/H20850 /H20853/H9255/H11509t+/H20851im/H9024/H20849r/H20850+ikV/H20849r/H20850−im/H9024/H20849r0/H20850−ikV/H20849r0/H20850/H20852F/H20849t/H20850/H20854b/H9258 =r/H9024/H11032/H20849r/H20850F/H20849t/H20850br+/H9255/H20851/H20849L−r−2/H20850b/H9258+2imr−2br/H20852, /H208499/H20850 where the Laplacian operator Lis deﬁned by L=/H11509r2+r−1/H11509r−r−2m2−k2. /H2084910/H20850 In the highly conducting limit /H9255/H112701 we adopt the smooth Ponomarenko dynamo scaling,8 m=/H9255−1 /3M,k=/H9255−1 /3K,r=r0+/H92551/3s,t=/H92552/3/H9270, /H2084911/H20850 where /H9270is a time scale on which the dynamo mode grows, intermediate between the O/H208491/H20850diffusive time scale and the O/H20849/H9255/H20850advective time scale. This scaling gives a magnetic mode localized at the radius r=r0and it is known that the ﬁnal formulas obtained with this choice of scaling give the“richest” asymptotic picture including both the case m,k =O/H208491/H20850and the peak growth rates, achieved at m,k =O/H20849/H9255 −1 /3/H20850. Settingbr/H20849r,t/H20850=/H92551/3bˆr0/H20849s,/H9270/H20850+¯, b/H9258/H20849r,t/H20850=bˆ/H92580/H20849s,/H9270/H20850+¯, /H2084912/H20850 F/H20849t/H20850=Fˆ/H20849/H9270/H20850, together with the /H9024/H20849r/H20850andV/H20849r/H20850expansion at r=r0, /H9024/H20849r/H20850=/H9024/H20849r0/H20850+/H92551/3s/H9024/H11032/H20849r0/H20850+1 2/H92552/3s2/H9024/H11033/H20849r0/H20850+¯, /H2084913/H20850 V/H20849r/H20850=V/H20849r0/H20850+/H92551/3sV/H11032/H20849r0/H20850+1 2/H92552/3s2V/H11033/H20849r0/H20850+¯, /H2084914/H20850 we obtain from Eqs. /H208498/H20850and /H208499/H20850at leading order in /H9255 /H20851/H11509/H9270+c0+ic2s2Fˆ/H20849/H9270/H20850−/H11509s2/H20852bˆr0=−2 iMr0−2bˆ/H92580, /H2084915/H20850 /H20851/H11509/H9270+c0+ic2s2Fˆ/H20849/H9270/H20850−/H11509s2/H20852bˆ/H92580=r0/H9024/H11032/H20849r0/H20850Fˆ/H20849/H9270/H20850bˆr0, /H2084916/H20850 with c0=r0−2M2+K2,c2=1 2/H20851M/H9024/H11033/H20849r0/H20850+KV/H11033/H20849r0/H20850/H20852. /H2084917/H20850 From Eqs. /H2084915/H20850and /H2084916/H20850we immediately see how the dy- namo works: The differential rotation /H9024/H11032/H20849r0/H20850stretches the radial ﬁeld bˆr0to generate bˆ/H92580, and the diffusion of bˆ/H92580in cylindrical geometry then regenerates bˆr0. For the ﬂows con- sidered below c2/H110220, and for simplicity, we will take this to be the case in what follows: There are insigniﬁcant changesif this quantity is negative. These equations are solved by an exact ansatz involving time-dependent, complex Gaussian functions. We put thesein, at the same time rescaling using constants a r,a/H9258,a/H9270, and ah, to eliminate as many parameters as possible, with /H20875bˆr0/H20849/H9270,s/H20850 bˆ/H92580/H20849/H9270,s/H20850/H20876exp/H20849c0/H9270/H20850=/H20875arf¯/H20849/H9270¯/H20850 a/H9258g¯/H20849/H9270¯/H20850/H20876exp/H20851−ahh¯/H20849/H9270¯/H20850s2/H20852,/H2084918/H20850 where /H9270¯=a/H9270/H9270is a new time scale. We now ﬁx the constants with ar=2M,a/H9258=r02c21/2,a/H9270=ah=c21/2, /H2084919/H20850 and we are left with the following system of ODEs in /H9270¯to solve /H11509/H9270¯h¯+4h¯2=iF¯/H20849/H9270¯/H20850, /H11509/H9270¯f¯+2h¯f¯=−ig¯, /H2084920/H20850 /H11509/H9270¯g¯+2h¯g¯=−DF¯/H20849/H9270¯/H20850f¯, where the constant D=−2M/H9024/H11032/H20849r0/H20850/r0c2and the time- dependent factor is F¯/H20849/H9270¯/H20850=Fˆ/H20849/H9270/H20850./H20851Note that modes with more radial structure can be studied, taking the form of two time- dependent polynomials times a Gaussian in Eq. /H2084918/H20850, but these will be subdominant. /H20852 The equation for h¯is nonlinear, while those for f¯andg¯ are linear. The exponential growth rate /H9253¯of the magnetic ﬁeld components f¯andg¯depends only on the parameter D,122104-2 Peyrot, Gilbert, and Plunian Phys. Plasmas 15, 122104 /H208492008 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 08:14:18which captures the local geometry of the helical streamlines,8 and the form of the time-dependence F¯/H20849/H9270¯/H20850. From Eqs. /H208496/H20850and /H2084917/H20850we have D=−4 r0/H20875/H9024/H11033/H20849r0/H20850 /H9024/H11032/H20849r0/H20850−V/H11033/H20849r0/H20850 V/H11032/H20849r0/H20850/H20876−1 , /H2084921/H20850 showing that Ddepends only on the geometry of the velocity ﬁeld. The function h¯/H20849/H9270¯/H20850gives the Gaussian envelope, with Reh¯/H110220 required for exponential localization of the mode. So far the system /H2084920/H20850applies to any time-dependence F¯/H20849/H9270¯/H20850. Under the rescaling, the two given in Eq. /H208493/H20850corre- spond to F¯/H20849/H9270¯/H20850= cos/H9275¯/H9270¯/H20849ZM/H20850,F¯/H20849/H9270¯/H20850=1+/H9267cos/H9275¯/H9270¯/H20849NZM /H20850, /H2084922/H20850 where the frequencies are linked by /H9275=/H9255−2 /3c21/2/H9275¯/H11013/H208531 2/H9255−1/H20851m/H9024/H11033/H20849r0/H20850+kV/H11033/H20849r0/H20850/H20852/H208541/2/H9275¯. /H2084923/H20850 For the rescaled system /H2084920/H20850, after a transient, h¯will become periodic with the same frequency /H9275¯as the forcing F¯ and the linear equations for the magnetic ﬁeld components /H20849f¯,g¯/H20850will take a Floquet form: The solution will have expo- nential growth superposed on periodic behavior. The overall growth rate may be measured as /H9253¯/H20849D,/H9275¯/H20850/H20849suppressing /H9267in the nonzero mean case /H20850. This is linked to the growth rate /H9253of the magnetic ﬁeld in the original systems /H208498/H20850and /H208499/H20850/H20851or Eq. /H208491/H20850/H20852with /H9253=/H9255−2 /3/H20851c21/2/H9253¯/H20849D,/H9275¯/H20850−c0/H20852 /H11013/H208531 2/H9255−1/H20851m/H9024/H11033/H20849r0/H20850+kV/H11033/H20849r0/H20850/H20852/H208541/2/H9253¯/H20849D,/H9275¯/H20850−r0−2m2−k2, /H2084924/H20850 or, using the deﬁnition of Din Eq. /H2084921/H20850, /H9253=/H20851−2m/H9024/H11032/H20849r0/H20850//H9255r0D/H208521/2/H9253¯/H20849D,/H9275¯/H20850−r0−2m2−k2. /H2084925/H20850 In the stationary case F/H20849t/H20850=1, from Eq. /H2084920/H20850we ﬁnd h¯ =/H11006i1/2/2 and/H9253¯=−2h¯/H11006/H20849iD/H208501/2. Then from Eq. /H2084924/H20850and tak- ingh¯with Re h¯/H333560, we obtain the real part of the growth rate as Re/H9253=/H9255−1 /2/H20851r0−1/H20841m/H9024/H11032/H20849r0/H20850/H20841/H208521/2 −1 2/H9255−1 /2/H20851/H20841m/H9024/H11033/H20849r0/H20850+kV/H11033/H20849r0/H20850/H20841/H208521/2−r0−2m2−k2, /H2084926/H20850 as previously found.8–10Together with this goes the purely geometrical criterion for Ponomarenko dynamo action inhighly conducting stationary ﬂow, that /H20841D/H20849r 0/H20850/H20841/H110221 at the resonant radius r0. Note that formulas /H2084923/H20850–/H2084926/H20850, although derived using the scaling equation /H2084911/H20850, are in fact valid for allmandklinked by the resonance condition /H208496/H20850. The ex- pansions would give equivalent results had we taken m,k =O/H208491/H20850, and/H9255→0, though this would not immediately cap- ture the fastest growing modes which are of the scale m,k =O/H20849/H9255−1 /3/H20850as/H9255→0. The key assumption in the expansion is that at small /H9255the magnetic ﬁeld localizes in a thin layer. From Eqs. /H2084911/H20850and /H2084918/H20850the width of the layer isr−r0=O/H20849/H92551/3c2−1 /4h¯−1 /2/H20850 =O/H20851/H92551/4/H208491 2/H20851m/H9024/H11033/H20849r0/H20850+kV/H11033/H20849r0/H20850/H20852/H20850−1 /4h¯−1 /2/H20852, /H2084927/H20850 and this goes to zero as /H9255→0; however we should note that this is assuming that /H9275¯is ﬁxed as we take the limit, so that the magnitude of h¯is of order unity in the limit. If instead we allow /H9275¯/H20849or other parameters or wavenumbers /H20850to vary as well, then we need to be careful to check the condition thatthe width given by Eq. /H2084927/H20850is small, to validate the asymptotic theory. For example, if we ﬁx /H9275as/H9255→0 we have /H9275¯=O/H20849/H92551/2/H9275/H20850→0 from Eq. /H2084923/H20850and as h¯turns out to be bounded in this limit /H20849of low frequencies, so similar to the stationary case /H20850this condition is veriﬁed. III. RESULTS To test the above asymptotic results, we use the ﬂow equation /H208492/H20850with radial proﬁle /H9024/H20849r/H20850=1− rand V/H20849r/H20850 =/H9003/H208491−r2/H20850, where /H9003is a helicity factor, for which r0= −m/2k/H9003andD=4, independent of radius. We begin by checking the stationary case, followed by the examples ofzero-mean and nonzero mean ﬂows in Eq. /H208493/H20850. The growth rate of magnetic ﬁeld for the asymptotic theory is obtained by simulating Eq. /H2084920/H20850using a fourth order Runge–Kutta scheme, with frequencies and growth rateslinked by Eqs. /H2084923/H20850and /H2084924/H20850. For our given ﬂow we have /H9275=/H20849−k/H9003/H9255−1/H208501/2/H9275¯,/H9253=/H20849−k/H9003/H9255−1/H208501/2/H9253¯−/H208491+4/H90032/H20850k2. /H2084928/H20850 Typically for /H9003=2 and k=−0.5, we have /H9275=/H9255−1 /2/H9275¯and/H9253 =/H9255−1 /2/H9253¯−4.25. This growth rate will be compared to the one obtained with direct numerical simulation, solving the induc-tion equation /H208491/H20850without asymptotic approximation, using a Galerkin method for the radial discretization and again aRunge–Kutta scheme for the time evolution. 3Note that for the full problem the ﬁeld settles into a Floquet form withB/H20849t+T/H20850=exp /H20849 /H9253BT/H20850B/H20849t/H20850, where Tdenotes the period of F/H20849t/H20850. From Eqs. /H208494/H20850and /H208497/H20850it is given by /H9253B=/H9253−i/H20851m/H9024/H20849r0/H20850+kV/H20849r0/H20850/H20852T−1/H20885 0T F/H20849t/H11032/H20850dt/H11032, /H2084929/H20850 where the phase factor has been reintroduced for a good comparison of frequency measurements. A. Stationary ﬂow ForF/H20849t/H20850=1 and our given ﬂow, we have /H9253¯=/H208491+i/H20850//H208812. From Eq. /H2084928/H20850we obtain /H9253B=/H9253−i/H20851m/H9024/H20849r0/H20850+kV/H20849r0/H20850/H20852with /H9253=/H9255−1 /2/H208811 2/H20841k/H20841/H9003/H208491+i/H20850−/H208491+4/H90032/H20850k2. /H2084930/H20850 In Fig. 1, the growth rate /H9253Bis plotted against /H9255−1for two different values of r0. The curves show a good agreement between both asymptotic and simulation growth rates andfrequencies provided the magnetic Reynolds number Rm=/H9255 −1is sufﬁciently large. For /H9255−1/H33356103the difference is less than 5% for the growth rate and 0.6% for the frequency. InFig. 2, the modulus of each magnetic ﬁeld component is plotted for several values of /H9255 −1. Clearly, increasing /H9255−1con-122104-3 Oscillating Ponomarenko dynamo … Phys. Plasmas 15, 122104 /H208492008 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 08:14:18centrates the magnetic ﬁeld in a thinner layer at r0, and the asymptotic formulation becomes increasingly accurate. De-ﬁning the layer thickness /H9254as the width over which the mag- netic energy falls to half of its peak value, we ﬁnd that /H9254 /H11229O/H20849/H92550.27/H110060.03/H20850from the simulation. An estimate from the asymptotic expressions /H2084918/H20850, and using Eqs. /H2084911/H20850and /H2084917/H20850, leads to /H9254=O/H20849/H92551/4/H20850form=1 which is in good agreement. However this quantity goes to zero quite slowly with /H9255and so convergence is slow. B. Periodic ﬂow with zero mean We now consider the second case, with zero mean in the original time-dependence equation /H208493/H20850or in the rescaled ver- sion Eq. /H2084922/H20850. We ﬁrst work with the asymptotic system and solve Eq. /H2084920/H20850to obtain the growth rate /H9253¯/H20849D,/H9275¯/H20850as a function of/H9275¯for different values of D, as plotted in Fig. 3/H20849left/H20850. For D/H110211 we obtain pure decay Re /H9253¯/H110210; for D=1.5 the sign of Re/H9253¯depends on /H9275¯/H20849positive at small /H9275¯/H20850, whereas for D/H333562 we ﬁnd that Re /H9253¯/H333560 for all /H9275¯. Recall that in the stationary case /H20841D/H20841/H110221 is necessary and sufﬁcient for dynamo action in the highly conducting limit. We can investigate this result further by taking an addi- tional limit of solving Eq. /H2084920/H20850for/H9275¯/H112711. For that we use a new time coordinate u=/H9275¯/H9270¯and set a small parameter /H9256 =/H9275¯−1/H112701. Then we have, without approximation, from Eq. /H2084920/H20850and dropping the bars to ease notation,/H9256−1/H11509uh+4h2=icosu, /H2084931/H20850 /H9256−1/H11509uf+2hf=−ig, /H2084932/H20850 /H9256−1/H11509ug+2hg=−Dfcosu, /H2084933/H20850 where f,g, and hare now functions of u. We set /H20851f/H20849u/H20850,g/H20849u/H20850/H20852=exp /H20849/H9262u/H20850/H20851f/H20849u/H20850,g/H20849u/H20850/H20852with/H9262a constant Flo- quet exponent and require fandgto be strictly periodic functions of u. We have /H9256−1/H11509uh+4h2=icosu, /H9256−1/H9262f+/H9256−1/H11509uf+2hf=−ig, /H2084934/H20850 /H9256−1/H9262g+/H9256−1/H11509ug+2hg=−Dfcosu. Expanding /H9262,f,g, and hin powers of /H9256, /H20849/H9262,f,g,h/H20850=/H20849/H9262,f,g,h/H208500+/H9256/H20849/H9262,f,g,h/H208501 +/H92562/H20849/H9262,f,g,h/H208502+¯, /H2084935/H20850 we solve the system /H2084934/H20850order by order, using the terms /H92620, /H92621,... to enforce periodicity. This leads to f=A0+/H9256A1+/H92562/H20851iA0/H208492−D/H20850cosu+A2/H20852+¯, /H2084936/H20850 FIG. 2. Modulus of each magnetic ﬁeld component plotted against rfor/H20849a/H20850/H9255−1=500, /H20849b/H208501000, /H20849c/H208502000, /H20849d/H208504000, /H20849e/H208506000, /H20849f/H2085010000, for m=1,/H9003=2,k =−0.5, and r0=0.5. FIG. 1. The magnetic growth rate Re /H9253B/H20849left/H20850and frequency Im /H9253B/H20849right /H20850plotted against /H9255−1in the stationary case F/H20849t/H20850=1, for m=1,/H9003=2, and k=−0.7, r0=0.35 /H20849black /H20850, and k=−0.5, r0=0.5 /H20849gray /H20850. The asymptotic solution and the simulation correspond to dashed and full curves, respectively.122104-4 Peyrot, Gilbert, and Plunian Phys. Plasmas 15, 122104 /H208492008 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 08:14:18g=/H9256DA0/H20849− sin u/H110062−1 /2i/H20850+/H92562/H20849−DA1sinu+B2/H20850 +¯, /H2084937/H20850 h=/H9256/H208492−1 /2+isinu/H20850+/H92562C2 +/H92563/H20849C3− sin 2 u+4/H208812icosu/H20850+¯, /H2084938/H20850 /H9262=/H92562/H20849−21/2/H110062−1 /2D/H20850+¯, /H2084939/H20850 where the Ai,Bi, and Ciare integration constants and we have imposed Re h/H110220./H20849The Aiare arbitrary; the BiandCi can be determined in terms of the Aiat higher orders of the expansion. /H20850Then the growth rate of f¯andg¯is given by /H9253¯ =/H9262/H9275¯with /H9253¯/H11229/H9275¯−12−1 /2/H20849−2/H11006D/H20850. /H2084940/H20850 This conﬁrms that /H9253¯/H333560 only if D/H333562, in the high frequency limit, as seen in Fig. 3/H20849left/H20850. In addition to the results of system /H2084920/H20850,/H9253¯is plotted versus /H9275¯using the asymptotic ex- pansion /H2084940/H20850. We ﬁnd a good agreement between both, even for moderate values of /H9275¯. In Fig. 3/H20849right /H20850, we plot the growth rate /H9253Bfor different values of /H9275, from both the asymptotic ODEs /H2084920/H20850and from simulations of the primitive equations /H208498/H20850and /H208499/H20850, showing good agreement provided /H9255−1 is sufﬁciently large. Note that formula /H2084940/H20850indicates a growth rate /H9253¯that increases as the frequency /H9275¯→0, and this perhaps suggests fast dynamo action. The dynamo here would be fast if thegrowth rate on the short, advective time scale, here given by/H9255 /H9253, remains bounded above zero as /H9255→0, holding the ﬂow ﬁxed. In our ﬂow D=4 which, from Eq. /H2084940/H20850, leads to /H9253¯ /H11011/H208812//H9275¯and from Eq. /H2084928/H20850to /H9253/H11229−/H208812k/H9003/H9255−1/H9275−1−/H208491+4/H90032/H20850k2. /H2084941/H20850 In the limit of small /H9255, given that − k/H9003/H110220, this at ﬁrst sight appears to be a fast dynamo. This formula was derived on theassumption that /H9275¯−1/H112701, but as /H9255→0 for a ﬁxed ﬂow, which includes a ﬁxed /H9275, the assumption becomes violated. As /H9255 →0,/H9275¯→0 from Eq. /H2084928/H20850and so we move towards the left in Fig.3/H20849left panel /H20850: If the asymptotic curves /H20849dashed /H20850contin-ued to grow to the left, the dynamo would be fast. However the computed values /H20849solid /H20850saturate for small /H9275¯and the dynamo is in the slow camp, as expected. /H20849As line elements are only stretched linearly the ﬂow fails to have Lagrangianchaos, technically positive topological entropy, a requirementfor fast dynamo action in a smooth ﬂow. 13/H20850 C. Critical values for Rm in ﬂows with zero mean The asymptotic theory gives an estimate for the critical value of Rm or /H9255for the onset of dynamo instabilities, namely from Eq. /H2084924/H20850, /H9255c1/2=R e/H20849/H208531 2/H20851m/H9024/H11033/H20849r0/H20850+kV/H11033/H20849r0/H20850/H20852/H208541/2/H9253¯/H20849D,/H9275¯/H20850/H20850/H20849r0−2m2+k2/H20850−1, /H2084942/H20850 or from Eq. /H2084925/H20850, /H9255c1/2=R e /H20853/H20851−2m/H9024/H11032/H20849r0/H20850/r0D/H208521/2/H9253¯/H20849D,/H9275¯/H20850/H20854/H20849r0−2m2+k2/H20850−1. /H2084943/H20850 For the mode m=1, the agreement with numerics at on- set is poor in Fig. 1because the critical magnetic Reynolds number is not large enough /H20849contrast the situation in Ref. 10/H20850. In the zero-mean case the agreement seems to be better as seen in Fig. 3/H20849right /H20850. In fact, we generally expect agree- ment for critical values to improve if there is some otherasymptotic parameter to push the critical Rm into thesmall-/H9255, large-Rm regime, provided the condition of thin layer width, Eq. /H2084927/H20850, is satisﬁed. One possibility could be to take the limit when the axial ﬂow V/H20849r/H20850is weak or strong compared with the angular velocity /H9024/H20849r/H20850/H20849as measured by the helicity factor /H9003/H20850. Unfortunately direct simulations for /H9003 /H112701o r/H9003/H112711 are difﬁcult to achieve. Instead we consider the limit of increasing frequency /H9275. We have already seen that the asymptotic analysis leads to Eq. /H2084941/H20850provided /H9275¯is large enough. In this case, the threshold should scale as /H9255c−1/H11008/H9275 and/H9275¯tends to inﬁnity from Eq. /H2084923/H20850, which is necessary for consistency. Carrying out simulations in order to determine/H9255 cfor values of /H9275in the range 100 /H33355/H9275/H333551000, we found that /H9275/H9255c=0.209 /H110060.004. This is the correct scaling as predicted FIG. 3. Zero mean case: growth rate /H9253¯vs/H9275¯/H20849left/H20850for/H20849from bottom to top /H20850D=1,2,3,4. The full curves give growth rates from integration of Eq. /H2084920/H20850; dashed curves and symbols give growth rates from Eq. /H2084940/H20850. Growth rate Re /H9253Bvs/H9255−1/H20849right /H20850form=1,/H9003=2,k=−0.5, r0=0.5, and /H9275=200 /H20849black /H20850/H9275=500 /H20849gray /H20850.T h e asymptotic results are shown by dashed curves, simulations by full curves.122104-5 Oscillating Ponomarenko dynamo … Phys. Plasmas 15, 122104 /H208492008 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 08:14:18by Eq. /H2084941/H20850but the constant is not that predicted, 0.3328. The resolution of this paradox is that in this limit with /H9275 =O/H20849/H9255−1/H20850, we have /H9275¯=O/H20849/H9255−1 /2/H20850→/H11009from Eq. /H2084923/H20850. This means that from Eq. /H2084938/H20850,h¯=O/H20849/H9275¯−1/H20850=O/H20849/H92551/2/H20850and so the layer width, Eq. /H2084927/H20850, is of order unity and does not go to zero. The increasing frequency /H9275¯is tending to increase the width at the same time as the decreasing /H9255is tending to localize the mode, and the two effects cancel out completely.The theory gives the correct scaling law but is not asymp-totically correct as the magnetic ﬁeld is not localized. Thisindicates the care that has to be taken with double limits andthe usefulness of the condition that Eq. /H2084927/H20850is small. /H20851There may be an asymptotic theory appropriate to the limit /H9255→0 with/H9255 /H9275=O/H208491/H20850, based on a reduced, ﬁnite number of modes in time /H9270¯similar to those in Eqs. /H2084936/H20850–/H2084938/H20850, but retaining full radial dependence; we leave this for further investigation. /H20852 D. Periodic ﬂow with nonzero mean Here we consider the case with nonzero mean ﬂow, NZM, in Eqs. /H208493/H20850and /H2084922/H20850. The growth rate /H9253¯from the asymptotic ODEs /H2084920/H20850is plotted against /H9267in Fig. 4/H20849left/H20850for /H9275¯=10 and different values of D. Taking /H9267=0 corresponds to the stationary case. Then increasing the ﬂuctuation level /H9267 may increase the growth rate depending on whether Dis sufﬁciently large, the transition being for Dbetween 2 and 3. This shows that ﬂuctuations may increase the dynamo efﬁ-ciency /H20849at least in the scalings we are using /H20850. A different conclusion has been obtained at the dynamo threshold whichgenerally increases with the ﬂuctuation rate. 3When /H9267is in- creased to large values, the mean part of the ﬂow becomessmall compared to the ﬂuctuations. We conﬁrmed that thislimit recovers the results of the previous zero-mean case withappropriate rescaling for /H9253¯and/H9275¯. In Fig. 4/H20849right /H20850the growth rate Re /H9253Bis plotted against /H9255−1for different values of /H9275. The curves show a good agree- ment between the growth rates from the asymptotic ODEsand from simulation of the full system provided /H9255 −1is sufﬁ- ciently large. The difference is less than 4% for /H9255−1/H33356400.IV. ANALYSIS FOR TIME-VARYING RESONANT RADIUS We now brieﬂy indicate how theory is extended to the more general time dependence, U/H20849r,t/H20850=/H208530,r/H9024/H20849r/H20850/H20851F/H20849t/H20850+qG/H20849t/H20850/H20852,V/H20849r/H20850/H20851F/H20849t/H20850−qG/H20849t/H20850/H20852/H20854. /H2084944/H20850 Here qis a parameter that controls the difference in time- dependence between the axial and azimuthal componentsandF/H20849t/H20850,G/H20849t/H20850are functions of time, of order unity. For ex- ample we could take a general, single frequency, zero-mean case, F/H20849t/H20850= cos /H9275t,G/H20849t/H20850= cos /H20849/H9275t−/H9021/H20850. /H2084945/H20850 Now generally the resonant radius becomes a function of time, r0/H20849t/H20850, with m/H9024/H11032/H20849r0/H20850/H20851F/H20849t/H20850+qG/H20849t/H20850/H20852+kV/H11032/H20849r0/H20850/H20851F/H20849t/H20850−qG/H20849t/H20850/H20852=0 , /H2084946/H20850 but such variation is found to have a strong damping effect on the ﬁeld.3At the resonant radius the shear in the ﬂow is aligned with the helical ﬁeld lines in the magnetic mode; inthe stationary case, as one departs from this radius, the shearchanges direction, tending to introduce ﬁne radial scales andenhanced diffusion. Moving the resonant radius with a time-dependent ﬂow then leads to a damping effect on modes,which are strongly suppressed when the ﬁeld concentrationis distant from r 0/H20849t/H20850/H20851noting that the ﬁeld cannot readily dif- fuse in radius to follow r0/H20849t/H20850/H20852. For these reasons, in our asymptotic framework we will take qto tend to zero in magnitude as /H9255→0. This makes r0 ﬁxed at leading order and we deﬁne r0by Eq. /H208496/H20850as we did originally. Going through the previous calculations we ob-tain, in place of Eqs. /H2084915/H20850and /H2084916/H20850, the equations /H20851 /H11509/H9270+c0+ic1sGˆ/H20849/H9270/H20850+ic2s2Fˆ/H20849/H9270/H20850−/H11509s2/H20852bˆr0=−2 iMr0−2bˆ/H92580, /H2084947/H20850 FIG. 4. Nonzero mean case: growth rate /H9253¯vs/H9267/H20849left/H20850,f o r/H9275¯=10. The curves /H20849a/H20850,/H20849b/H20850,/H20849c/H20850,/H20849d/H20850correspond to D=1,2,3,4, respectively. Growth rate Re /H9253Bvs /H9255−1/H20849right /H20850form=1,/H9003=2,k=−0.5, r0=0.5,/H9275=100 and from bottom to top /H9267=1,4,8,10. The asymptotic results correspond to dashed curves and simulations to full curves.122104-6 Peyrot, Gilbert, and Plunian Phys. Plasmas 15, 122104 /H208492008 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 08:14:18/H20851/H11509/H9270+c0+ic1sGˆ/H20849/H9270/H20850+ic2s2Fˆ/H20849/H9270/H20850−/H11509s2/H20852bˆ/H92580 =Fˆ/H20849/H9270/H20850r0/H9024/H11032/H20849r0/H20850bˆr0, /H2084948/H20850 with the new linear term deﬁned by G/H20849t/H20850=Gˆ/H20849/H9270/H20850,q=/H92551/3Q,c1=Q/H20851M/H9024/H11032/H20849r0/H20850−KV/H11032/H20849r0/H20850/H20852. /H2084949/H20850 Substituting /H20875bˆr0/H20849/H9270,s/H20850 bˆ/H92580/H20849/H9270,s/H20850/H20876exp/H20849c0/H9270/H20850=/H20875arf¯/H20849/H9270¯/H20850 a/H9258g¯/H20849/H9270¯/H20850/H20876exp/H20851−ajj¯/H20849/H9270¯/H20850s−ahh¯/H20849/H9270¯/H20850s2/H20852 /H2084950/H20850 and setting Gˆ/H20849/H9270/H20850=G¯/H20849/H9270¯/H20850andaj=c21/4gives the system /H11509/H9270¯h¯+4h¯2=iF¯/H20849/H9270¯/H20850, /H11509/H9270¯j¯+4h¯j¯=iQG¯/H20849/H9270¯/H20850, /H2084951/H20850 /H11509/H9270¯f¯−j¯2f¯+2h¯f¯=−ig¯, /H11509/H9270¯g¯−j¯2g¯+2h¯g¯=−DF¯/H20849/H9270¯/H20850f¯. We now have a new parameter that quantiﬁes the difference in the time-dependence of azimuthal and axial ﬂows, and sothe variation in resonant radius, given by Q=c 1c2−3 /4/H11013q/H9255−1 /4m/H9024/H11032/H20849r0/H20850−kV/H11032/H20849r0/H20850 /H208531 2/H20851m/H9024/H11033/H20849r0/H20850+kV/H11033/H20849r0/H20850/H20852/H208543/4. /H2084952/H20850 As for system /H2084920/H20850, system /H2084951/H20850applies to any time- dependence F¯/H20849/H9270¯/H20850andG¯/H20849/H9270¯/H20850. Again this system can be solved numerically to obtain a growth rate. With the speciﬁc time- dependence equation /H2084945/H20850, the growth rate will be a function /H9253¯/H20849D,/H9275¯,Q,/H9021/H20850: The phase angle /H9021quantiﬁes the polarization of the axial and azimuthal components of the ﬂow, in a loose sense. In the system /H2084951/H20850, we see that changing Qto −Qonly changes j¯to −j¯without affecting the other variables. There- fore it is sufﬁcient to consider positive values of Q. Our numerical investigations, which we summarize rather thanpresenting graphically, indicate that compared to the curvesgiven in Fig. 3forQ=0, changing Qand/H9021systematically leads to lower values of /H9253¯, without changing much the shape of the /H9275¯andDdependencies. We can show that /H9253¯is /H9266-periodic in /H9021. We ﬁnd that /H9253¯is a maximum for /H9021=0, a minimum for /H9021=/H9266/2 and that /H9253¯/H20849/H9021=/H9266/4/H20850=/H9253¯/H20849/H9021=3/H9266/4/H20850. Fi- nally/H9253¯is found to be monotonically decreasing with Qand it would be interesting to obtain a proof conﬁrming thisobservation. We can investigate these results further by taking the additional limit of large /H9275¯as in Sec. III B. We ﬁnd that Eq. /H2084940/H20850holds even for Qnonzero. In other words the leadingorder growth rate /H20849proportional to /H9275¯−1/H20850is independent of Q and/H9021in the limit /H9275¯→/H11009and is given in Eq. /H2084940/H20850. This is very clear in the expansion /H2084951/H20850: The j¯2f¯andj¯2g¯terms come in at one order below what is needed to obtain Eq. /H2084940/H20850. They are asymptotically smaller than h¯f¯andh¯g¯in the same equa- tions, as h¯andj¯are both of size /H9256at leading order. V. CONCLUSIONS We extended the theory of the Ponomarenko dynamo in the asymptotic limit of large Rm, to the case of a nonstation-ary ﬂow. We considered only a very simple model for ﬂuc-tuating ﬂow, but one that is nevertheless revealing. Withinthis class of ﬂows it highlights the basic mechanisms fordynamo action, the effect of the motion of the resonant ra-dius in suppressing ﬁeld generation, and the parameter com-binations that are relevant at large Rm. Our results includecriteria for dynamo action at large Rm involving the purelygeometrical quantity D, linked to the rate of change of pitch of the velocity shear. For example, we ﬁnd from Fig. 3that the geometrical condition /H20841D/H20841/H110222 at a given radius is needed for dynamo action with a mode localized there, at high fre-quencies in the zero-mean case for large Rm. Note that for stationary ﬂow the corresponding criterion is/H20841D/H20841/H110221 for magnetic ﬁeld ampliﬁcation. To see the rel- evance of these results, consider the spiral Couette ﬂow,which is simply the general solution for differentially rotat-ing ﬂow forced by rotating, translating cylindrical bound-aries /H9024/H20849r/H20850=A 1+A2r−2,V/H20849r/H20850=A3+A4logr. /H2084953/H20850 This corresponds to /H20841D/H20841=2, and so satisﬁes the condition in the stationary case; dynamo action was observed in Ref. 15. If the motion of the boundaries is now periodic with zeromean, and sufﬁciently slow that the above functions are justmultiplied by F/H20849t/H20850=cos /H9275t, then Fig. 2shows that dynamo action becomes marginal provided /H9275¯is large. Of course the full picture for any boundary forcing is complicated by thedevelopment of Stokes’ layers unless it is slow comparedwith viscous time scales. Nonetheless the key point is thatﬂows with larger values of /H20841D/H20841over a range of radii are likely to be more efﬁcient as dynamos in nonstationary as well asstationary ﬂows, and this consideration could be important inoptimizing experiments and understanding experimental ornumerical results. We also considered ﬂow ﬂuctuations inwhich components are out of phase, leading to a time-dependent resonant radius. The results show that this inhibitsthe dynamo action, conﬁrming previous results obtained fora cellular type of ﬂow. 14 The ﬂuctuations considered in this paper have a very simple structure, and it would be a natural extension to con-sider ﬂuctuating ﬂows that carried the ﬁeld across stream-lines, in other words depending also on /H9258andz. This would lose the separation of variables employed here and make thestudy more numerical, unless averaging is done analytically,122104-7 Oscillating Ponomarenko dynamo … Phys. Plasmas 15, 122104 /H208492008 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 08:14:18which would generally give an alpha effect.16Other possible interesting developments would be to consider random timedependence, which increases the complexity of the system,especially close to the threshold, 17and to include some of the effects of nonlinear feedback on the ﬂow ﬁeld.11 ACKNOWLEDGMENTS We are grateful to the European Network on Electro- magnetic Processing of Materials /H20849COST Action P17 /H20850which supported a visit of M.P. to Exeter in 2007, where this re-search project commenced. A.G. is grateful for a LeverhulmeTrust Research Fellowship held during the completion of thispaper. F.P. is grateful to the Dynamo Program at KITP /H20849sup- ported in part by the National Science Foundation underGrant No. PHY05-51164 /H20850for completion of the paper. We thank Professor Andrew Soward and Dr. Matthew Turner forhelpful comments during this research.1Yu. B. Ponomarenko, J. Appl. Mech. Tech. Phys. 6, 755 /H208491973 /H20850. 2C. Normand, Phys. Fluids 15, 1606 /H208492003 /H20850. 3M. Peyrot, F. Plunian, and C. Normand, Phys. Fluids 19, 054109 /H208492007 /H20850. 4A. Gailitis, O. Lielausis, E. Platacis, S. Dementiev, A. Cifersons, G. Ger- beth, Th. Gundrum, F. Stefani, M. Christen, and G. Will, Phys. Rev. Lett. 86, 3024 /H208492001 /H20850. 5R. Stieglitz and U. Müller, Phys. Fluids 13, 561 /H208492001 /H20850. 6R. Monchaux, M. Berhanu, M. Bourgoin, M. Moulin, Ph. Odier, J.-F. Pinton, R. Volk, S. Fauve, N. Mordant, F. Pétrélis, A. Chiffaudel, F. Da-viaud, B. Dubrulle, C. Gasquet, L. Marié, and F. Ravelet, Phys. Rev. Lett. 98, 044502 /H208492006 /H20850. 7P. Frick, V. Noskov, S. Denisov, S. Khripchenko, D. Sokoloff, R. Stepanov, and A. Sukhanovsky, Magnetohydrodynamics 38,1 4 3 /H208492002 /H20850. 8A. D. Gilbert, Geophys. Astrophys. Fluid Dyn. 44, 241 /H208491988 /H20850. 9A. A. Ruzmaikin, D. D. Sokoloff, and A. M. Shukurov, J. Fluid Mech. 197,3 9 /H208491988 /H20850. 10A. Gilbert and Y. Ponty, Geophys. Astrophys. Fluid Dyn. 93,5 5 /H208492000 /H20850. 11W. Dobler, A. Shukurov, and A. Brandenburg, Phys. Rev. E 65, 036311 /H208492002 /H20850. 12E. G. Blackman and J. C. Tan, Astrophys. Space Sci. 292,3 9 5 /H208492004 /H20850. 13I. Klapper and L. S. Young, Commun. Math. Phys. 173,6 2 3 /H208491995 /H20850. 14F. Pétrélis and S. Fauve, Europhys. Lett. 76, 602 /H208492006 /H20850. 15A. A. Solovyev, Izv. Akad. Nauk. SSSR, Fiz. Zemli 12,4 0 /H208491985 /H20850. 16A. M. Soward, Geophys. Astrophys. Fluid Dyn. 53,8 1 /H208491990 /H20850. 17N. Leprovost and B. Dubrulle, Eur. Phys. J. B 44, 395 /H208492005 /H20850.122104-8 Peyrot, Gilbert, and Plunian Phys. Plasmas 15, 122104 /H208492008 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 08:14:18
1.2400058.pdf	Large cone angle magnetization precession of an individual nanopatterned ferromagnet with dc electrical detection M. V. Costache, S. M. Watts, M. Sladkov, C. H. van der Wal, and B. J. van Wees Citation: Applied Physics Letters 89, 232115 (2006); doi: 10.1063/1.2400058 View online: http://dx.doi.org/10.1063/1.2400058 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/89/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetization switching detection of a single permalloy nanomagnet using magneto-transport measurements J. Appl. Phys. 115, 053912 (2014); 10.1063/1.4864216 Optically induced ultrafast magnetization dynamics in two-dimensional ferromagnetic nanodot lattices AIP Conf. Proc. 1512, 1325 (2013); 10.1063/1.4791550 Angular dependence of ferromagnetic resonance and magnetization configuration of thin film Permalloy nanoellipse arrays J. Appl. Phys. 110, 053921 (2011); 10.1063/1.3633214 Magnetization switching in a mesoscopic NiFe ring with nanoconstrictions of wire J. Appl. Phys. 99, 08C506 (2006); 10.1063/1.2159425 Magnetization pattern of ferromagnetic nanodisks J. Appl. Phys. 88, 4437 (2000); 10.1063/1.1289216 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.139.173.111 On: Wed, 03 Dec 2014 01:58:01Large cone angle magnetization precession of an individual nanopatterned ferromagnet with dc electrical detection M. V. Costache,a/H20850S. M. Watts, M. Sladkov, C. H. van der Wal, and B. J. van Wees Physics of Nanodevices, Materials Science Center, University of Groningen, Nijenborgh 4, 9747 AG Groningen, the Netherlands /H20849Received 5 September 2006; accepted 24 October 2006; published online 8 December 2006 /H20850 The on-chip resonant driving of large cone-angle magnetization precession of an individual nanoscale Permalloy element is demonstrated. Strong driving is realized by locating the element inclose proximity to the shorted end of a coplanar strip waveguide, which generates a microwavemagnetic ﬁeld. A frequency modulation method is used to accurately measure resonant changes ofthe dc anisotropic magnetoresistance. Precession cone angles up to 9° are determined with betterthan 1° of resolution. The resonance peak shape is well described by the Landau-Lifshitz-Gilbertequation. © 2006 American Institute of Physics ./H20851DOI: 10.1063/1.2400058 /H20852 The microwave-frequency magnetization dynamics of nanoscale ferromagnetic elements is of critical importance toapplications in spintronics. Precessional switching usingferromagnetic resonance /H20849FMR /H20850of magnetic memory elements, 1and the interaction between spin currents and magnetization dynamics are examples.2For device applica- tions, methods are needed to reliably drive large angle mag-netization precession and to electrically probe the precessionangle in a straightforward way. We present here strong on-chip resonant driving of the uniform magnetization precession mode of an individualnanoscale Permalloy /H20849Py/H20850strip. The precession cone angle is extracted via dc measurement of the anisotropic magnetore-sistance /H20849AMR /H20850, with angular resolution as precise as 1°. An important conclusion from these results is that large preces-sion cone angles /H20849up to 9° in this study 3/H20850can be achieved and detected. Moreover, measurements with an offset angle be-tween the dc current and the equilibrium direction of themagnetization show dc voltage signals even in the absence ofapplied dc current, due to the rectiﬁcation between inducedac currents in the strip and the time-dependent AMR. Recently we have demonstrated the detection of FMR in an individual, nanoscale Py strip, located in close proximityto the shorted end of a coplanar strip waveguide /H20849CSW /H20850,b y measuring the induced microwave voltage across the strip inresponse to microwave power applied to the CSW. 4How- ever, detailed knowledge of the inductive coupling betweenthe strip and the CSW is required for a full analysis of theFMR peak shape, and the precession cone angle could not bequantiﬁed. In other recent experiments, dc voltages havebeen measured in nanoscale, multilayer pillar structures thatare related to the resonant precessional motion of one of themagnetic layers in the pillar. 5,6In one case the dc voltage is generated by rectiﬁcation between the microwave current ap-plied through the structure and its time-dependent giant mag-netoresistance effect. 6Similar voltages have been observed for a long Py strip that intersects the shorted end of a CSW,which was related in part to rectiﬁcation between microwavecurrents ﬂowing into the Py strip and the time-dependentAMR. 7Figure 1/H20849a/H20850shows the schematic diagram of the device used in the present work. A Py strip is located adjacent to theshorted end of a CSW and contacted with four in-line Ptleads. The CSW, Py strip, and Pt leads were fabricated on aSi/SiO 2substrate in separate steps by conventional e-beam lithography, e-beam deposition, and lift-off techniques. TheCSW consisted of 150 nm Au on 5 nm Ti adhesion layer.Figure 1/H20849b/H20850shows a scanning electron microscopy image of the 35 nm thick Py /H20849Ni 80Fe20/H20850strip, with dimensions 3 /H110030.3/H9262m2and the 50 nm thick Pt contacts /H20849the Py surface was cleaned by Ar ion milling prior to Pt deposition to insuregood metallic contacts /H20850. Pt was chosen so as to avoid picking up voltages due to the spin pumping effect.8,9An AMR re- sponse of /H110111.7% was determined for the strip by four-probe measurement of the difference /H9004Rbetween the resistances when an external magnetic ﬁeld is applied parallel to thecurrent and when it is applied perpendicular. This calibrationof the AMR response will allow accurate determination ofthe precessional cone angle, as described below. Microwave power of 9 dBm was applied from a genera- tor and coupled to the CSW /H20849designed to have a nominal 50/H9024impedance /H20850via electrical contact with a microwave probe. This drives a microwave frequency current of the or-der of 10 mA through the CSW, achieving the highest cur-rent density in the terminating short and thereby generating amicrowave magnetic ﬁeld h 1of the order of 1 mT normal to the surface at the location of the strip. A dc magnetic ﬁeld h0 is applied along the axis of the strip, perpendicular to h1.I n a/H20850Electronic mail: m.v.costache@rug.nl FIG. 1. /H20849a/H20850Schematic diagram of the device. /H20849b/H20850Scanning electron micro- scope image of device with four contacts. /H20849c/H20850The AMR of a typical device.APPLIED PHYSICS LETTERS 89, 232115 /H208492006 /H20850 0003-6951/2006/89 /H2084923/H20850/232115/3/$23.00 © 2006 American Institute of Physics 89, 232115-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.139.173.111 On: Wed, 03 Dec 2014 01:58:01this geometry we have previously shown that we can drive the uniform FMR precessional mode of the Py strip.4All measurements were performed at room temperature. In the AMR effect /H20851see Fig. 1/H20849c/H20850/H20852, the resistance depends on the angle /H9258between the current and the direction of the magnetization as R/H20849/H9258/H20850=R0−/H9004Rsin2/H9258, where R0is the resis- tance of the strip at h0=0. When the dc current and the equilibrium magnetization direction are parallel and the mag-netization of the Py undergoes circular, resonant precessionabout the equilibrium direction, the dc resistance will de-crease by /H9004Rsin 2/H9258c, where /H9258cis the cone angle of the pre- cession. Since the shape anisotropy of our Py strip causesdeviation from circular precession, /H9258cis an average angle of precession. We have used a frequency modulation method in order to better isolate signals due to the resonance state, removing the background resistance signal due to R0and dc voltage offsets in the ampliﬁer. In this method, the frequency of themicrowave ﬁeld is alternated between two different values5 GHz apart, while a dc current is applied through the outercontacts to the strip. A lock-in ampliﬁer is referenced to thefrequency of this alternation /H20849at 17 Hz /H20850, and thus measures the difference in dc voltage across the inner contacts betweenthe two frequencies, V=V/H20849f high/H20850−V/H20849flow/H20850. Only the additional voltage given by the FMR-enhanced AMR effect will be measured when one of the microwave frequencies is inresonance. Figure 2/H20849a/H20850shows a series of voltage vs ﬁeld curves in which both f lowandfhighare incremented in 1 GHz intervals at a dc current of 400 /H9262A. The curves feature dips and peaks at magnetic ﬁeld magnitudes corresponding to the magneticresonant condition with either fhighorflow, respectively. In Fig.2/H20849b/H20850we focus on the peak for f=10.5 GHz, and show curves for different currents ranging from −300 to+300 /H9262A. The peak height scales linearly with the current as expected for a resistive effect /H20851Fig.2/H20849c/H20850/H20852. From the slope of 1.325 m /H9024we obtain an average cone angle /H9258c=4.35° for this frequency.3Interestingly, in Fig. 2/H20849b/H20850a small, somewhat off- center dip is observed even for zero applied current, givingan intercept of −30 nV in Fig. 2/H20849c/H20850. We will discuss this in detail later in this letter. To extract information about the magnetization dynam- ics from the peak shape, we use the Landau-Lifshitz-Gilbert/H20849LLG /H20850equation, dm/dt=− /H9253m/H11003/H92620H+/H20849/H9251/ms/H20850m/H11003dm/dt, where H=/H20849h0−Nxmx,−Nymy,h1−Nzmz/H20850includes the demag- netization factors Nx,Ny, and Nz/H20849where Nx+Ny+Nz=1/H20850, /H9253=2//H926628 GHz/T is the gyromagnetic ratio, /H9251is the dimen- sionless Gilbert damping parameter, and msis the saturation magnetization of the strip. Due to the large aspect ratio of thestrip, N xcan be neglected. In the small angle limit /H20849dmx/dt =0, such that mx/H11229ms/H20850the LLG equation can be linearized. In response to a driving ﬁeld h1cos/H9275twith angular fre- quency /H9275, we express the solutions as a sum of in-phase and out-of-phase susceptibility components, so my =/H9273y/H11032/H20849/H9275/H20850h1cos/H9275t+/H9273y/H11033/H20849/H9275/H20850h1sin/H9275tand mz=/H9273z/H11032/H20849/H9275/H20850h1cos/H9275t +/H9273z/H11033/H20849/H9275/H20850h1sin/H9275t. The components for myare as follows: /H9273y/H11032/H20849/H9275/H20850=−ms 2hc+ms/H9251 /H20849/H9253/H92620//H9275/H208502/H20849h0−hc/H208502+/H92512, /H9273y/H11033/H20849/H9275/H20850=ms 2hc+ms/H20849/H9253/H92620//H9275/H20850/H20849h0−hc/H20850 /H20849/H9253/H92620//H9275/H208502/H20849h0−hc/H208502+/H92512. /H208491/H20850 The components of mzare related to those of myby/H9273z/H11032 =/H9251/H9273y/H11032−/H20849/H9253/H92620//H9275/H20850/H20849hc+Nyms/H20850/H9273y/H11033and/H9273z/H11033=/H9251/H9273y/H11033+/H20849/H9253/H92620//H9275ht/H20850/H20849hc +Nyms/H20850/H9273y/H11032. The resonance ﬁeld hcfor the uniform preces- sional mode is related to /H9275by Kittel’s equation, /H92752=/H92532/H926202/H20849hc+/H208491−Ny/H20850ms/H20850/H20849hc+Nyms/H20850. /H208492/H20850 The precession angle /H9258c/H20849t/H20850is determined from the relation sin2/H9258c/H20849t/H20850/H11229/H9258c2/H20849t/H20850=/H208491/ms2/H20850/H20849my2+mz2/H20850. We ﬁnd that /H9258c2can be written as the sum of a time-independent term and terms with time dependence at twice the driving frequency, /H9258c2 =/H9258dc2+/H9258c2/H208492/H9275t/H20850, where /H9258dc2=/H208491/2/H20850/H20849h1/ms/H208502/H20849/H9273y/H110322+/H9273y/H110332+/H9273z/H110322 +/H9273z/H110332/H20850. The dc voltage is calculated to be V=A1 /H20849/H9253/H92620//H9275/H208502/H20849h0−hc/H208502+/H92512, /H208493/H20850 where A=/H208491/2/H20850Idc/H9004R/H20849h1/2hc+ms/H208502/H208491+/H20849/H9253/H92620//H9275/H208502/H20849hc +Nyms/H208502/H20850. Each peak from the data shown in Fig. 2/H20849b/H20850has been averaged with peaks at the same frequency, and replot- ted in Fig. 3/H20849a/H20850as a function of /H9253/H92620h0//H9275. The solid lines are ﬁts of Eq. /H208493/H20850to the data, in which Aand hcare free ﬁt parameters for each curve, and we have required /H9251to be the same for all of the peaks, resulting in a best-ﬁt value /H9251=0.0104. A plot of the frequency versus the center position of each peak hcis shown in Fig. 3/H20849b/H20850. The excellent ﬁt of Eq. /H208492/H20850to the data veriﬁes that this is the uniform precessional mode, and yields values of /H92620ms=1.06 T and Ny=0.097 as ﬁt parameters. With these values we can extract the drivingﬁeld h 1from the peak ﬁt parameter A/H20851Fig.3/H20849c/H20850/H20852. In agree- ment with our initial estimates, the ﬁeld is of the order of1 mT, but drops off by roughly a factor of 2 between 10 and FIG. 2. /H20849a/H20850dc voltage measured at Idc=400 /H9262A as a function of h0 using the frequency modulation technique, where each curve represents V=V/H20849fhigh/H20850−V/H20849flow/H20850, with flow,high increasing in 1 GHz increments, and fhigh−flowalways 5 GHz. The curves are offset by 700 nV for clarity. /H20849b/H20850 The peak at flow=10.5 GHz for a number of currents between −300 and 300/H9262A./H20849c/H20850The peak height from the data in /H20849b/H20850plotted vs the current. The line is a linear ﬁt to the data.232115-2 Costache et al. Appl. Phys. Lett. 89, 232115 /H208492006 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.139.173.111 On: Wed, 03 Dec 2014 01:58:0120 GHz, consistent with frequency dependent attenuation of our microwave cables and probes. We now discuss the observation of dc voltages in the absence of any applied dc current. In our device, there aremicrowave currents induced in the strip and detection circuitdue to capacitive and inductive couplings to the CSW struc-ture, and thus there is the possibility for rectiﬁcation betweenthe time-dependent AMR and induced microwave currents.We express the induced current as a sum of in-phase andout-of-phase components, I in=I1cos/H9275t+I2sin/H9275t. For rectiﬁcation to occur, the resistance must also have ﬁrst harmonic components. As mentioned earlier, the ellipti-cal precession of the magnetization gives a time dependent term for the cone angle /H9258c2/H208492/H9275t/H20850, but this is only at the second harmonic and cannot produce rectiﬁcation. However, if there is an offset angle /H9278between the applied ﬁeld and the long axis of the strip, then the AMR term is approximately−/H9004R/H20849sin 2/H9278+/H20849my/H20849t/H20850/ms/H20850sin 2/H9278/H20850, obtained by taking a small angle expansion of /H9258y/H20849t/H20850=my/H20849t/H20850/msabout /H9278. Multiplying with the current yields a dc voltage term V=−1 2h1 ms/H9004R/H20849I1/H9273y/H11032+I2/H9273y/H11033/H20850sin 2/H9278. /H208494/H20850 Figure 4shows resonance peaks at f=17.5 GHz and at f=12.5 GHz for ﬁve different angles between the applied ﬁeld and the long axis of the strip. The zero angle /H20849/H9278=0/H20850is with respect to the geometry of our setup; however, it is possible that there is some offset angle /H92780at this position. For the 17.5 GHz data, rotating the ﬁeld by −5° causes thepeak to practically disappear. At −10° the peak reverses sign.This is in agreement with Eq. /H208494/H20850, with an offset angle /H92780= −5°. However, the data at f=12.5 GHz show almost no peak signal already at /H9278=0 even though there has been no change in the setup. Moreover, in this data we more clearly seecontribution from a dispersive line shape, corresponding to theI 2/H9273y/H11033term in Eq. /H208494/H20850. For each frequency, we ﬁt Eq. /H208494/H20850to all the curves simultaneously, where we have used the pa-rameters for the magnetization extracted earlier and allowedonly I 1,I2, and an offset angle /H92780to be free parameters. For the 17.5 GHz data, we obtain /H92780=−6.1°, I1=−28 /H9262A, and I2=8/H9262A. For the 12.5 GHz data, we obtain /H92780=−1.5°, I1=23/H9262A, and I2=11/H9262A. The large difference in /H92780between 12.5 and 17.5 GHz is likely due to a frequency dependenceof the induced currents and how they ﬂow through the Ptcontacts to the Py. The method and analysis presented here are valid for any ferromagnet under the conditions that it exhibits AMR and auniform FMR precession mode. Additional anisotropy ﬁeldsin hard ferromagnets will modify the Kittel equation /H20851Eq. /H208492/H20850/H20852. Situations in which a uniform precession mode cannot be obtained, such as when there are multiple domains and/orresonant modes, 10could also be detected by AMR but will require a more sophisticated analysis. For the purposes ofthis experiment we have used a relatively long strip geom-etry with four in-line contacts. However, two contacts aresufﬁcient and there is no particular limit to how small theferromagnetic element can be, as long as it can be electri-cally contacted and dc current applied along the equilibriummagnetization direction. In terms of resolution, we estimatethat precessional cone angles as precise as 1° can beresolved. This work was ﬁnancially supported by the Dutch Orga- nization for Fundamental Research on Matter /H20849FOM /H20850. The authors acknowledge J. Jungmann for her assistance in thisproject. 1S. Kaka and S. E. Russek, Appl. Phys. Lett. 80,2 9 5 8 /H208492002 /H20850. 2M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, V. Tsoi, and P. Wyder, Nature /H20849London /H20850406,4 6 /H208492000 /H20850. 3At 13 dBm of applied power, we have measured an angle of /H9258c=9.0° at f=10.5 GHz. We focus here on the 9 dBm data, however, since we can apply this power over the entire 10 to 25 GHz bandwidth. 4M. V. Costache, M. Sladkov, C. H. van der Wal, and B. J. van Wees, Appl.Phys. Lett. 89, 192506 /H208492006 /H20850. 5A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and S. Yuasa, Nature/H20849London /H20850438, 339 /H208492005 /H20850. 6J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. Ralph, Phys. Rev. Lett. 96, 227601 /H208492006 /H20850. 7A. Yamaguchi, T. Ono, Y. Suzuki, S. Yuasa, A. Tulapurkar, and Y. Naka- tani, e-print cond-mat/0606305. 8A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin, Phys. Rev.B66, 060404 /H20849R/H20850/H208492002 /H20850. 9M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der Wal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603 /H208492006 /H20850. 10J. Grollier, M. V. Costache, C. H. van der Wal, and B. J. van Wees, J. Appl. Phys. 100, 024316 /H208492006 /H20850. FIG. 3. /H20849a/H20850Resonant peaks at various frequencies ranging from 10.5 to 21.5 GHz in 1 GHz steps as a function of the ﬁeld h0normalized to the frequency. Solid lines are the ﬁts of Eq. /H208493/H20850to the data. /H20849b/H20850The fre- quency of the peak vs its center position hc. The line is a ﬁt of Eq. /H208492/H20850to the data. /H20849c/H20850The ﬁeld h1calculated from the ﬁt coefﬁcients to the data in /H20849a/H20850and /H20849b/H20850as a function of frequency. FIG. 4. Voltage peaks at f=17.5 GHz /H20849left/H20850and at f=12.5 GHz /H20849right /H20850with- out any applied dc current for different angles /H9278between the h0and the long axis of the strip. Solid lines are ﬁts of Eq. /H208494/H20850to the data.232115-3 Costache et al. Appl. Phys. Lett. 89, 232115 /H208492006 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.139.173.111 On: Wed, 03 Dec 2014 01:58:01
1.373000.pdf	Interlayer coupling within individual submicron magnetic elements David J. Smith, R. E. Dunin-Borkowski, M. R. McCartney, B. Kardynal, and M. R. Scheinfein Citation: J. Appl. Phys. 87, 7400 (2000); doi: 10.1063/1.373000 View online: http://dx.doi.org/10.1063/1.373000 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v87/i10 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 04 May 2013 to 142.150.190.39. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsInterlayer coupling within individual submicron magnetic elements David J. Smith,a),b),f)R. E. Dunin-Borkowski,b),c)M. R. McCartney,b)B. Kardynal,d),e) and M. R. Scheinfeina) Arizona State University, Tempe, Arizona 85287 ~Received 22 November 1999; accepted for publication 10 February 2000 ! The interlayer coupling and magnetization reversal of patterned, submicron Co/Au/Ni nanostructures, shaped as diamonds, ellipses, and rectangles, have been investigated using off-axiselectron holography and micromagnetic simulations. Antiferromagnetic coupling between theferromagnetic layers, attributed to the strong Co demagnetization ﬁeld, was visualized directly.Simulated hysteresis loops overall showed reasonable agreement with the experimental results.Local structural imperfections may be responsible for small discrepancies between the observedmagnetization states of the patterned elements and the simulations. © 2000 American Institute of Physics. @S0021-8979 ~00!04010-X # I. INTRODUCTION Quantitative characterization of magnetization reversal mechanisms in submicron-sized magnetic elements is essen- tial for the future development of high-density, magnetic in-formation storage systems. The behavior of thin continuousﬁlms, which can be studied by bulk characterization meth-ods, is rarely a reliable guide for predicting the properties ofsmall magnetic elements of well-deﬁned size and shape. 1 The formation of edge domains in patterned submicron ele-ments strongly inﬂuences their switching ﬁelds, 2while the shape anisotropy of magnetic tunnel junctions ~MTJs !domi- nates their hysteretic response.3The magnetic interactions between two thin, closely separated, ferromagnetic ~FM!lay- ers within individual lithographically deﬁned structures, suchas MTJs or spin valves, can also inﬂuence their switchingmode and coercive ﬁeld. Changes in the separation of theFM materials and their parallel or antiparallel magneticalignment can cause substantial changes in electrical resis-tance when the magnitude and direction of an externally ap-plied magnetic ﬁeld is varied. This effect is commonlytermed giant magnetoresistance ~GMR !. 4 We have previously shown that off-axis electron holog- raphy is a powerful experimental tool for the characterizationof magnetic interactions associated with patterned Conanostructures. 5,6Here, we apply this technique to investi- gate magnetization reversal in magnetically asymmetric, Co/Au/Ni patterned trilayer structures with approximate lateraldimensions of between 100 and 300 nm. We also compareour experimental measurements with simulated domain dis-tributions determined from solutions to the Landau– Lifshitz–Gilbert ~LLG!equations. 7 II. EXPERIMENTAL DETAILS The elements examined here consisted of Co/Au/Ni trilayers patterned into diamonds, ellipses and rectangularbars. They were prepared directly onto self-supporting 55-nm-thick silicon nitride membranes using electron-beam li-thography and lift-off processes. Individual trilayer elementswere well separated laterally in order to reduce inter-elementmagnetic interactions. 6Electron holograms were recorded at 200 kV using a Philips CM200 transmission electron micro-scope equipped with a ﬁeld-emission electron gun. In addi-tion to an electrostatic biprism for generating electron holo-grams, the instrument was equipped with an additional~Lorentz !minilens which allowed holograms to be recorded at magniﬁcations of up to ;70kxand resolutions of ;2n m with the objective lens switched off and the sample locatedin nearly ﬁeld-free conditions. 8The objective lens could also be excited slightly so that magnetization processes and hys-teresis loops could be followed in situby tilting the specimen in a known, previously calibrated, magnetic ﬁeld. 5Figure 1~a!shows a low magniﬁcation bright-ﬁeld image of one array of patterned shapes, while the schematic diagram inFig. 1 ~b!shows the nominal cross-sectional structure of each element. The representative electron hologram in Fig. 1 ~c! illustrates the typical lateral dimensions of the nanostruc-tures, which are thin rectangular bars in this example. Thecorresponding smaller and larger diamonds and ellipses hadsimilar heights as the bars, and widths of 120 or 160 nm. Inall of the experimental results reported below, the contribu-tion of the mean inner potential to the holographic phase wassubtracted from the holograms in order to obtain the mag-netic contribution of primary interest. 9 Micromagnetic simulations incorporated room tempera- ture simulation parameters for Co ~and Ni !, including the exchange stiffness, A51.55 ~and 0.80 for Ni !merg/cm and the saturation magnetization, Ms51414 ~and 440 for Ni ! emu/cm3. The magnetocrystalline anisotropy constant, K,i n our polycrystalline layers ~below 10 nm grain size !was seta!Department of Physics and Astronomy, Arizona State University, Tempe, AZ85287-1504. b!Center for Solid State Science, Arizona State University, Tempe, AZ85287-1704. c!Present Address: Department of Materials, Parks Road, OxfordOX13PH, UK. d!Center for Solid State Electronics Research, Arizona State University, Tempe, AZ85287. e!Present Address: Clarendon Laboratory, Parks Road, OxfordOX13PU, UK. f!Electronic mail: david.smith@asu.eduJOURNAL OF APPLIED PHYSICS VOLUME 87, NUMBER 10 15 MAY 2000 7400 0021-8979/2000/87(10)/7400/5/$17.00 © 2000 American Institute of Physics Downloaded 04 May 2013 to 142.150.190.39. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsto zero. A value for Kof 0 is consistent with the observation that the coercivity in our elements ~ascribed to shape anisot- ropy!is much larger than that typical of bulk ﬁlms of the same thickness, implying that magnetocrystalline anisotropyplays a minor role in the energetics of switching. A gyro-magnetic frequency, g517.6MHz/Oe, and a damping con-stant, a51, were used in the LLG calculations, and the ef- fects of temperature ﬂuctuations were not included. In-plane,discrete moments ~representing a continuous magnetization distribution !were 5.0 nm on each side, and a single layer of moments was used for each magnetically active layer. Thedemagnetization ﬁeld was computed to all orders, couplingthe moments in all cells with each other. Magnetization re-versal processes were followed by assigning an initial do-main structure and then integrating the LLG equations in aﬁxed external ﬁeld until equilibrium was reached. The exitcriteria corresponded to the largest change in the residualdirection cosine of all discretized moments in the grid chang-ing by less than 2 310 25. III. RESULTS AND DISCUSSION Representative results for a selection of the elements are tabulated in three sets of four columns in Fig. 2. The leftcolumn of each set shows a selection of the experimentalresults in the form of the magnetic contributions to the ex-perimental holographic phases during a complete hysteresiscycle. In these observations, the in-plane ﬁeld ~shown at left ! was varied along the vertical direction of the ﬁgure between61930 Oe ~corresponding to 630° sample tilt !in an out- of-plane ﬁeld of 3600 Oe. The different directions of themeasured in-plane magnetization are represented by a con-tinuous color wheel, in which the directions in the plane ofthe ﬁlm are blue ~up!, red ~right!, yellow ~down !, and green ~left!. The intensity of the colors reﬂects the magnitude of the in-plane magnetization. The holographic phase contours,which have a spacing of 0.064 pradians, follow lines of con- stant magnetic induction ~B-ﬁeld strength !: Their separation is proportional to the in-plane component of the magneticinduction integrated in the incident beam direction. Althoughthe data are noisy due to the underlying silicon nitride mem-brane, the contours can still be followed both inside the ele-ments and in the surrounding magnetic leakage ﬁelds. Theemergence of phase contours from the sides of the diamondsand ellipses over substantial portions of the hysteresis cycleindicates that interactions between neighboring elementswould occur if they were placed in close proximity. The experimental results in Fig. 2 show that the switch- ing ﬁelds needed for complete magnetization reversal of thediamond- and elliptical-shaped elements are smaller than forthe rectangular bar. A solenoidal vortex state is visible forthe elliptical shape during both forward and reverse cycles~at2168 and 1336 Oe !. Simulations were unable to repli- cate this vortex structure despite extensive trial-and-error at-tempts, suggesting that structural imperfections may havebeen a contributing factor. 10Vortex states were also seen in the smaller diamond-shaped elements but they were neverobserved in the rectangular bars, presumably because of thenarrow dimensions and the dominant inﬂuence of shape an-isotropy on the magnetic response. 3Signiﬁcantly, the bars are also too thin and narrow to form end-domains, whichgovern the reversal of larger rectangular elements. 11–13In- stead, the phase contours typically curve at their ends by amaximum angle of ;45° just before magnetization reversal FIG. 1. Experimental conﬁguration. ~a!Low magniﬁcation image showing array of small patterned elements in the form of diamonds, rectangles, andellipses; ~b!schematic cross section showing nominal vertical dimensions of Co/Au/Ni trilayer structure. The thin Al overlayer was intended to provideprotection against oxidation and to prevent charging of the sample duringobservation in the electron microscope. ~c!Electron hologram of two rect- angular bars showing typical lateral dimensions.7401 J. Appl. Phys., Vol. 87, No. 10, 15 May 2000 Smith et al. Downloaded 04 May 2013 to 142.150.190.39. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions~see, for example, the 1336 Oe image of the rectangular bar at the bottom of Fig. 2 !. The experimental phase contours have two distinct spac- ings in each element ~narrower at higher applied ﬁelds and wider close to remanence !. These different spacings are as- sociated with the presence of ferromagnetic and antiferro-magnetic coupling between the Ni and Co layers, as dis-cussed below. Measurement of the phase-contour separationsfor both of these two coupled states implies that the thick-nesses of each of the magnetically active layers was close to3 nm, rather than the nominal 10 nm expected from calibra-tion of the electron-beam evaporator used for ﬁlm deposi-tion. Processing the holograms to extract the mean inner po-tential contribution to the holographic phase conﬁrmed thatthe thicknesses of the individual layers were approximatelycorrect, so that over half of each magnetic layer was mag-netically dead. Oxidation during lithographic processing is the most likely origin of this discrepancy. The closest match with the experimental results was achieved for FM layer thicknesses of 3.5 nm, in close agree-ment with the thickness estimates based on the experimentalphase contours. The corresponding simulations for each setof element shapes for 3.5-nm-thick magnetic ﬁlms are shownin the remaining columns of Fig. 2. The columns labeled‘‘Co’’ and ‘‘Ni’’ track the magnetization states of the indi- vidual FM layers within each element during the hysteresiscycle, while those labeled ‘‘total’’ show the computed holo-graphic phase shifts, which can be compared directly withthe experimental data. Changes in the total contour spacingsare apparent between ﬁelds at which the Ni layer has re-versed but the Co layer is still unchanged; similar behavior is FIG. 2. ~Color !Comparison of experimental and simulated magnetization states during complete hysteresis cycles for patterned Co/Au/Ni spin-valve elements in the form of diamonds, ellipses, and rectangular bars. Applied in-plane ﬁelds ~in vertical direction on page !are shown at left. Experimental phase contours are separated by 0.064 pradians. Columns labeled ‘‘Co’’ and ‘‘Ni’’ are simulations for the individual FM layers, and those labeled ‘‘total’’ are simulations for the composite Co/Au/Ni structure.7402 J. Appl. Phys., Vol. 87, No. 10, 15 May 2000 Smithet al. Downloaded 04 May 2013 to 142.150.190.39. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsalso visible in each experimental hysteresis cycle ~see dis- cussion below !. Careful examination of the simulations for the individual FM elements as well as for the composite structures providesfurther insight into the magnetization reversal process. Themost important result is that the Ni layer in each elementreverses its magnetization well before the external ﬁeldreaches 0 Oe, conﬁrming that an antiferromagneticallycoupled state is the normal remanent state that would beobtained after saturation of the element followed by removalof the external ﬁeld. This antiferromagnetic ~AFM !coupling must be due to the ﬂux closure associated with the strongdemagnetization ﬁeld of the closely adjacent and magneti-cally more massive Co layer ~higherM stproduct, where tis the layer thickness !. The darker yellow color of the Ni layer relative to that of the Co indicates that its magnetization isbeing pulled out-of-plane both by the externally applied ﬁeldand by the strong Co demagnetization ﬁeld. The domain structures observed here in these extremely small, coupled magnetic structures clearly differ markedlyfrom those seen in larger elements and single ﬁlm structures.For example, the switching ﬁelds of the Ni and Co elementsare sensitive to their thickness and shape, as well as thesaturation magnetization of each layer. The occurrence ofﬂux closure associated with an antiferromagnetic remanent state contributes to a lack of end domains. This contrastswith the behavior observed in thicker single layer ﬁlms inwhich end domains help to eliminate the external stray ﬁeldsthat would lead to signiﬁcantly higher free energies. Wehave previously reported that larger Co nanostructures of 30nm thickness have solenoidal domain structures over a widerange of applied ﬁelds, 6and similar magnetization vortices have been reported to cause anomalous switching behaviorin 20-nm-thick NiFeCo submicron arrays. 14Conﬁgurations resembling the remanent ‘‘ S’’ and ‘‘C’’ states simulated in Ref. 2 for a single Co layer can be recognized for the bar-shaped element, for example, at 2336 and 1168 Oe, respec- tively, but these conﬁgurations are not the remanent statesfor this coupled system. The switching behavior of several of the trilayer ele- ments is compared in the form of hysteresis loops in Figs. 3and 4. The experimental loops in Fig. 3 were obtained byplotting the magnetic contribution to the total phase differ-ence across the mid-point of each element, as measured di-rectly from the holographic phase contours. For both thediamond- and elliptical-shaped elements, there is a small butnoticeable decrease in the phase soon after the external mag-netic ﬁeld is reduced in strength. This phase decrease is con- FIG. 3. Experimental phase differences measured across patterned Co/Au/Ni trilayer structures during complete hysteresis cycle: ~a!diamonds, ~b!ellipses, ~c! rectangular bars. Solid/dashed lines correspond to larger/smaller elements, respectively. FIG. 4. Hysteresis loops derived from micromagnetic simulations for patterned elements: ~a!diamonds, ~b!ellipses, ~c!rectangular bars. Solid/dashed lines correspond to larger/smaller elements, respectively.7403 J. Appl. Phys., Vol. 87, No. 10, 15 May 2000 Smithet al. Downloaded 04 May 2013 to 142.150.190.39. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionssistent with the magnetization of the Ni layer being gradually pulled out of the plane before ﬁeld reversal occurs. Note alsothat the major Co switching ﬁeld for the bar-shaped elementsis considerably larger than for the other element shapes,which is presumably due to the difﬁculty of nucleating re-versal in a thin narrow element in the absence of end-domainstructures. 11 The simulated loops in Fig. 4 show the fractional mag- netization M/Ms~whereMsis the saturation magnetization in the direction of the applied ﬁeld !, as computed for both smaller and larger elements of each shape. The drop in mag-nitude that occurs in these simulations before 0 Oe corre-sponds to the reversal of the Ni layers but the drop is laterand more abrupt than observed experimentally, perhaps re-ﬂecting some variability in grain size and orientation in theexperimental Ni ﬁlm that helps to facilitate earlier reversal.The square shape for the Co switching in both the experi-mental and simulated loops provides further evidence for theabsence of end domains which are reported to affect the be-havior of large rectangular elements. 13It is also interesting that the smaller of the two diamond-shaped elements has asubstantially larger switching ﬁeld, similar to that of the bar-shaped element. Similar increases in switching ﬁeld with de-creasing element width have been reported previously forthin Co nanoelements, 11and have been attributed to the in- creased difﬁculty of nucleating magnetization reversal. A signiﬁcant outcome of this study is the agreement be- tween the computed and measured phase contour maps, par-ticularly in the ferromagnetically and antiferromagneticallycoupled regimes. The Co element reverses its magnetizationexperimentally at the correct coercive ﬁeld, although the Niswitches earlier and more gradually than expected from thesimulations. A slight left-right asymmetry is also observedexperimentally, possibly due to slight irregularities in theelement shapes or thicknesses. The importance of acquiringhigh quality experimental data is highlighted by the sensitiv-ity of the simulations to a large number of variables. Thediamond- and elliptical-shaped elements examined werelarge enough to support vortices experimentally but we wereunable to form them in the computations without artiﬁcialmeans ~roughness, large magnetization ﬂuctuations, etc. ! during evolution from the saturated state. Experimental fac-tors such as crystal grain size and orientation are likely tohave an increasing inﬂuence on domain conﬁgurations infuture generations of even smaller elements. The reproduc-ibility of the domain structure in successive hysteresis cycleswill also become an important consideration, and may pos-sibly be overcome by careful attention to element shape andaspect ratio. The remanent AFM coupling of closely spacedFM layers will also be of particular relevance in practicaldevice applications. ACKNOWLEDGMENTS This work was partly supported by an IBM subcontract on the DARPA Advanced MRAM Project under ContractNo. MDA-972-96-C-0014. The authors thank James Speidellat IBM for providing Si 3N4membranes, and they acknowl- edge use of facilities in the Center for High Resolution Elec-tron Microscopy at Arizona State University. 1J. N. Chapman, P. R. Aitchison, K. J. Kirk, S. McVitie, J. C. S. Kools, and M. F. Gillies, J. Appl. Phys. 83, 5321 ~1998!. 2Y. Zheng and J.-G. Zhu, J. Appl. Phys. 81, 5470 ~1997!. 3Y. Lu, R. A. Altman, A. Marley, S. A. Rishton, P. L. Trouilloud, G. Xiao, W. J. Gallagher, and S. S. P. Parkin, Appl. Phys. Lett. 70,2 6 1 0 ~1997!. 4S. S. P. Parkin, Annu. Rev. Mater. Sci. 25,3 5 7 ~1995!. 5R. E. Dunin-Borkowski, M. R. McCartney, B. Kardynal, and D. J. Smith, J. Appl. Phys. 84, 374 ~1998!. 6R. E. Dunin-Borkowski, M. R. McCartney, B. Kardynal, D. J. Smith, and M. R. Scheinfein, Appl. Phys. Lett. 75, 2641 ~1999!. 7M. R. Scheinfein, J. Unguris, J. L. Blue, K. J. Coakley, D. T. Pierce, R. J. Celotta, and P. J. Ryan, Phys. Rev. B 43, 3395 ~1991!. 8M. R. McCartney, D. J. Smith, R. F. C. Farrow, and R. F Marks, J. Appl. Phys.82, 2461 ~1997!. 9R. E. Dunin-Borkowski, M. R. McCartney, D. J. Smith, and S. S. P. Parkin, Ultramicroscopy 74,6 1~1998!. 10Y. Zheng and J.-G. Zhu, J. Appl. Phys. 85, 4776 ~1999!. 11M. Ruhrig, B. Khamsepour, K. J. Kirk, J. N. Chapman, P. R. Aitchison, S. McVitie, and C. D. Wilkinson, IEEE Trans. Magn. 32, 4452 ~1996!. 12K. J. Kirk, J. N. Chapman, and C. D. W. Wilkinson, Appl. Phys. Lett. 71, 539~1997!. 13J. Shi, T. Zhu, M. Durlam, E. Chen, S. Tehrani, Y. F. Cheng, and J.-G. Zhu, IEEE Trans. Magn. 34, 997 ~1998!. 14J. Shi, S. Tehrani, T. Zhu, Y. F. Zheng, and J.-G. Zhu, Appl. Phys. Lett. 74, 2525 ~1999!.7404 J. Appl. Phys., Vol. 87, No. 10, 15 May 2000 Smithet al. Downloaded 04 May 2013 to 142.150.190.39. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
1.4985662.pdf	Magnetic domain wall engineering in a nanoscale permalloy junction Junlin Wang , , Xichao Zhang , , Xianyang Lu , , Jason Zhang , , Yu Yan , , Hua Ling , , Jing Wu , , Yan Zhou , and , and Yongbing Xu Citation: Appl. Phys. Lett. 111, 072401 (2017); doi: 10.1063/1.4985662 View online: http://dx.doi.org/10.1063/1.4985662 View Table of Contents: http://aip.scitation.org/toc/apl/111/7 Published by the American Institute of Physics Articles you may be interested in Spin pumping torque in antiferromagnets Applied Physics Letters 110, 192405 (2017); 10.1063/1.4983196 Excitation of coherent propagating spin waves in ultrathin CoFeB film by voltage-controlled magnetic anisotropy Applied Physics Letters 111, 052404 (2017); 10.1063/1.4990724 Electric-field tuning of ferromagnetic resonance in CoFeB/MgO magnetic tunnel junction on a piezoelectric PMN-PT substrate Applied Physics Letters 111, 062401 (2017); 10.1063/1.4997915 Magneto-ionic effect in CoFeB thin films with in-plane and perpendicular-to-plane magnetic anisotropy Applied Physics Letters 110, 012404 (2017); 10.1063/1.4973475 Spin-orbital coupling induced four-fold anisotropy distribution during spin reorientation in ultrathin Co/Pt multilayers Applied Physics Letters 110, 022403 (2017); 10.1063/1.4973884 Device-size dependence of field-free spin-orbit torque induced magnetization switching in antiferromagnet/ ferromagnet structures Applied Physics Letters 110, 092410 (2017); 10.1063/1.4977838Magnetic domain wall engineering in a nanoscale permalloy junction Junlin Wang,1,2Xichao Zhang,3Xianyang Lu,4Jason Zhang,4YuY an,2Hua Ling,2 Jing Wu,4Ya nZhou,3and Y ongbing Xu1,2,a) 1York-Nanjing International Center of Spintronics (YNICS), Collaborative Innovation Center of Advanced Microstructures, School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China 2Spintronics and Nanodevice laboratory, Department of Electronics, The University of York, York YO10 5DD, United Kingdom 3School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518067, China 4Department of Physics, The University of York, York YO10 5DD, United Kingdom (Received 31 May 2017; accepted 4 August 2017; published online 14 August 2017) Nanoscale magnetic junctions provide a useful approach to act as building blocks for magnetoresistive random access memories (MRAM), where one of the key issues is to control themagnetic domain conﬁguration. Here, we study the domain structure and the magnetic switching in the Permalloy (Fe 20Ni80) nanoscale magnetic junctions with different thicknesses by using micromagnetic simulations. It is found that both the 90-/C14and 45-/C14domain walls can be formed between the junctions and the wire arms depending on the thickness of the device. The magnetic switching ﬁelds show distinct thickness dependencies with a broad peak varying from 7 nm to 22 nm depending on the junction sizes, and the large magnetic switching ﬁelds favor the stability ofthe MRAM operation. VC2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http://creativecommons.org/ licenses/by/4.0/ ).[http://dx.doi.org/10.1063/1.4985662 ] The magnetoresistive random access memory (MRAM)1–6 based on the tunneling magnetic resistance (TMR) effect has the potential to replace all existing memory devices in a computer or other hard disk d rives as it could provide a high read/write operation speed and is also nonvolatile.7–9 On the other hand, the magnetic domain wall gives a ﬂexi- ble approach in the data storage and the logic circuit.10–13 Compared with the TMR-based MRAM devices, a domain wall motion-based magnetic junction could have a single layer structure, which might have great advantages in termsof fabrication and application. 14,15The magnetic junction shows several types of the magnetoresistance effect by applying the magnetic ﬁeld.16,17 The magnetic switching in the junction structure can be controlled by either the external magnetic ﬁeld or the applied electrical current.18,19Recently, it has been reported that the magnetic switching induced by the spin-transfer tor- que (STT) can enable the junction to work as a STT-MRAM device.20,21There are also reports about the spin-polarized current that can induce the junction to generate spin waves.22–24It is found that the Permalloy junction has sev- eral metastable magnetization states, which can be used tostore the information. 18Thus, the reliable control of the magnetic domain conﬁguration in the magnetic junction is an important task. In this letter, we present a micromagneticstudy of the domain structures and the magnetic switching in the nanoscale Permalloy junctions within magnetic cross structures with different thic knesses. The numerical simulations are carried out by using the Object Oriented MicroMagnetic Framework (OOMMF) software. 25It is found that the junc- tion thickness has distinct effects on the domain wall conﬁg-uration, the initial magnetic switching, and the coercivity. Inthe initial states, both 45 /C14and 90/C14domain walls are found to be formed in the studied model. Both the initial magneticswitching and the coercivity show the nonlinear dependences on the thickness, indicating the importance of controlling the thickness for the writing process when the nanoscalePermalloy junction works as a building block for informationstorage devices. The micromagnetic simulations are performed using the standard micromagnetic simulator OOMMF software, 25which stands on the Landau-Lifshitz-Gilbert equation26,27 dM dt¼/C0 j cjM/C2Heffþa MSM/C2dM dt/C18/C19 ; (1) whereMis the magnetization of the magnetic layer, MSis the saturation magnetization, cis the Gilbert gyromagnetic ratio, and ais the damping constant. Heffis the effective ﬁeld, which is derived from the magnetic energy density Heff¼/C0 l/C01 0de dM; (2) where econtains the Heisenberg exch ange, anisotropy, applied magnetic ﬁeld, and demagnetization energy terms. The magnetic material used for the micromagnetic simu- lation is Permalloy, i.e., Fe 20Ni80alloy, which has a low coer- civity and a high permeability. The size of the cross structure of the nanoscale junction is deﬁned as 10 nm /C210 nm with a varying thickness from 2.5 nm to 25 nm, and the length of allthe wire arms is ﬁxed at 200 nm. The saturation magnetiza-tion is equal to 8.6 /C210 5A/m. The exchange stiffness and the crystalline anisotropy constant are set at 13 /C210–12J/m and 0 J/m3, respectively. The simulation cell size is set as 2.5 nm /C22.5 nm /C22.5 nm, which is compared with the exchange length (5.3 nm) of Permalloy.a)Electronic mail: yongbing.xu@york.ac.uk 0003-6951/2017/111(7)/072401/4 VCAuthor(s) 2017. 111, 072401-1APPLIED PHYSICS LETTERS 111, 072401 (2017) The initially relaxed magnetization distributions around the junctions of the cross structures with different thick- nesses are shown in Fig. 1(a), which are obtained by relaxing the cross structures with random magnetization distributions.The spins in the wire arms are all aligned in parallel along the wire directions due to the strong shape anisotropy. There are three types of domain conﬁgurations around the junc- tions. For the junctions with the thicknesses of 2.5 nm, 7.5 nm, and 12.5 nm, the spins in the junction are aligned inparallel with the spins in one of the wires, and the 90 /C14 domain walls form between another wires. For the caseswith the thicknesses of 5 nm and 15 nm, the spins in the junc- tions are aligned largely in parallel, which can be described as a single magnetic domain or a coherent spin block (CSB), and the spin direction of these CSBs is in 45 /C14with those in both wires. For the cases with thickness of 10 nm, the spinsin the junction are form a 90 /C14domain wall within the junc- tion. Indeed, the initial magnetization distribution in the cross structure can also be controlled by the applying an external magnetic ﬁeld. As shown in Fig. 1(b), the initial magnetization distribution in the cross structure can be modi-ﬁed to be 45 /C14domain by applying a magnetic ﬁeld pointing at an angle of 45/C14to the þx-direction. The required ampli- tude of the magnetic ﬁeld corresponding to different thick- ness is given in Fig. 1(c). The applied magnetic ﬁeld changes the domain structure to the coherence switching mode fromthe initially relaxed states, where the required magnetic ﬁeld is different for samples with different initially relaxed mag- netization distribution and thickness. Note that the magneti- zation distribution conﬁguration in the thickness direction is uniform (see supplementary material , Fig. S1).In the following, we study the magnetization switching process driven by an external magnetic ﬁeld for the junctions with different thicknesses. The simulated hysteresis loops arethe same in different layers of the device. Figure 2shows the result for the 2.5-nm-thick junction. The magnetic ﬁeld is ﬁrst applied along the þx-direction, of which the amplitude ﬁrst increases from 0 Oe to 2000 Oe and reduces to 0 Oe. Then, the magnetic ﬁeld changes in the same manner but along the –x-direction. The simulated hysteresis loop is given in Fig. 2, and the magnetization conﬁgurations illustrated in Fig. 3are corresponding to the marked states in the hysteresis loop given in Fig. 2, which represent the magnetic switching pro- cess in the nanoscale junction. The initial magnetization con- ﬁguration in the nanoscale junction is given in Fig. 3. The magnetic ﬁeld of the ﬁrst magnetic switching from the ini- tially relaxed state to the state with a 45 /C14domain wall is deﬁned as the initial magnetization switching ﬁeld ( Hi), which is indicated in Fig. 2(a). As the applied magnetic ﬁeld increases from 0 Oe to 2000 Oe, the direction of the magneti- zation in the junction is switched where the amplitude of the critical switching ﬁeld, i.e., the coercivity ( Hc) of the junction as indicated in Fig. 2(b), is equal to 1050 Oe. The snapshots of the switching process given in Fig. 3 further show that the switching of the magnetization in thejunction is coherent. The angle between the x-axis and the spins at the junction is deﬁned as h. Before applying the magnetic ﬁeld, the spins in the junction are in parallel with they-axis, and the his equal to –90 /C14. By increasing the applied magnetic ﬁeld above 2000 Oe, hincreases and then reaches 0/C14. When the applied magnetic ﬁeld is reduced to 0 Oe, hdecreases to –45/C14. That means in the remanence FIG. 1. (a) The magnetic domain con- ﬁgurations in the cross structure relaxed from random magnetization distribu- tions for different thicknesses. (b) The magnetic domain conﬁgurations obtained by applying a magnetic ﬁeld. (c) The amplitude of the magnetic ﬁeld which changes the domain conﬁgura-tions shown in (a) to those shown in (b).072401-2 Wang et al. Appl. Phys. Lett. 111, 072401 (2017)state, the spins in the junction are aligned 45/C14away from the wire direction, and 45/C14domain walls are formed between the junction and the wires. When the applied mag- netic ﬁeld decreases from 0 Oe to –2000 Oe in the x-direc- tion, the spins change to be paralleled with the x-direction, and the hincreases from –45/C14to 0/C14. It is found that the whole magnetization switching process in the junction iscoherent and reversible. The typical spin conﬁgurations during the magnetic switching process of the junctions with different thicknessesare shown in Fig. 4. From Fig. 4(a), we found that the junc- tion with a different thickness usually has a different relaxed state. Before using the magnetic ﬁeld to achieve reversible magnetic switching in the cross structure, the spins in the cross structure have to be tuned to the coherent switchingmode. The coherent switching mode is deﬁned as the states in Fig. 3(b) which shows 45 /C14domain walls. The processes to enable the coherence switching modes are shown in Fig. 4. Unlike the magnetic switching processes shown in Fig. 3, the relaxed magnetization conﬁguration at 0 Oe is different fromthe junctions with the thicknesses of 5 nm, 7.5 nm, 10 nm, 12.5 nm, and 15 nm. The spin conﬁgurations of the junctions with the thicknesses of 5 nm and 15 nm are similar to the spin conﬁgurations in Fig. 3(d) which are in the coherent switching mode. The junction with a thickness of 7.5 nm hasthe parallel spins in the cross structure with a has –90 /C14 between the x-direction. As the magnetic ﬁeld increases to 3000 Oe, hdecreases to 0/C14in the junction. When themagnetic ﬁeld reduces back to 0 Oe, hincreases to 45/C14, and the CSB is formed. From the junction with a thickness of10 nm, the initial 90 /C14DW within the junction can be elimi- nated, and the CSB can be formed by controlling the magne- tization process. When the thickness of the junction is12.5 nm, the spins in the cross structure requires a large mag- netic ﬁeld up to 3825 Oe to reverse the magnetization direc- tion. For the whole cross structures with the thicknesses of5 nm, 7.5 nm, and 10 nm, the spin conﬁgurations in the y-arms have not been changed, and the spin conﬁguration in thex-arms can be switched. However, for the cross structures with thicknesses of 12.5 nm and 15 nm, the spin directions rotate along the y-axis as well. The magnetic switching ﬁeld of the junction can be affected by the thickness of the junction. Figure 5shows H c as a function of the thickness for the junctions with differentFIG. 2. The hysteresis loop for the cross structure with the thickness of 2.5 nm. The initial switching magnetic ﬁeld ( Hi) and the coherence switching magnetic ﬁeld ( Hc) are indicated in (a) and (b), respectively. FIG. 3. The magnetic domain conﬁgurations of the 2.5-nm-thick junction at different applied magnetic ﬁelds. The magnetic ﬁeld is applied along the x-direction. The labels correspond to the states indicated in Fig. 2, i.e., (a) H¼0 Oe, (b) H¼600 Oe (before switching), (c) H¼2000 Oe (after switch- ing), (d) H¼0 Oe, (e) H¼– 1025 Oe (before switching), (f) H¼– 1050 Oe (after switching), and (g) H¼– 2000 Oe. FIG. 4. The magnetic domain conﬁgurations of the junctions at different applied magnetic ﬁelds for different thicknesses. The magnetic ﬁeld is applied along the x-direction.072401-3 Wang et al. Appl. Phys. Lett. 111, 072401 (2017)lateral sizes. The size of the device is varying from 100 nm /C2100 nm to 400 nm /C2400 nm, i.e., the length of the cross structure is varying from 5 nm to 20 nm. Hcincreases ﬁrst with increasing thickness and then decreases, showing abroad peak from 7 nm to 22 nm depending on the junction sizes. The reason is that for the junction with a certain cross section size, the magnetization switching is coherent. Thus,H cis proportional to the total magnetization, which also means Hcincreases with the thickness as the total magnetiza- tion is proportional to the thickness. However, when thethickness is larger than a certain critical value, multiple domains can be formed during the magnetization switching, leading to incoherent magnetization switching. In such acase, H cdecreases with the thickness, as thicker junction is more likely to form multiple domains, due to the demagneti- zation effect. Besides, the critical value of the thicknessincreases with increasing lateral dimensions of the junction. The reason is that the amplitude of the demagnetization effect resulting in the incoherent switching is proportional tothickness and is inversely proportional to the lateral dimen- sions of the junction (see supplementary material , Figs. S2–S4). We note that there are large magnetic switchingﬁelds of around 2500 Oe, which favor the stability of the MRAM operation. The large magnetic switching ﬁelds, how- ever, may lead to large current needed for the switching ofthe junction. One may need to explore the spin torque trans- fer or thermal assistant switching for future applications. In conclusion, we have carried out a full micromagnetic study on the magnetization conﬁguration and the magnetic switching process in a nanoscale Permalloy junction. The relationship between the magnetic switching ﬁelds and thethickness of the nanoscale junction has been investigated. While different types of domain walls can be formed in the initially relaxed states depending on the speciﬁc thicknesses,the junction acts as a single CSB where the spins are aligned in parallel during the magnetization process. The magnetiza- tion direction can be controlled and switched coherentlyby applying an external magnetic ﬁeld. Both the initial mag- netization ﬁeld and the coercivity are found to depend on the thickness, and the large coercivity could enhance thestability of the device operation. Our work shows that the nanoscale magnetic junction has the potential to be used as abuilding block for future spin-based data storage or logiccomputing technologies. Seesupplementary material for magnetization distribu- tion in device at different layers and side views of the mag-netization distribution in device. This work was supported by State Key Program for Basic Research of China (Grant Nos. 2014CB921101 and2016YFA0300803), NSFC (Grants Nos. 61427812 and11574137), Jiangsu NSF (No. BK20140054), JiangsuShuangchuang Team Program, Shenzhen FundamentalResearch Fund under Grant Nos. JCYJ20160331164412545and the UK EPSRC (EP/G010064/1). 1J. A˚kerman, Science 308, 508 (2005). 2E. Chen, D. Apalkov, Z. Diao, A. Driskill-Smith, D. Druist, D. Lottis, V. Nikitin, X. Tang, S. Watts, S. Wang, S. A. Wolf, A. W. Ghosh, S. Lu, J.W. Poon, W. H. G. S. Stan, M. Butler, C. K. A. Mewes, T. Mewes, and P.B. Visscher, IEEE Trans. Magn. 46, 1873 (2010). 3J.-G. Zhu, Proc. IEEE 96, 1786 (2008). 4T. Pramanik, U. Roy, L. F. Register, and S. K. Banerjee, IEEE Trans. Nanotechnol. 14, 883 (2015). 5Y. Yan, C. Lu, H. Tu, X. Lu, W. Liu, J. Wang, L. Ye, I. Will, B. Kuerbanjiang, V. K. Lazarov, J. Wu, W. Johnny, B. You, J. Du, R. Zhang,and Y. Xu, AIP Adv. 6, 095011 (2016). 6B. Liu, X. Ruan, Z. Wu, H. Tu, J. Du, J. Wu, X. Lu, L. He, R. Zhang, and Y. Xu, Appl. Phys. Lett. 109, 042401 (2016). 7J. J. Nahas, T. W. Andre, B. Garni, C. Subramanian, H. Lin, S. M. Alam, K. Papworth, and W. L. Martino, IEEE J. Solid-State Circuits 43, 1826 (2008). 8S. Tehrani, J. Slaughter, E. Chen, M. Durlam, J. Shi, and M. DeHerren,IEEE Trans. Magn. 35, 2814 (1999). 9M. Krounbi, V. Nikitin, D. Apalkov, J. Lee, X. Tang, R. Beach, D. Erickson, and E. Chen, ECS Trans. 69, 119 (2015). 10D. A. Allwood, G. Xiong, C. Faulkner, D. Atkinson, D. Petit, and R. Cowburn, Science 309, 1688 (2005). 11S. S. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 12S. Lepadatu and Y. Xu, Spintronic Mater. Technol. 2006 , 179. 13S. Li, A. Samad, W. Lew, Y. Xu, and J. Bland, Phys. Rev. B 61, 6871 (2000). 14J. A. Currivan, Y. Jang, M. D. Mascaro, M. A. Baldo, and C. A. Ross,IEEE Magn. Lett. 3, 3000104 (2012). 15S. Tehrani, B. Engel, J. Slaughter, E. Chen, M. DeHerrera, M. Durlam, P. Naji, R. Whig, J. Janesky, and J. Calder, IEEE Trans. Magn. 36, 2752 (2000). 16W. J. Gallagher and S. S. Parkin, IBM J. Res. Dev. 50, 5 (2006). 17Y. Xu, C. Vaz, A. Hirohata, C. Yao, W. Lee, J. Bland, F. Rousseaux, E. Cambril, and H. Launois, J. Appl. Phys. 85, 6178 (1999). 18Y. Xu, C. Vaz, A. Hirohata, H. Leung, C. Yao, J. Bland, E. Cambril, F. Rousseaux, and H. Launois, Phys. Rev. B 61, R14901 (2000). 19G. D. Chaves-O’Flynn, G. Wolf, D. Pinna, and A. D. Kent, J. Appl. Phys. 117, 17D705 (2015). 20U. Roy, T. Pramanik, M. Tsoi, L. F. Register, and S. K. Banerjee, J. Appl. Phys. 113, 223904 (2013). 21U. Roy, H. Seinige, F. Ferdousi, J. Mantey, M. Tsoi, and S. Banerjee, J. Appl. Phys. 111, 07C913 (2012). 22T. Pramanik, U. Roy, M. Tsoi, L. F. Register, and S. K. Banerjee, J. Appl. Phys. 115, 17D123 (2014). 23J. Z. Sun, Phys. Rev. B 62, 570 (2000). 24Y. Zhou, J. A ˚kerman, and J. Z. Sun, Appl. Phys. Lett. 98, 102501 (2011). 25M. J. Donahue and D. G. Porter, OOMMF User’s Guide (US Department of Commerce, Technology Administration, National Institute of Standardsand Technology, 1999). 26K. I. Bolotin, F. Kuemmeth, and D. Ralph, Phys. Rev. Lett. 97, 127202 (2006). 27M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53, 2497 (1984).FIG. 5. The coercivity Hc, i.e., the switching ﬁeld, as a function of the thick- ness for the junctions with different cross section sizes.072401-4 Wang et al. Appl. Phys. Lett. 111, 072401 (2017)
1.881375.pdf	Modeling Oceanic and Atmospheric Vortices David G. Dritschel Bernard Legras Citation: Physics Today 46, 3, 44 (1993); doi: 10.1063/1.881375 View online: http://dx.doi.org/10.1063/1.881375 View Table of Contents: http://physicstoday.scitation.org/toc/pto/46/3 Published by the American Institute of PhysicsMODELING OCEANIC AND ATMOSPHERIC VORTICES Violent storms in the English Channel, the growth of the ozone hole and the Grea t Red Spot of Jupiter are all problems in planetary fluid dynamics that challenge our most powerful computers . David G. Dritschel and Bernard Legras Many of the physical systems in the universe are fluids. Our understanding of the Earth, the planets, stars and even galaxie s depends crucially on fluid dynamics. This mathematical discipline has been instrumental in the development of meteorology, oceanography and, to a lesser extent, astrophysics. From the beginning, observations and calculations have made apparent the rich dynamical structure of astronomical fluids: Satellite observations told us that the Great Red Spot of Jupiter was simply the most prominent object within the turbulent Jovian atmosphere, which exhibits hundreds of observable vorti- ces embedded in a complex structure of bands or jets extending from one pole to the other. Similar structures have been observed, though in less detail, on the other Jovian planets. Complex dynamical structures are also believed to characterize condensing nebulae, evolving galaxies, stellar convective layers and the accretion disks of neutron stars and black holes. The most easily observable of all astrophysical fluid systems is the liquid and gaseous envelope of the Earth, which offers a fascinating variety of coherent structures superimposed on the mean general circulation of winds and oceanic currents. By "coherent structure" we mean here any observable pattern occurring repetitively with a lifetime greatly exceeding its own internal circulation time scale. This pattern makes it very unlikely that coherent structures are just turbulent fluctuations; they deman d a more specific explanation. A coherent structure is associated with the collective motion of part of a fluid; it necessaril y involves long-range interaction. Understand- ing the cause of such collective motion and exploiting it for David Dritschel is an Advanced Research Fellow of the UK Natural Environmental Research Council and a fellow of St. Catherine's College, Cambridge. Bernard Legras is a Research Director at the CNRS Laboratory for Dynamic Meteorology at the Ecole Normale Superieure in Paris.the computer modeling of fluid behavior remains a challenging research goal. In spite of our rapid progress in understanding weakly turbulent systems, it is the existence of these coherent structures in fluids that has made the theory of turbulence so difficult and elusive. The problem is that the coherent structures largely determine the statistical behavior of the fluid and thus of turbulence. This is particularly true in geophysical systems, where the interaction between differ- ent parts of the fluid is generally stronger than in isotropic, three-dimensional fluids, for which the theory of turbulence has been more successful (though they still exhibit coherent structures). In geophysical fluid systems, many of the coherent structures are vortices—local masses of rapidl y rotating fluid. We find vortices on all scales, from little dust devils on a hot summer day to ocean eddies a few tens of kilometers wide to low-pressure systems a thousand kilometers across. The polar vortex, thousands of kilo- meters in diameter, is the largest of these geophysical vortices. Numerical modeling and prediction The existence of such structures on all scales makes prediction difficult. And reliable prediction is, after all, the most sought-after goal of geophysical fluid dynamics. The practical need to know what to expect of the rapidly changing atmosphere has guided the development of the subject, and the complexity of this physical system has always called for the most powerful computers of the day. From the start, geophysical fluid dynamics has had to lean heavily on numerical computation. There are two reasons for this dependency: First of all, the conceptual math- ematical models that have been developed to account for geophysical flows are difficult to compare with observa- tions. Second, nonlinearities are everywhere, drastically limiting our ability to understand the kind of dynamical behavior they generate. The gigantic atmospheric vortex flowing around the North Pole, shown in figure la, is an 44 PHYSICS TODAY MARCH 1993 © 1993 American Insrirure of PhysicsPolar vortex over Arctic and north temperate zones on 27 January 1992. a: Observed mid-stratosphere distribution of potential vorticity q, which is essentially the scalar product of the thermal gradient and the curl of the wind field. Regions of highest q are shown in red; regions of lowest q in blue, b: Predicted q distribution for the same day obtained by "contour surgery" applied to the distribution observed 12 days earlier. (See box on page 49.) Movement of contours of constant q is calculated from winds initialized to the 1 2-day-old observations . This lets one see fine structure like filament s and the vortex's sharp edge, and it facilitates the study of ozone depletion. (See D. Waugh, R. Plumb, submitted to/. Atmos. Sci., 1993.) Figure 1 example of such nonlinear behavior. Figure lb shows a novel computer prediction and enhancement of the same polar vortex. In geophysical fluid dynamics, numerical simulations have always been regarded as a source of experimental data, and these "data" have in fact been used to develop much of the field's phenomenology. A lot of research on geophysical fluids is done with comprehensive computer models that try to include all the significant physical objects and processes, such as moun- tains, precipitation, solar heating, cloud formation, mois- ture and boundary-layer interaction s of the atmosphere with land and sea. Moisture and surface interactions , for example, are truly microscale processes whose macroscale effects are not well understood; comprehensive models treat them very approximately, and in some cases questionably. Even the basic fluid dynamical equations themselves are treated only within a mathematical and— more importantly—a numerical approximation. Ther e appears to be no alternative to this approach if one is to understand systems as complex as the atmo- sphere and the oceans well enough to make useful predictions. But this strategy is still drastically limited by computer resources and by our ability to model phenome- na occurring on scales smaller than the grid size of a particular model. The most advanced models used for numerical weather forecasting have a horizontal resolu- tion of the order of 100 km, which does not resolve, for in- stance, convective cells, fronts, small-scale turbulenc e or even drag by mountain ranges. Consequently the model- ing of these crucial processes is based not on a directrepresentation but rather on a parameterization of their effects on large-scale motion. Such parameterizations are valid only to the extent that the large-scale flow entirely determines behavior on a scale finer than the grid. This implies the existence of mathematical procedures that allow one to compute the feedback from small to large scales, thus shortcutting the explicit representation of sub- grid-scale phenomena. Moreover, such procedures would have to be computationally efficient. In general, however, no such mathematical procedures exist, and therefore the parameterizations must rely on empirical or ad hoc assumptions. For instance, sub-grid-scale turbulence is generally modeled as a purely diffusive process acting on the large- scale flow, even though this contradicts what we actually know from high-resolution experiments. Ther e are also problems with the way the atmospher e is divided for numerical calculation into a series of horizontal layers. These problems are most acute in atmospheric models used for climate studies, where one integrates over much longer durations than are interesting foi weather forecast- ing. At present such studies have a horizontal resolution of 500 km at best. The situation is no better for the ocean, despite its greater physical simplicity. Ther e are, for example, no clouds in the ocean, but one needs very fine horizontal resolution—down to a few kilometers—if one is to catch the most energetic marine processes and deal with coastal boundary effects and atmospheric stress. With all these limitations, general circulation models are nonethe - less very costly to run. This necessitates sacrifices: One PHYSICS TODAY MARCH 1993 45Potential vorticity map from the North Pole to the tropic of Cancer averaged over five days in February 1986. Highest positive q is shown in blue. Near the tropics q approache s zero (red). On this map a strong penetration of subtropical air to Scandinavia blocks perturbations coming from the Atlantic. This low-<7 blocking cente r is stabilized by a high-<7 center over eastern Europe. (Courtesy of Gilbert Brunet, Ecole Normal Superieure.) Figure 2 can't explore parameter space or verify statistical signifi- cance as thoroughly as one would like. Often one cannot even identify the basic cause of an observed effect because there are so many competing processes. That's where simplified, idealized mathematical mod- els can play a role. Such models are stripped-down versions of comprehensive models. Sometimes they con- centrate exclusively on the underlying fluid dynamical processes and sometimes they are restricted to a small number of physical processes in simplified form. Simula- tions done with these simplified models are generally much less expensive than the more comprehensive simula- tions. They let one concentrate on specific physical processes and explore the effect of numerical resolution on accuracy and predictability. They can be used to test and thus improve the parameterization of sub-grid-scale turbu- lence used in the comprehensive models. Other models, concentrating on small-scale processes while neglecting large-scale horizonta l processes, are used to investigate heat transfer effects or local three-dimensional turbulence of the kind one finds in the atmospheric boundary layer. Our understanding of atmospheric and oceanic dy- namics is in fact based on a dual approach. On one side we have the comprehensive strategy of global general circula- tion models, and on the other side we have a hierarchy of simplified models that can study individual processes in isolation. As usual in science, reduction is often a prerequisite to understanding. It is fortunate that many of the self-organization properties of geophysical fluids are exhibited by simple models. Simplified models also permit a radical new approach to numerical simulation, visualization and diagnosis. (Seethe article by Norman Zabusky, Deborah Silver, Richard Pelz and colleagues on page 24). It is, for instance, sometimes possible to use alternative mathematical for- mulations of the fluid dynamical equations to achieve extremely fine spatial resolution. The results from such models have led to reduced-degree-of-freedom models of fundamental fluid processes that have helped us interpret the results from the comprehensive models.1 Idealized systems Let us examine several idealized systems that show real promise as guides to understanding certain fundamental aspects of fluid flow in the atmosphere, in the oceans and even on other planets. For the most part these flows are rapidly rotating and strongly stratified by density. These two propertie s force the fluid to move predominantly within horizontal strata, particularly at large scales. In general the fluid motion in one stratum differs from that in other strata, but the motion depends on interactions between strata. The simplest idealized model of this layerwise two- dimensional system is the "quasigeostrophic " model, in which the horizontal pressure gradient is nearly balanced by the Coriolis force. This model simplifies the full fluid dynamical equations by considering the weak vertical motion only in leading order. Perturbative departures of pressure, density and temperature from standard vertical profiles are deduced from the weak displacement of density surfaces caused by the vertical motion. An important atmospheric field variable in the quasigeostrophic model as well as in the full fluid dynamical equations is q, the so-called potential vorticity. It depends on the vorticity <o, which is just the curl of the velocity field, the density/? and the entropy. The potential vorticity is so important because in real geophysical flows, the global distribution of q on surfaces of constant entropy in the atmosphere (or constant density in the oceans) largely determines the entire fluid motion. Furthermore, the redistribution of q by the velocity field leaves it largely undiluted. That is to say, every parcel of fluid retains its potential vorticity. In the quasigeostrophic model poten- tial vorticity is precisely conserved by parcels of fluid, and it determines all the fluid motion. In defining potential vorticity it is convenient to replace the entrop y by the closely related "potential temperature" 6, which is the temperature that a fluid parcel would acquire if it were adiabatically compressed from its actual pressure p to a standard atmospheric pressurep0. [The ideal gas law gives 0 = T{p o/p)2'7.] Then q = a-Wd/p For a dissipationless fluid in the absence of external heating, both q and 8 remain constant in a small moving parcel of fluid. (In the ocean, potential temperature is replaced by "potential density," the density a fluid parcel would acquire if moved adiabatically to p0.) Fluid elements move along surfaces of constant potential temperature while preservin g their potential vorticity. If one smooths out variations on scales of less than 2 km (the maximum size of individual convective clouds), one finds that potential temperature increases monotonically with height and that it forms quasihorizonta l surfaces. One 46 PHYSICS TODAY MARCH 1993can think of potential temperature as denning a new vertical coordinate; then the fluid motion is entirely horizontal or layerwise two-dimensional. There are notable exceptions to the conservation of potential vorticity, particularly in the atmosphere, where heat transfer processes such as solar heating, condensa- tion or radiative cooling dilute or concentrat e the q distribution. Nonetheless, "potential vorticity thinking," as it has been called,2 is remarkably helpful in reinterpret- ing and clarifying dynamical processes in the atmosphere and the oceans.3 When it can be regarded as nearly conserved, potential vorticity provides a clear image of the three- dimensional structure and motion of air masses. From mid-latitudes to the poles, the stratification of the troposphere above 2000 m (where the vertical wind shear is large) is such that the main contribution to q arises from the vertical component of vorticity, which is dominated by the Earth's rotation . Because stratification gets stronger toward the poles, the absolute value of q increases, on average, with latitude. High-latitude regions of the Northern and Southern Hemispheres are thus, respective- ly, reservoirs of strongly positive and negative potential vorticity, corresponding to the opposite senses of their Coriolis vortices. Vorticity is generally weak in the tropics, except when you're actually in a cyclone. The tropical atmosphere is characterized by a planetary convective cell that carries angular momentum and heat away from the equator. Stormy weather The boundary between the tropical and mid-latitude circulations (about 30° on either side of the equator) is marked by the tropospheric westerly jet stream at apressure of 0.2 atmospheres. This jet stream is present throughout the year, but it is much more intense in the winter hemisphere. The jet stream is associated with a strong positive gradient of potential vorticity toward the pole at 0.2 atmospheres and a negative surface tempera- ture gradient. Together these gradients create the condi- tions for the development of atmospheric disturbances such as mid-latitude cyclones or "lows." The usual scale of a mid-latitude cyclone is about 2000 km. In its mature stage it develops cold and warm frontal regions near the ground. Sometimes smaller cyclones on the order of a few hundred kilometers can develop in less than one day and reach very large amplitudes. These "explosive cyclones" typically develop when a concentration of positive q moves over a region of high temperature contrast at ground level.4 Such a high temperature contrast exists, for instance, near the Gulf Stream or between inland water- ways and the open ocean. Explosive moist convection occurs on the warm side. The great storm of October 1987 that devastated Brittany and the south of England was such an event, apparently initiated by a remnant of a tropical cyclone transported across the Atlantic. The Gulf Stream and the analogous Kuroshio Current off Japan are regions of intense potential-vorticity gradients in the ocean. Such currents are the most dynamically active of all oceanic regions.5 The spatia l scale of the dominant instabilities in the ocean is about 50 km, much smalle r than the correspond- ing 2000-km scale of the atmosphere. That's why vortices in the ocean are smaller and far more numerous. Furthermore, dissipation is weaker in the ocean than in the atmosphere, and the smalle r size of oceanic eddies makes them less sensitive to dispersion by large-scale q gradients. Therefore they can usually be followed for a Evolution of a simulated polar vortex subjected to strong forcing by simulated mountains.6 The horizontal potential-vorticity distribution shown here is integrated over height, which is indicated by color: Lowest- altitude parts of the vortex (12 km) are shown in orange; highest parts (up to 60 km) are blue and violet. The vortex, which was perfectly circular to start with, is shown here after 4 days (a), 7.5 days (b) and 8.5 days (c). Figure 3 PHYSICS TODAY MARCH 1993 47Simulated Jovian atmosphere calculated by contour surgery for a single-layer planetary atmosphere startin g with the observed zonal winds of Jupiter.10 The overall strong potential-vorticity gradient from pole to pole (from positive to negative q) is characteristic of rapid, almost rigid rotation of the atmosphere. Superposed on this global gradient are numerous latitudinal striations indicating zonal gradient reversals, some of which give rise here to nonlinear instabilities. Figure 4Self-organization into a coherent system of interacting vortices and filaments is evident in this computer simulation of turbulence excited in a two-dimensional fluid in a box. Pairings of clockwise and counterclockwise vortices (shown lighter and darker, respectively) are evident. Just above the cente r there is even a tripole of vortices, and above its right side is one clockwise vortex being torn apart by another. Figure 5 month or more, wherea s the time scale of a typical atmospheric perturbation is less than a week. In the troposphere, potential vorticity can also manifest itself by anomalously cold or warm weather. That happens when low q from the subtropics penetrates to high latitudes and stays put for a week or so. This phenomenon, known as "atmospheric blocking" can be seen in figure 2, which is a map of potential vorticity in the north temperate and polar zones as recorded in February 1986. Atmospheric blocking occurs preferentiall y over the eastern Atlantic and Pacific Oceans and north of the Urals. The anticyclonic circulation associated with low q diverts humid westerly air toward the north on its west side and pulls in dry continental air southward and westward over western Europe or the North American Great Plains. As a result, very cold weather in winter and very hot weather in summer can install itself over large continental areas. This anticyclonic circulation is most often associated with a cyclonic center of high potential vorticity on its south flank. On weather maps the whole structure looks like a gigantic vortex dipole. It is generally believed that the coupling between the two centers explains the exceptional stability of the structure. It has been known to last a full month. The preferred locations for Pacific blocks in the American West seems to depend on both the interaction of the mean atmospheric flow with the Rocky Mountains and the feedback from atmospheric perturbations traveling across the Pacific. Perturbations develop on vorticity and temperature gradi- ents, and they tend to reduce the gradients. When a perturbation has traveled eastward from its generatingarea toward a region of lower gradients, the mature perturbation can reverse the gradients, thereby initiating a blocking event. For this very interesting phenomenon, the effect of perturbations on the mean flow is quite the op- posite of that which can be represented by simple diffusion. Another important vortex structure in the atmo- sphere dominates the stratosphere in winter. Observa- tions have revealed the existence of a giant cylindrical vortex straddling the winter pole, lying predominantly poleward of 60° latitude and riding on top of the tropopause (the transition from the lower atmospher e to the much more strongly stratified middle atmosphere). A remarkable propert y of this polar vortex is its sharp definition, which one can see particularly well in figure lb. The vortex boundary is characterized by extremely sharp q gradients associated with the eastward stratospheric jet stream. Tropospheric blocking events are often associated with a rapid warming of the lower stratosphere. This warming causes giant deformations in the shape of the polar vortex and weakens it. With the coming of spring, solar heating and the upward propagation of large-scale tropospheric disturbances lead to the destruction of the vortex. Summer brings a revers e westward circulation because the stratosphere is actually warmer at the summer pole than at the equator. There is an asymmetr y between the stratospheric winter polar vortices of the two hemispheres. The Antarctic vortex is much less disturbed by tropospheric waves, essentially because less land and fewer mountains 48 PHYSICS TODAY MARCH 1993border the Antarctic zone. The Antarctic vortex is therefore much more stable, intense and cold than its Arctic counterpart. This is a key factor in the much more effective springtime destruction of the ozone layer over the Antarctic. Contour dynamics We will now illustrate the effect of tropospherically generated disturbances on the polar vortex with the quasigeostrophic model and an unconventional but prom- ising numerical method called "contour dynamics." (See the box at right.) In mid-winter, the polar vortex is most intense and least affected by heat transfer processes. Its sharp edge is actually ideal for contour dynamics, which starts from the premise that the q field is piecewise- uniform, like a terraced hill, with discontinuities at its contour lines. Therefore in this approximation gradients of q are never dissipated. Spatial resolution is governed by the number of discrete points used to represent a contour and by the number of contours used to represent the global distribution of q. The simplest conceivable model represents the polar vortex by a single jump in q for each of a discrete number of horizontal layers.6 A major issue is the stability, or robustness, of the polar vortex in the face of disturbances propagating up from the troposphere, for example, pertur- bations caused by topographic features.7 The polar vortex proves to be remarkably stable when it is subject to moderate topographic forcing in this simple model, retaining a high degree of vertical coherence. But at larger levels of forcing, the vortex is severely disrupted. Figure 3 shows the evolution of a simulated polar vortex subjected to strong forcing by mountains.6 The cente r of the vortex, which sits at the lowest altitudes (down to 12 km), is split by the topography into two pieces, and the weaker piece gets sheared away. The higher-altitud e outer edges of the vortex eject long arms of potential vorticity to great distances. These higher-altitude parts, which go as high as 60 km, are completely at the mercy of the much denser core of the vortex. Therefore they behave rather like passive tracers during the evolution of the flow. A remarkable feature of this and other simulations is that the vortex retains significant vertical coherence.6 Such vertica l coherence had previously been observed in the absence of forcing.8 It motivates an even simpler idealized "two-dimensional" model in which all the fluid strata move together. In fact, the effect of forcing on the polar vortex was first studied with a two-dimensional model.79 The observation of strong vertical coherence in the three-dimensional model lends credibility to such a simplified approach. Figure 4 illustrates the state of the art of two- dimensional modeling. It is a snapshot from a simulation of a Jupiter-like atmospher e by contour dynamics on a sphere. This simulation , recently carried out by Dritschel and Lorenzo Polvani (Columbia University), was done without significant dissipation or forcing.10 The level of detail evident in figure 4 is well beyond the reach of conventional, grid-based models. The figure is the result of 110 Jupiter days of simulate d evolution from a slightly and randomly disturbed initial state constructed fromThe Equotions of Contour Dynamics Contour dynamics, originally developed for strictly two- dimensional fluid motion,15 can be formulated for any fluid that possesses a generalized vorticity invariant (call it q and think of potential vorticity) that remains constant within infinitesimal local volumes of fluid as they move around, and for which the instantaneous distribution of q alone determines the velocity field.10 So q simply rearranges itself without dilution. This rearrangement depends on the distribution of q throughout the fluid. The motion at a particular point x in the fluid is determined by a weighted sum of q at all points x', the weight being the Green's function C(x',x), which depends only on the boundary conditions and on the operator relationship between q and the velocity field. For a two-dimensional fluid (or for one layer of a quasigeostrophic fluid), the velocity field (u,v) is given by the skewed gradi ent of a scalar field if>(x), dy' dx with the stream function i/i given by the Green's function integral 4>=[\ dx' dy' C7(x',x) q(x') In many fluid systems, C depends only on the distance r= \|x' — x\|. For a two-dimensional fluid C = (log r)/2v; for a quasigeostrophic fluid C = — K0(yr)/2ir, where Ko is a modified Bessel function and y is a parameter called the inverse radius of deformation. One arrives at contour dynamics by taking the distribution of q to be piecewise-uniform and using the Lagrangian form of the fluid dynamical equations: The plane is divided up like a terraced hill into annular regions Slk of uniform qk, bounded by closed contour lines ¥>k, where the index k = 1,2,3, .... In the Lagrangian formalism one has simply dx dt= u(x) This is really just a definition of the motion of a fluid element. For piecewise-uniform q, the area integrals over x' and y' can be reduced to contour integrals around each contour 1>k by Stokes's theorem: u(x) = - C(r k) dxk where kqk is the jump in q across ^k, \k is a point on the /rth contour and rk = \xk — x\|. For x on one of the contours, these two equations constitute a closed, one- dimensional dynamical system. That's an enormous saving, even though the one-dimensional system does have an infinite number of degrees of freedom. The most advanced numerical algorithm implement- ing contour dynamics is called "contour surgery." It involves the selective removal of essentially passive filamentary structures at a spatial scale much finer than one can reach with even the most advanced convention- al (grid-based) nimerical models. (For details see refer- ence 10.) PHYSICS TODAY MARCH 1990 49Comparison of numerical techniques for simulating evolution of neighboring vortices of unequal size.11 The sequence at left is calculated by a conventional pseudospectral method; the one at right by contour surgery. The obvious difference is due mostly to the rapid erosion of the smaller vortex in the conventional model. (Courtesy of Hongbing Yao, Rutgers University.) Figure 6zonal averages of the two-dimensional winds observed by the Voyager missions. We simulate d the Jovian atmo- sphere by using the simplest fluid model of two-dimension- al (barotropic) flow. What is novel here is the absence of significant dissipation. That's essential for preserving the multitude of jets observed in the real data. Some of our simulate d features are strikingly similar to the real observations. Other real features, most notably the Great Red Spot, are conspicuously absent from the simulation. It may be that the fluid model is too idealized to capture this and other large vortices, or else it may just require much more computer time for such great features to emerge. Geophysical turbulence For many years, two-dimensional turbulence has been a paradigm for geophysical fluid dynamics. It is a striking fact that for any type of random initial state or external forcing, a two-dimensional fluid will rapidly organize itself into a system of coherent, interacting vortices swimming through a sea of passive filamentary structures produced from earlier vortex interactions. (See, for example, figure 5.) A given vortex basically sees a distant vortex as if its total vorticity were concentrated at a single point. The resulting velocity field can, however, vary across the vortex. The field's average value simply translates the vortex without deforming it. The vortex is also rotated and deformed by the first moments of the velocity field over its surface. This evolution can be made irreversible by pulling out filaments of vorticity, similar to what was seen in figure 3. If one gives the vortex a distributed profile, as distinguished from the single vorticity layer at each level assumed for the simulation in figure 3, the deformation leaves the core of the vortex intact and generates a new edge with very high vorticity gradients. In the denouement of this more complex computer model the emitted filaments behave essentially as passive structures; they are stretched and folded by the velocity field induced by the dominant vortices. Within conventional numerical models, finite resolu- tion necessitates the numerical dissipation of small-scale structures if one is to achieve numerical stability. This numerical dissipation is equivalent to a parameterization of unresolved scales that has the effect of reducing vorticity gradients. (Real fluid dissipation by molecular diffusion is negligible.) When a vortex is resolved with high accuracy, the true dynamics can be preserved in this way. But in most cases, and in particular for atmospheric and oceanic models, each vortex is spanned by only a few tens of grid points. That's not enough for high local gradients. The result is an artificial damping of the vortex that can lead to its premature destruction. When two vortices rotating with the same sense get to within about three radii of one another, they begin to rotate around each other and then undergo a partial or total merger." Little vortices generated during such events and embedded in the velocity field of the dominant vortex are likely to disappear rapidly in conventional numerical simulations as a result of numerical dissipa- tion. (See figure 6.) Sometimes two vortices rotating in opposite senses bind into a stable propagating dipole, or even into a more complicated tripolar structure that exhibits high stability and provides at least a qualitative 50 PHYSICS TODAY MARCH 1993explanation of the persistence of some observed structures such as atmospheric blocking.12 Although two-dimensional turbulence has been stud- ied for many years, it is remarkable that most of the basic statistical issues are still controversial. Ther e is, for instance, a famous prediction that the energy spectrum should scale with wavenumber k as k~3. We are still far from knowing whether this assertion is true. The very existence of some universa l asymptotic laws for small- scale motion in two-dimensional turbulenc e is often questioned. In three dimensions, by the way, such laws are well established. Why coherent vortices are so common is itself a puzzle. One could argue that nonlinear interactions and dissipation simply select stable struc- tures, but there is also growing evidence that in many circumstances nonlinear instabilities occur that can generat e large-scale vortical motion in turbulent flows, provided certain symmetries are broken.13 Outlook It is becoming imperative to exploit the eye's great skill at distinguishing structure in a complicated medium like a fluid. Therefore we need to make computer output visualizable, and we need to quantify what we see so that we are not mere passive onlookers but active collaborators with the computer. (See the articl e by Zabusky and his coauthors on page 24.) At a more advanced level, we need to teach the computer some of the skills we already have. In this way, the fluid continuum might be reduced to its most important structures, each of them distinguished by a small number of descriptors. In the same way, we may also come to understan d collisions or close interactions between structures like vortices in turbulent flow. We need to develop models that use these descriptors as their basic elements. Not only will this minimize the output of unnecessary information; it will also aid in the practical prediction of weather and climate. For such practical applications, simpler models ought to be built within conventional ones. A simple model alone cannot replace a conventional one, but it can greatly augment the full model's predictive power by enhancing accuracy in crucial places. For example, the "contour advection" scheme used to create the computer simulation shown in figure lb brings out features that one could not have seen in the conventional data analysis. It has become a truism that reliable metereological prediction much beyond a week is impossible because minuscule chaotic perturbations (like the proverbial butterfly in China) left out of an otherwise supremely accurate numerical model can initiate global distur- bances. The issue is rather academic, because we are very far from having a supremely accurate numerical model. Nor do we really understand the physics of clouds or the chemical processes that bear on the heatin g and cooling of the atmosphere. In any case there is no hope of gettin g the supremely accurate observational network that would have to provide the initial data. The butterfly in China is hardly likely to be a limiting factor on our ability to predict the weather. Nonetheless, rapid progress is being made. Forecast- ing models are already using elaborate initialization methods to incorporat e the imperfect observational data.14 We contend that further advances may come from a more accurate representation of coherent structures by numeri-cal models, and from explicit use of their bulk properties as input parameters. To solve the difficult problems in geophysical fluid dynamics, we must entertain the possibility of alternative modeling techniques—even of alternative ways of think- ing. For example, meteorologists for many years have forecast the weathe r by looking at pressure and winds. Only recently has there been a shift toward thinking in terms of potential vorticity. Unlike pressure , potential vorticity is often an approximately conserved quantity. That lets one make a reasonable forecast simply by letting q move with the wind. In this way one can directly infer the subsequent positions of frontal systems and the stratospheric jet stream at the edge of the polar vortex. This new attention to potential vorticity has greatly increased our knowledge of the polar atmosphere , and specifically of the factors that affect ozone depletion. The conjunction of different approaches—such as comprehensive and idealized modeling, and conventional and novel numerical techniques—offers our best hope for solving practical problems. The importance of these problems to global environmental issues makes such conjunctions imperative. We thank Alan Plumb for his comments and access to his most re- cent work, Lorenzo Polvani for processing and providing several of the color images shown here, R. Saravanan for developing and permitting use of the software for visualizing contour dynamics calculations, Darryn Waugh for sharing his recent work and providing figures, Hongbing Yao also for providing a figure, and Norman Zabusky for his suggestions and encouragement. Many of Dritschel's results were generated at the Rutherford-Appleton Laboratory on the Cray X-MP/48 and the Cray Y-MP/8. Legras thanks Gilbert Brunet for preparing figure 2, and the Centre de Calcul Vectoriel pour la Recherche, which provided computa- tional resources on its Cray 2. References 1. B. Legras, D. Dritschel, Phys. Fluids A 3, 845 (1991); Fluid Dyn. Res. (1993), in press. 2. B. Hoskins, M. Mclntyre, A. Robertson, Q. J. Meteorol. Ill, 877; 113, 402 (1985). 3. M. Mclntyre, in Atmospheric Dynamics, Proc. Int. Sch. Phys. "Enrico Fermi," CXV Course, J. C. Gille, G. Visconti, eds. North Holland, New York (1991). 4. F. Sanders, J. Gyakum, Mon. Weather Rev. 108, 1589 (1980). 5. J. Nycander, D. Dritschel, G. Sutyrin, to be published in Phys. Fluids A (1993). 6. D. Dritschel, R. Saravanan, submitted to Q. J. R. Meteorol. Soc. (1993). 7. L. Polvani, R. Plumb, to be published in J. Atmos. Sci. (1993). 8. L. Polvani, G. Flierl, N. Zabusky, J. Fluid Mech. 205, 215 (1989). J. McWilliams, J. Fluid Mech. 198, 199 (1989). 9. D. Waugh, "Single-Layer Geophysical Vortex Dynamics," PhD thesis, U. of Cambridge (1992), p. 207. 10. D. Dritschel, Comput. Phys. Rep. 10, 77 (1989). 11. D. Dritschel, D. Waugh, Phys. Fluids A 4, 1737 (1992). 12. B. Legras, P. Santangelo, R. Benzi, Europhys. Lett 5 37 (1988). 13. M. Vergassola, submitted to Phys. Rev. Lett. (1993). 14. J. Tribbia, D. Baumhefner, J. Atmos. Sci. 45, 2306 (1988). 15. N. Zabusky, M. Hughes, K. Roberts, J. Comput. Phys 30 96 (1979). ' m PHYSICS TODAY MARCH 1993 51
1.3520144.pdf	Spin-transfer torque efficiency measured using a Permalloy nanobridge M. C. Hickey, D.-T. Ngo, S. Lepadatu, D. Atkinson, D. McGrouther, S. McVitie, and C. H. Marrows Citation: Applied Physics Letters 97, 202505 (2010); doi: 10.1063/1.3520144 View online: http://dx.doi.org/10.1063/1.3520144 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/97/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spin-transfer force acting on vortex induced by current gradient in a planar polarizer geometry J. Appl. Phys. 114, 113911 (2013); 10.1063/1.4822020 Low current density spin-transfer torque effect assisted by in-plane microwave field Appl. Phys. Lett. 99, 032502 (2011); 10.1063/1.3611446 Proposal for a standard problem for micromagnetic simulations including spin-transfer torque J. Appl. Phys. 105, 113914 (2009); 10.1063/1.3126702 Coupling of spin-transfer torque to microwave magnetic field: A micromagnetic modal analysis J. Appl. Phys. 101, 053914 (2007); 10.1063/1.2435812 Frequency modulation of spin-transfer oscillators Appl. Phys. Lett. 86, 082506 (2005); 10.1063/1.1875762 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.235.251.160 On: Thu, 18 Dec 2014 00:31:09Spin-transfer torque efﬁciency measured using a Permalloy nanobridge M. C. Hickey,1,a/H20850D.-T . Ngo,2,b/H20850S. Lepadatu,1D. Atkinson,3D. McGrouther,2S. McVitie,2 and C. H. Marrows1,c/H20850 1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom 2School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom 3Department of Physics, Durham University, Durham DH1 3LE, United Kingdom /H20849Received 16 August 2010; accepted 4 November 2010; published online 19 November 2010 /H20850 We report magnetoresistance, focused Kerr effect, and Lorentz microscopy experiments performed on a nanoscale Permalloy bridge connecting microscale pads. These pads can be switched from aparallel to antiparallel state through the application of small ﬁelds, causing a detectablemagnetoresistance. We show that this switching ﬁeld H swis modiﬁed by the application of a high current density /H20849Jdc/H20850through spin-transfer torque effects, caused by the spin-current interacting with the magnetization gradients generated by the device geometry, yielding an estimate for the spin-transfer torque efﬁciency /H9264=dHsw/dJdc=0.027 /H110060.001 Oe /MA cm−2.© 2010 American Institute of Physics ./H20851doi:10.1063/1.3520144 /H20852 It has become a commonplace that spin-polarized cur- rents, ﬂowing through ferromagnets containing magnetiza-tion gradients, give rise to spin-transfer torques that act lo-cally on the magnetization. 1,2The experimentally observed consequences include domain wall /H20849DW/H20850motion,3–8 depinning,9–12resonance,13–16and transformation.17This ef- fect has applications in solid state storage class memories18 and is the basis for a magnetic logic gate design.19 Here we report on the effects of a spin-polarized current on well-characterized and controlled magnetization gradients that are generated through selection of device geometry. Weused a structure based on one originally designed by Jubertet al. , 20where a magnetization gradient is generated in a nanoconstriction that forms a bridge between two microscalepads, which have their shapes chosen to give differing coer-cive ﬁelds. The application of small ﬁelds can hence switchthe pads into lateral parallel /H20849P/H20850or antiparallel /H20849AP/H20850magnetic states. We subsequently reﬁned that design to give a largerdifference in coercivity between the two pads, 21and mea- sured the magnetoresistance /H20849MR/H20850associated with switching. Here we show that the current density ﬂowing through thebridge affects the switching ﬁeld as detected by MR, andextract the spin-transfer torque efﬁciency /H9264, deﬁned as the effective magnetic ﬁeld per unit current density. The samples were fabricated by either electron beam li- thography, sputter deposition o fa7n m thick Permalloy /H20849Py/H20850 layer capped with 2 nm Au, and liftoff, either on thermallyoxidized Si substrates, for MR and magneto-optical measure-ments, or on an electron transparent Si 3N4membrane for domain imaging measurements. The bridge connecting thetwo pads was 300 nm wide and 900 nm long. Ti/Au contactsfor transport were added by a further optical lithography lift-off step, and MR measurements were carried out using astandard lock-in detection method. The ac excitation currentwasI ac=50/H9262A at 1333 Hz, with a dc bias current Idcadded for certain measurements. The geometry is such that highcurrent densities are localized at the bridge. Local magne- tometry was carried out using focused magneto-optic Kerreffect /H20849MOKE /H20850measurements, with an elliptical spot size of /H110117/H110035 /H9262m2, using a diode laser. Imaging was carried out using Lorentz scanning transmission electron microscopy,with vector maps of the magnetic induction in the regionsurrounding the bridge obtained by differential phase con-trast /H20849DPC /H20850imaging. In Fig. 1/H20849a/H20850we show our device geometry. The extended shape of the two pads deﬁnes a magnetic easy axis. The a/H20850Present address: Department of Physics, University of Massachusetts Low- ell, One University Avenue, Lowell, MA 01854, USA. b/H20850Present address: Information Storage Materials Laboratory, Toyota Tech- nological Institute, Nagoya 468-8511, Japan. c/H20850Electronic address: c.h.marrows@leeds.ac.uk. 2.5 µmLIA ac + dcH (a) (b)PyTi/Au pointed pad switchingelliptical pad switchingJ -2 .5-2 .0-1 .5-1 .0-0 .500. 51. 0 -50 -4 0 -3 0 -2 0 -1 0 0 10 20 30 40 5000.0 50.1 00.1 5 H(Oe)Normalized Ke rr Int en sity ΔR/R (%) FIG. 1. /H20849Color online /H20850Device geometry and switching. /H20849a/H20850A scanning elec- tron micrograph of a completed device with surrounding schematic showingthe full device outline and measurement circuit, with ﬁeld axis marked. /H20849b/H20850 Focused MOKE /H20849circles /H20850and MR /H20849squares /H20850hysteresis loops, with solid symbols for the positive-going ﬁeld sweep, open symbols for the negative-going sweep. The larger area of the elliptical pad means it yields a largersignal in MOKE. The black solid points /H20849/L50098/H20850are scaled values of cos 2/H9258from the DPC imaging.APPLIED PHYSICS LETTERS 97, 202505 /H208492010 /H20850 0003-6951/2010/97 /H2084920/H20850/202505/3/$30.00 © 2010 American Institute of Physics 97, 202505-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.235.251.160 On: Thu, 18 Dec 2014 00:31:09elliptical pad is designed to have a lower switching ﬁeld than the narrower element,21which has pointed ends to increase its coercivity.22Conventional current ﬂowing across the bridge from the pointed to the elliptical pad is deﬁned aspositive, with electrons ﬂowing in the other direction. Themeasured device resistance R/H11011180/H9024. A focused MOKE hysteresis loop is displayed in Fig. 1/H20849b/H20850for a similar device of this geometry with the laser spot overlapping both pads. We see separate switching events forthe two pads, which we conﬁrmed by shifting the laser spotto cover only one or the other. P and AP states for this struc-ture can be deﬁned by analogy to a spin-valve structure,according to whether the magnetization in the two pads ispointing in the same or opposite directions. During a majorloop, the AP state occurs between the two switching ﬁelds ofthe pads, and manifests itself as a plateau in the MOKE loopbetween the switching events, in this case between H=5.2 and 8.2 Oe. The MR loop shows magnetic switching from ahigh to low resistance state as the ﬁeld is swept, with asubsequent return to a low resistance state at high ﬁeld /H20849be- yond the range of the data shown here /H20850. It is noteworthy that the switch into the high resistance state occurs before theﬁeld has passed through zero, contrary to our expectationbased on the switching into the AP state observed by focusedMOKE. Our magnetotransport measurements are very sensitive to the magnetic state in the region of the bridge. To study thismore closely, Lorentz microscopy was carried out on a simi-lar Py nanostructure. The magnetic induction vector mapsconstructed from DPC images are shown in Fig. 2. The DPC measurement allows the direction of the magnetic inductionto be calculated, in the bridge region the average angle of theinduction is calculated along with its measured variance. Anegatively magnetized P state is shown in panel /H20849a/H20850. While the pads are uniformly magnetized in the negative direction,the shape anisotropy in the bridge gives rise to a magnetiza-tion texture in its vicinity. Analysis of the DPC image showsthat on average the magnetization in the bridge is canted atan angle of /H9258=−48/H1100610° to the vertical, leading to regions of DW-like rotation of the magnetization at either end, withthicknesses /H9004of several tens of nm. The canting away from /H9258=0 reduces the exchange energy costs in these regions.The subsequent state at remanence /H20849H=0/H20850is shown in panel /H20849b/H20850. While the pads remain unswitched, the magnetiza- tion in the bridge has relaxed to lie more closely along itsaxis: now /H9258=−2/H110066° and horn-shaped regions of canted magnetization extend from the bridge into the pads. On ap-plying a forward ﬁeld the magnetization in the softer ellipti-cal pad forms a rippled state /H20849not shown here /H20850and then re- verses with a 180° DW sweeping through the ellipse andpinning at the junction with the bridge, shown in panel /H20849c/H20850. The magnetization in the bridge remains closely aligned withits axis, with /H9258=+6/H110065°. On the scale of the Kerr laser spot the system is in the AP state, but these images show thatlocally the situation is more complex. Upon increasing H further, the DW is depinned from the bridge and the ellipticalpad fully reverses, shown in panel /H20849d/H20850, and now the magne- tization begins to twist away from the bridge axis again, with /H9258=+18/H110066°. For still higher Hthe pointed pad reverses, again preceded by a rippled state /H20849not shown /H20850, returning the device to a positively magnetized P state, shown in panel /H20849e/H20850, with/H9258=+39/H110069°. At H=20 Oe, /H9258=44/H110069°/H20849this image is not shown /H20850. The values for /H9258are consistent, within error bars, with those obtained by micromagnetic modeling.23 It is known from our previous experiments that MR can be observed due to the magnetic switching of such devices,which arise not from intrinsic DW MR, 24but are due to the anisotropic magnetoresistance /H20849AMR /H20850effect.21,25Most of the resistance arises in the bridge itself, and so this measurementessentially acts as a nanomagnetometer, 26measuring the lo- cal magnetization direction in the bridge. The AMR givesrise to a drop in resistance wherever the magnetization direc-tion is rotated away from collinearity with the current den-sity. We would therefore expect a resistance contribution/H11008cos 2/H9258. Scaled values of this quantity are overlaid on the MR curve in Fig. 1/H20849b/H20850. Consideration of the DPC images shows that it is not strictly the change from P to AP state thatcontrols the MR, but the magnetization angle in and imme-diately around the bridge: switching into a low /H9258state occurs before the ﬁeld passes through zero, consistent with the up-ward jump in MR observed in Fig. 1/H20849b/H20850. While the down- ward jump is not properly reproduced, it should be noted thatwe are comparing different samples. We now turn to the effects of dc current offsets on the MR data, our key experiment in this report. As shown abovewe can determine the switching ﬁeld H swof the bridge from a MR measurement, by ﬁrst applying a large reverse ﬁeld/H20849/H11002200 Oe /H20850to saturate the sample into a P state with both pads magnetized in the negative direction, then sweeping itpositive and observing at what ﬁeld we get an abrupt upwardstep in resistance. In Fig. 3we show a plot constructed from a series of such normalized MR /H20849/H9004/H20850sweeps carried out with different values of negative and positive J dc. The total device resistance was found to rise as Jdc2due to Joule heating. The bridge resistance rose by /H1101110/H9024at the highest current den- sities used. In Fig. 3we plot the MR ratio /H9004R/Rseparately normalized for each run, and so this background effect is notevident. The MR switching ﬁeld H swforJdc=0 is 1.5 /H110060.5 Oe, corresponding to a change from red to blue contrast in theplot. We can see from Fig. 3that the effect of a ﬁnite value of dc offset current is to shift the observed switching ﬁeld. Thephase boundary for low R→high Rswitching is marked by a dotted black line in Fig. 3, and the switching ﬁeld is seen to (a) (b) (c) (d) (e)H = -20 Oe = -48 θ ° H=9O e =1 8 θ ° H=1 2 Oe= 39θ °H=0O e =- 2 θ ° H=5 . 6O e =6 θ ° θ H 1µ m FIG. 2. /H20849Color online /H20850Magnetic induction vector maps constructed from scanning DPC images, which may be interpreted using the color wheel. Thesoft elliptical pad is uppermost. The average magnetization direction withinthe bridge is given for each frame along with the value of applied ﬁeld.202505-2 Hickey et al. Appl. Phys. Lett. 97, 202505 /H208492010 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.235.251.160 On: Thu, 18 Dec 2014 00:31:09follow a linear relationship with Jdc. This is a signature of spin-transfer torque phenomena, any thermal activation ef-fect would appear even in current. The spin-transfer torquetakes effect by displacing some of the magnetization textureat the ends of the bridge. The rate of change of switching ﬁeld with current den- sity can be deﬁned as a spin-transfer efﬁciency, measuredhere as /H9264=dHsw/dJdc=0.027 /H110060.001 Oe /MA cm−2from a linear ﬁt to all the switching ﬁelds. Although measured by adifferent method, this is of the same order as that measuredfor Py by Vernier et al. , 0.05 Oe /MA cm −2.6As a secondary result, we can estimate from /H9264the spin-torque nonadiabatic- ity parameter from standard theory describing spin-torque atDWs, 27/H20849related to, but distinct from, the Slonczewski torques in multilayer nanopillars28/H20850using the formula /H9252 =2eMs/H9004/H92620/H9264/P/H6036/H9266.8For Py, the magnetization Ms =0.83 MA /m,29and polarization P=0.5.12Analysis of the DPC images leads to DW widths /H9004/H11011100 nm, yielding /H9252 /H110110.04, the same as previous estimates by us for Py in other depinning studies,11,12and approximately ﬁve times the Gil- bert damping constant /H9251/H110150.008 in our Py ﬁlms prior to patterning.30We note, however, that this formula for /H9252is only valid when the energy barrier to be overcome is linearinH, and we are not certain that this is the case here.31 To summarize, we have studied the effect of high current densities on the micromagnetic state of a nanoscale bridgeconnecting two microscale Py pads by measuring the char-acteristic AMR signal that indicates switching of the magne-tization direction in the bridge. The switching ﬁeld wasfound to have a linear dependence on the current density.The canting angle /H9258is given by equilibrium between the torques exerted on the bridge magnetization that arise fromthe following energy terms: shape anisotropy, Zeeman, andexchange coupling to the magnetization in the pads. Thespin-transfer torque adjusts this equilibrium point, with J dc /H110220 providing an equivalent negative effective ﬁeld, delayingthe relaxation of the bridge magnetization to point along its length. This work was carried out under the auspices of the Spin@RT consortium, funded by the EPSRC, Grant Nos.EP/D000661/1 and EP/D062357/1 /H20849Leeds /H20850, EP/D003199/1 /H20849Glasgow /H20850, and EP/D50578X/1 /H20849Durham /H20850, and the European Science Foundation EUROCORES project Spincurrent. 1L. Berger, J. Appl. Phys. 55, 1954 /H208491984 /H20850. 2C. H. Marrows, Adv. Phys. 54, 585 /H208492005 /H20850. 3P. P. Freitas and L. Berger, J. Appl. Phys. 57, 1266 /H208491985 /H20850. 4L. 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Coey, Magnetism and Magnetic Materials /H20849Cambridge University Press, New York, 2010 /H20850. 30S. Lepadatu, J. S. Claydon, C. J. Kinane, T. R. Charlton, S. Langridge, A. Potenza, S. S. Dhesi, P. S. Keatley, R. J. Hicken, B. J. Hickey, and C. H.Marrows, Phys. Rev. B 81, 020413 /H208492010 /H20850. 31J.-V. Kim and C. Burrowes, Phys. Rev. B 80, 214424 /H208492009 /H20850.- 8 0- 6 0- 4 0- 2 0 0 2 0 4 0 6 0 8 0-15-10-5051015 0.010.020.030.04ΔR/R (%) low R statehigh R state Jdc(MA/cm2)H( O e ) FIG. 3. /H20849Color online /H20850dc current offset effects on the MR. The bitmap is constructed from a series of MR ﬁeld sweeps carried out for different valuesofJ dc, carried out after reverse saturation. The ﬁeld sweep direction from low to high Rstates is shown, and the phase boundary between them is marked with a dotted black line. Small but unavoidable drifts in some of theMR sweeps leads to some vertical streaking in the plot at higher values ofJ dc.202505-3 Hickey et al. Appl. Phys. Lett. 97, 202505 /H208492010 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.235.251.160 On: Thu, 18 Dec 2014 00:31:09
1.5063456.pdf	Asymmetric velocity and tilt angle of domain walls induced by spin-orbit torques Manuel Baumgartner , and Pietro Gambardella Citation: Appl. Phys. Lett. 113, 242402 (2018); doi: 10.1063/1.5063456 View online: https://doi.org/10.1063/1.5063456 View Table of Contents: http://aip.scitation.org/toc/apl/113/24 Published by the American Institute of Physics Articles you may be interested in Anomalous spin Hall magnetoresistance in Pt/Co bilayers Applied Physics Letters 112, 202405 (2018); 10.1063/1.5021510 Readable racetrack memory via ferromagnetically coupled chiral domain walls Applied Physics Letters 113, 152401 (2018); 10.1063/1.5049859 Study of spin-orbit torque induced magnetization switching in synthetic antiferromagnet with ultrathin Ta spacer layer Applied Physics Letters 113, 162402 (2018); 10.1063/1.5045850 Maximizing the quality factor to mode volume ratio for ultra-small photonic crystal cavities Applied Physics Letters 113, 241101 (2018); 10.1063/1.5064468 Vortex spin-torque diode: The impact of DC bias Applied Physics Letters 113, 242403 (2018); 10.1063/1.5064440 Angle dependent magnetoresistance in heterostructures with antiferromagnetic and non-magnetic metals Applied Physics Letters 113, 202404 (2018); 10.1063/1.5049566Asymmetric velocity and tilt angle of domain walls induced by spin-orbit torques Manuel Baumgartnera)and Pietro Gambardellab) Department of Materials, ETH Zurich, H €onggerbergring 64, CH-8093 Zurich, Switzerland (Received 27 September 2018; accepted 24 November 2018; published online 11 December 2018) We present a micromagnetic study of the current-induced domain wall motion in perpendicularly magnetized Pt/Co/AlO xracetracks. We show that the domain wall velocity depends critically on the tilt angle of the wall relative to the current direction, which is determined by the combined action of the Dzyaloshinskii-Moriya interaction, damping-like, and ﬁeld-like spin-orbit torques. The asymmetry of the domain wall velocity can be controlled by applying a bias-ﬁeld perpendicu-lar to the current direction and by the current amplitude. As the faster domain walls are expelled rapidly from the racetrack boundaries, we argue that the domain wall velocity and tilt measured experimentally depend on the timescale of the observations. Our ﬁndings reconcile the discrepancybetween time-resolved and quasi-static domain wall measurements in which domain walls with opposite tilts were observed and are relevant to tune the velocity of domain walls in racetrack struc- tures. Published by AIP Publishing. https://doi.org/10.1063/1.5063456 The propagation of domain walls (DWs) plays a funda- mental role in determining the efﬁciency and speed of current-induced switching of magnetic devices. 1–10In the context of spin-orbit torques (SOTs),11DW propagation has been extensively studied by analytical12–15and micromag- netic models,16–19magneto-optical Kerr effect (MOKE),4–9 nitrogen-vacancy magnetometry,20and x-ray imaging.10,21 An important conclusion drawn from this extended body of work is that the DWs in perpendicular magnetized layers, such as Pt/Co/AlO xand Ta/CoFeB/MgO, are chiral N /C19eel walls stabilized by the Dzyaloshinskii-Moriya interaction(DMI). The N /C19eel wall magnetization points in-plane, perpen- dicular to the DW, and hence parallel to the current direc- tion, which maximizes the amplitude of the current-induced damping-like SOT and promotes very large DW displace- ment velocities v DW, of the order of 100 m s/C01for a current density j¼108Ac m/C02. This large vDWallows for high speed DW displacements in racetrack structures4,8,9and for sub-ns reversal of ferromagnetic dots.10,22 Two prominent effects of the DMI in perpendicularly magnetized layers are the tilting of the DW10,16,23–27and the asymmetric vDWrelative to the current direction.9,26,28These two effects are related by the DW dynamics under the com- bined action of DMI and damping-like SOT.16,25–27Tilted DWs were ﬁrst observed in Pt/Co/Ni/Co layers by imaging the magnetic domains after a sequence of current pulsesusing MOKE microscopy 23and later reproduced by ana- lytical and micromagnetic models.16,25,27Figure 1(a) illus- trates the DW conﬁgurations reported in Ref. 23for the four combinations of current (black arrows) and up/down and down/up domains propagating in a racetrack. The tilt angle is indicated by w, and the propagation direction of the DW is given by the green arrows. These DW tilt symmetries aretypical of perpendicularly magnetized ﬁlms with a Ptunderlayer. Recent time-resolved x-ray microscopy measure- ments on Pt/Co/AlO xdots, however, reported DWs rotated by about 90/C14for the same current polarity and domain orien- tation,10as shown in Fig. 1(b). We suppose that these con- trasting observations may arise from the static vs.time- resolved nature of the experiments since in the ﬁrst case, the DWs are imaged after the injection of several current pulses, whereas in the second case, the DWs are imaged during cur- rent injection following a nucleation event. Furthermore, theﬁeld-like component of the SOT may also induce a tilt of the DW, similar to the effect of an in-plane ﬁeld orthogonal to the current. 16,25,29,30The aim of this work is to reconcile these controversial observations by elucidating the time-resolved dynamics of tilted DWs in racetrack structures and investigate the inﬂuence of DW tilt and ﬁeld-like torque on the velocity of the walls. We present a study of the current-driven dynamics of chiral DWs in heavy metal/ferromagnetic racetracks performed using micromagnetic simulations. As a model system, wechoose Pt/Co/AlO xstripes divided into 4 nm /C24n m/C21n m rectangular cells with the following material parameters: Co thickness, 1 nm; saturation magnetization, Ms¼900 kA m/C01; exchange coupling, Aex¼10/C011Jm/C01; effective uniaxial FIG. 1. Schematics of the current-induced tilted DWs for the four combina- tions of current and domain orientation measured by (a) static MOKEmicroscopy 23and (b) time-resolved x-ray microscopy10in perpendicularly magnetized Pt/Co bilayers. The black and green arrows indicate the current and the propagation direction of the DWs, respectively. The tilt angle w between the positive x-axis and the normal to the DW nis shown in (a).a)Author to whom correspondence should be addressed: manuel.baumgartner@ mat.ethz.ch b)Electronic mail: pietro.gambardella@mat.ethz.ch 0003-6951/2018/113(24)/242402/4/$30.00 Published by AIP Publishing. 113, 242402-1APPLIED PHYSICS LETTERS 113, 242402 (2018) anisotropy energy, Ku¼657 kJ m/C03; DMI constant, D ¼1.2 mJ m/C02; and damping a¼0.5. The magnitudes of the damping-like and ﬁeld-like SOTs are given in ﬁeld units per unitary magnetization as TDL¼18 mT and TFL¼10 mT per j¼1/C2108Ac m/C02, respectively. For simplicity, we neglect the effects of pinning and temperature,16,26which are not central to the results presented in this work. The simulationswere carried out using the object oriented micromagnetic framework (OOMMF) code 31including the DMI extension module32and an additional SOT module. We note that the outcome of the simulations does not change if we decrease the cell size to, e.g., 1 nm /C21n m/C21n m . Figure 2(a) shows the equilibrium conﬁguration of an up/down DW in Pt/Co/AlO x, which is a left-handed N /C19eel wall stabilized by the DMI. In order to illustrate the differentmechanisms that lead to the tilting of the DW, we report in Figs. 2(b)and2(c)the response of such a DW to a transverse magnetic ﬁeld B yand damping-like torque TDL, respectively. In Fig. 2(b),Byrotates the DW moments away from the lon- gitudinal direction towards þy, which causes a negative tilt of the DW in order to maintain the energetically favouredN/C19eel conﬁguration. The equilibrium tilt is determined by the balance between the external ﬁeld, DMI, and DW energy, which increases with the DW length and hence with the tiltangle. 16,25,29The effect of TDLdue to a positive electric cur- rent (electrons ﬂowing to the left) is shown in Fig. 2(c).I n order to understand the tilt of the DW in this case, we have toconsider the action of the current-induced SOTs on the DW magnetization. The damping- and ﬁeld-like torques have sym- metry T DL¼TDLm/C2ðy/C2mÞandTFL¼TFLm/C2y,r e s p e c - tively.33The Landau-Lifshitz-Gilbert (LLG) equation is then given bydm dt¼/C0jcj ð1þa2ÞX iTi/C0jcja ð1þa2Þm/C2X iTi;(1) withX iTi¼m/C2BeffþTDLþTFL; (2) where m¼M=Msis the unit magnetization vector, cthe electronic gyromagnetic ratio, l0the free space permeability, andBeff¼BextþBK/C01 MsdEDMI dm/C01 MsdEex dmthe effective mag- netic ﬁeld. Here, Bextis the external magnetic ﬁeld, BK¼ 2Ku=Msthe effective out-of-plane anisotropy ﬁeld (including the demagnetizing ﬁeld), and the last two terms are the effective DMI and exchange magnetic ﬁelds. We considerﬁrst only the effect of the damping-like torque. In this case, the LLG equation can be written in simpliﬁed form as dm=dt//C0T DL/C0am/C2TDL. Hence, the DW magnetization is deviated towards /C0yandþzby the damping-like torque, as shown schematically in Fig. 2(c). This dynamic process leads to the observed propagation (due to the z-component of dm=dt) and tilting of the DW (due to the y-component of dm=dt). A quantitative description of this process is given in terms of a one-dimensional model of DW propagation inRefs. 16,25, and 27. The effect of the ﬁeld-like torque can ﬁnally be understood in analogy with that of the magnetic ﬁeld B yso that the DW tilt angle at steady state depends on the ratio TFL=TDL, as shown in Fig. 2(d). In order to investigate the relationship between the DW tilt angle wand vDW, we simulate the dynamics of a DW consisting of one straight and two tilted sections in a square sample under the action of TDLalone (Fig. 3). The mag- netization on the left (right) side of the structure points alongþz(/C0z). We ﬁrst relax the DW magnetization, which leads to the emergence of left-handed N /C19eel walls. Due to the initial conditions, the three DWs have a tilt w¼/C045 /C14,0/C14, and 45/C14, shown in (a). Successive snapshots of the magnetic con- ﬁguration during current injection reveal that the different DW sections propagate with distinct velocities, as shown inFigs. 3(b) and3(c). This behaviour can be easily understood in terms of Eq. (1)asT DLrotates the DW magnetization against the effective DMI ﬁeld towards /C0y. As a result, for sufﬁciently large current, mxis the largest (smallest) for w¼/C045/C14ð45/C14Þ. Since vDW/ðdm=dtÞz/TDLand TDL/mx,vDWis the largest (smallest) for w¼/C045/C14ð45/C14Þ. Therefore, the different mxcomponents result in a pro- nounced asymmetry of the current-induced DW motion, as shown in (c). Alternatively, the difference in vDWcan be understood by an energy argument. Due to the presence of DMI, the energy is minimized if the DWs are of N /C19eel type. During current injection, the DWs tilted at w¼0/C14and/C045/C14 deform and acquire a mixed N /C19eel-Bloch character. These DWs propagate faster in order to reduce the total energy of the system by increasing the length of the energeticallyfavoured N /C19eel walls. The fastest direction of DW propaga- tion measured by time-resolved scanning transmission x-ray microscopy 10and the largest displacements reported in “oblique” Pt/Co/AlO xracetracks oriented at different angles with respect to the current9are consistent with this picture. A relevant consequence of the asymmetric DW velocity is that, in an elongated stripe, the faster DWs ðw¼0/C14;/C045/C14Þ FIG. 2. (a) Up/down DW in a Pt/Co/AlO xstripe at equilibrium. (b) Static DW tilt induced by a magnetic ﬁeld By¼20 mT. (c) Dynamic DW tilt due toTDLduring the injection of an electric current j¼2/C2108Ac m/C02. The schematics in (a)–(c) illustrate dm=dtaccording to Eq. (1)due to TDLand the resulting DW tilt. (d) Dependence of the dynamic DW tilt on the ampli-tude of T FLrelative to TDL¼18 mT per 108Ac m/C02for the same current density as in (c).242402-2 M. Baumgartner and P . Gambardella Appl. Phys. Lett. 113, 242402 (2018)are rapidly expelled from the sample, and the ﬁnal DW observed in steady state conditions is the slowest one with w¼45/C14[Fig. 3(d)]. This behavior has a compelling analogy with crystal growth, in which the crystal facets with the slow- est growth rate determine the ﬁnal crystal shape.34Similar arguments based on classical interface thermodynamicsexplain the faceting observed during the growth of chiral mag- netic bubbles subject to an applied ﬁeld. 35We thus conclude that the discrepancy between the DW conﬁgurations reported for quasi-static [Fig. 1(a)] and time-resolved measurements10,23 [Fig. 1(b)] is due to the different time-scales probed in these experiments, which correspond to the slower and faster DW in a racetrack, respectively. In time-resolved switching experi- ments, the initial conditions, namely, the shape of the DW after nucleation, also play a role in determining the tilt and velocity of the DW. The ﬁnal tilt angle is reached on a timescale of several ns, which increases with the stripe width. 25 The propagation velocity perpendicular to each DW front, vn DWðw¼45/C14Þ,vn DWðw¼0/C14Þ, and vn DWðw¼/C045/C14Þ, can be calculated by measuring the distance travelled by the DW as a function of time. Figure 4(a) shows that vn DWin- creases almost linearly with jfor all three DW components,however, with distinct slopes. Depending on w,vn DWfor the fastest and slowest DW can differ by more than a factor two. Furthermore, the asymmetry of vn DW, which we deﬁne as the ratio vn DWðw¼/C045/C14Þ=vn DWðw¼45/C14Þ, increases proportion- ally to jup to 3.5 /C2108Ac m/C02, as shown in Fig. 4(b). Finally, we study the effect of the ﬁeld-like torque on vn DW. For a positive current, TFLin Pt/Co/AlO xis equivalent to a magnetic ﬁeld Byopposite to the Oersted ﬁeld. Therefore, TFLcounteracts the rotation towards /C0yinduced by the damping-like torque. More importantly, TFL>0ð<0Þ leads to an additional ðdm=dtÞzcontribution which increases (decreases) vn DW. The amount of increase or decrease in vn DW due to the ﬁeld-like torque depends on wand hence on the damping-like torque and DMI. We ﬁnd that the ratio vn DWðTFL>0Þ=vn DWðTFL<0Þincreases linearly as a function ofjforw6¼/C045/C14. Although the increase is only about 10% at the highest j, this effect should not be neglected in devices with a signiﬁcant ﬁeld-like torque. These results are consis- tent with experiments in which an in-plane ﬁeld Bywas applied to reinforce the ﬁeld-like torque, thus assisting the magnetization reversal10and increasing the current-induced DW velocity.5 FIG. 3. (a) Initial magnetic conﬁgura- tion of a Pt/Co/AlO xsquare with one straight and two oppositely tilted DWs. The side of the square is 1.5 lm. The magnetization components mx,my, and mzare shown in color in different pan- els. The scheme on the right shows the in-plane magnetization and relative displacement of the DW. (b) and (c)Snapshot of the magnetic conﬁguration during injection of a positive current of amplitude j¼1.0/C210 8Ac m/C02and 4.5/C2108Ac m/C02, respectively, taken after 0.9 ns. The dotted lines show the initial DW position. (d) Snapshots of the DW propagation during injectionof a positive current j¼4.5/C210 8A cm/C02into a 4.5 lm long and 1.5 lm wide stripe. Note that after /C251.5 ns, the fastest DW is expelled from the stripe. As a consequence, the tilt angle at steady state corresponds to that of the slowest DW. FIG. 4. (a) Normal DW velocity vn DW as a function of current density for different tilt angles. The velocities are calculated for TDL¼18 mT and TFL ¼10 mT (full symbols), TFL¼0m T (dotted symbols), and TFL¼/C0 10 mT (open symbols) per j¼108Ac m/C02. (b) Asymmetry ratio vn DWðw¼/C045/C14Þ= vn DWðw¼45/C14Þ, plotted as a function of current density for the three values ofTFLshown in (a). Positive values ofTFL, as in Pt/Co/AlO x, reduce the asymmetry, whereas negative values increase it.242402-3 M. Baumgartner and P . Gambardella Appl. Phys. Lett. 113, 242402 (2018)In summary, we reported a comparative study of the tilt and velocity of DWs in perpendicularly magnetized Pt/Co/ AlO xlayers. Consistent with qualitative arguments derived from the LLG equation, our micromagnetic simulationsevidence that DWs with different tilt angles propagate at distinct speed, depending on the balance between DMI, damping-like, and ﬁeld-like torques, which determines them xcomponent of the DW magnetization. As a result of the asymmetric speed of tilted DWs, the fastest DW in racetrack structures is expelled from the track after a time of the order of 1.5 ns, which depends on the width of the track and initial shape of the DW. Thus, quasi-static measurements of theDW displacements induced by a sequence of current pulses probe the propagation and tilt of the slowest DW, 16,23,25,36 whereas time-resolved microscopy and “oblique” racetrack measurements probe the fastest DW.9,10As a side remark, we note that the fastest propagation direction of the DW cor- responds to the direction of motion of magnetic skyrmions,as described by the so-called “skyrmion Hall effect”. 37,38 Because a skyrmion is delimited by a DW with a tilt angle that varies continuously between w¼0/C14andw¼360/C14, the skyrmion Hall effect can be rationalized in terms of the pref- erential direction for DW propagation and the tendency of the skyrmions to retain their topologically protected shape.These ﬁndings allow for a better understanding and tuning of the DW motion and switching speed of magnetic memory elements of different shapes. We acknowledge funding by the Swiss National Science Foundation under Grant No. 200020-172775. 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1.5000245.pdf	Micromagnetic simulation of electric field-modulation on precession dynamics of spin torque nano-oscillator Congpeng Zhao , Xingqiao Ma , Houbing Huang , Zhuhong Liu , Hasnain Mehdi Jafri , Jianjun Wang , Xueyun Wang , and Long-Qing Chen Citation: Appl. Phys. Lett. 111, 082406 (2017); doi: 10.1063/1.5000245 View online: http://dx.doi.org/10.1063/1.5000245 View Table of Contents: http://aip.scitation.org/toc/apl/111/8 Published by the American Institute of Physics Articles you may be interested in Spin diffusion length of Permalloy using spin absorption in lateral spin valves Applied Physics Letters 111, 082407 (2017); 10.1063/1.4990652 Electric-field tuning of ferromagnetic resonance in CoFeB/MgO magnetic tunnel junction on a piezoelectric PMN-PT substrate Applied Physics Letters 111, 062401 (2017); 10.1063/1.4997915 Electric-field effect on spin-wave resonance in a nanoscale CoFeB/MgO magnetic tunnel junction Applied Physics Letters 111, 072403 (2017); 10.1063/1.4999312 Single shot ultrafast all optical magnetization switching of ferromagnetic Co/Pt multilayers Applied Physics Letters 111, 042401 (2017); 10.1063/1.4994802 Tuning the perpendicular magnetic anisotropy, spin Hall switching current density, and domain wall velocity by submonolayer insertion in Ta/CoFeB/MgO heterostructures Applied Physics Letters 111, 042407 (2017); 10.1063/1.4995989 Excitation of coherent propagating spin waves in ultrathin CoFeB film by voltage-controlled magnetic anisotropy Applied Physics Letters 111, 052404 (2017); 10.1063/1.4990724Micromagnetic simulation of electric field-modulation on precession dynamics of spin torque nano-oscillator Congpeng Zhao,1Xingqiao Ma,1,2,a)Houbing Huang,1Zhuhong Liu,1,2 Hasnain Mehdi Jafri,1Jianjun Wang,3Xueyun Wang,4and Long-Qing Chen3 1Department of Physics, University of Science and Technology Beijing, Beijing 100083, China 2Weak Magnetic Detection and Application Engineering Centre of Beijing, Beijing 100083, China 3Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA 4School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China (Received 6 May 2017; accepted 14 August 2017; published online 25 August 2017) Understanding electric ﬁeld effects on precession dynamics is crucial to the design of spin transfer torque devices for improving the performance in nano-oscillator. In this letter, the precession dynamics of a CoFeB/MgO multi-layer structured nano-oscillator under externally applied electric ﬁeld is predicted using a micromagnetic simulation. It is revealed that the electric ﬁeld can modifythe range of oscillation spectra in single frequency mode. With the increase in electric ﬁeld, there is a red-shift of the resonant frequency. When a positive electric ﬁeld pulse is applied, a phase lag of the spin precession is induced, which is proportional to the pulse amplitude and duration.The present work is expected to stimulate future experimental efforts on designing devices with electric-ﬁeld modulated spin transfer torque nano-oscillators. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.5000245 ] Manipulation of magnetization via electric ﬁeld in ferro- magnets attracts considerable attention due to the potentialapplications in spin devices such as spin transfer sensors, 1 spin transfer diodes,2,3magnetic random access memories,4,5 and spin torque oscillator (STO).6–8This has been achieved in multiferroic magneto-electric heterostructures, wherein the electric and magnetic order parameters can be coupled through interfacing mechanisms such as charge/orbital modu-lation, 9–11exchange coupling,12–14or/and elastic coupling via strain.14–18It can also be achieved via voltage-induced spin-polarized current by spin transfer torque (STT)19,20or spin-orbit torque.21,22Recently, electric ﬁeld induced interfa- cial perpendicular magnetic anisotropy (IPMA) was found both experimentally and theoretically in CoFeB/MgO (CM)bilayer. 23–27Both experimental and theoretical investigations on IPMA have been reported yet, which is originated from hybridization of Fe- dz2and O- pzorbitals.17Meanwhile, the IPMA depends on the thickness of CoFeB layer16,26and can be linearly changed by electric ﬁeld on MgO layer;26 thus, the manipulation of the status of spins achieved by electrical ﬁeld via IPMA is possbile, which has been demon- strated through simulation in magnetization reversal pro- cess.28For the precession process in STO, however, most of the previous work27,29focused on using magnetic ﬁeld (change ﬁeld angle or amplitude) to modulate precession, which is not easy to control for nano-integrated devices. Thismay be improved by introducing an electric ﬁeld between the electrodes of the nano-oscillator to replace the magnetic ﬁeld. Since modulation of precession is the key in designing andengineering high frequency nano-oscillators, the precession dynamics under electrical ﬁeld induced IPMA deserve inten- sive studies.In this letter, we studied precession dynamics in CM based STO under electric ﬁeld modulation through IPMA by micromagnetic simulation. Results show the modulation of both spectrum proﬁle types and frequency, by applying a voltage across MgO layer. The phase of the oscillator could also be modulated by applying square impulse voltage. Figure 1schematically illustrates a STO framework 28of combining spin transfer current and electric ﬁeld, which con- tains CoFeB-based free and pinned layers, barrier, MgO oxide layer, and two electrodes. Barrier layer separates the free and pinned layer, which is critical to introduce polarized electrons, while an extra oxide layer on the top of free layer is used to introduce IPMA. A free layer ( M) is magnetized along þz axis, while the pinned layer ( P) is magnetized along þyaxis. A negative spin current ( Istt) is expected to produce oscilla- tion30in the free layer where the IPMA could be manipulated under the extra electric ﬁeld. In our model (Fig. 1), we consid- ered varying electric ﬁeld on the top MgO layer from /C00.5 to 0.5 V/nm. To study the dynamic behavior under electric ﬁeld, we solved Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equa- tion by using a self-written code FIG. 1. STO-device schematic via electric ﬁeld modulation. The schematic illustrates a STO-device combining electric ﬁeld and spin-polarized current.a)Author to whom correspondence should be addressed: xqma@sas.ustb.edu.cn 0003-6951/2017/111(8)/082406/4/$30.00 Published by AIP Publishing. 111, 082406-1APPLIED PHYSICS LETTERS 111, 082406 (2017) dm dt¼/C0cm/C2Heffþam/C2dm dt/C18/C19 þTstt; (1) where arepresents Gilbert damping constant. m¼ðmx;my; mzÞis normalized magnetization of free layer deﬁned by m¼M Mswith saturation magnetization Ms.cis gyromagnetic ratio normalized by c¼c0 1þa2(c0is gyromagnetic ratio). tis reduced simulation time described by t¼c0Ms 1þa2t0(t0represents real time). Torque term Tsttcontains in-plane torque and ﬁeld-like torque31,32 Tstt¼J/C22hgm;pðÞ 2l0M2 sedfreem/C2p/C2m ðÞ þnm/C2p/C2/C3; (2) where p¼ðpx;py;pzÞ¼ð 0;1;0Þis normalized magnetiza- tion of pinned layer deﬁned by p¼P Ms.Janddfreeare spin current density and thickness of free layer, respectively. nis the ratio coefﬁcient, which describes ratio of in-plane andﬁeld-like torques. gðm;pÞ¼ g 1þg2m/C1p33,34is scalar function in MTJ which possess spin-torque asymmetry.35,36grepresents polarization constant. The effective ﬁeld ( Heff) takes into account the contributions of anisotropy, magnetostatic, and exchange interaction. Particularly, electric ﬁeld dependentanisotropy or IPMA of free layer, e.g., the IPMA for CoFeB/MgO interface, is introduced as follows: H ani¼2K1EðÞ l0M2 smzþ2K2 l0M2 smz1/C0m2 z/C0/C1 ; (3) where K1ðEÞ¼K01/C0EDK;K01,DK,K2are thickness dependent magnetic anisotropy constants.26Erepresents the electric ﬁeld of MgO barrier induced by voltage. Parametersused in our simulations are anisotropy constants K 01¼6:5 /C2105J=m3,K2¼0:1/C2105J=m3,DK¼0:15/C210/C04J=ðm2VÞ,26 ratio coefﬁcient n¼0:2, saturation magnetization Ms¼1:0 /C2106A=m, exchange constant A¼2:0/C210/C011J=m, and damping constant a¼0:01. The total simulation size is 89 :6 /C2179 :2n m2(total simulation grid is 64 /C2128/C21 with cell size: 1 :4/C21:4/C21:4n m3) and the thickness of free layer and top oxide layer are 1.4 and 0.8 nm, respectively. The spin oscillation proﬁle under different current densi- ties, i.e., from 0 to 10 /C2106A/cm2, was divided into fourregions: damping region (A), non-uniform precession region (B), stable precession region (C), and ﬂipping region (D) [Fig.2(a)], based on their different dynamic features. Four special current densities 1.2, 3.0, 6.0, and 9.8( /C210 6A/cm2)w e r e selected to show their oscillation behaviors of average magne-tization in each of the four regions [Fig. 2(a)]. In region A, the current density is too weak to excite oscillation, and oscil- lating amplitude of magnetization damped to near zero [Fig.2(b)]. In region B, oscillation is non-uniform, and the corre- sponding spectrum possesses multiple frequency modes [Fig.2(c)]. In region C, the uniform oscillating behavior demon- strates a spectrum with single frequency [Fig. 2(d)]. In region D, magnetization ﬂips and is stabilized in –ydirection under high current density [Fig. 2(e)]. In addition, Fig. 2(a)shows the boundary between regions B and C, which denotes thecritical current density between non-uniform (multiple dis- crete frequencies) and uniform precession (single frequency). The boundary indicates that output signals are qualitativelydifferent, which is important from the application point ofview (i.e., single frequency mode may be favorable as signalcarrier wave, compared with multiple frequency modes). Toinvestigate the inﬂuence of electric ﬁeld on signal quality, weshow the power spectra versus current density under differentelectric ﬁeld [Figs. 2(f)–2(h) ]. Output power P outwas normal- ized by Pout¼~PoutðfÞ=Pmax, where ~PoutðfÞandPmaxdenote frequency dependent power density and the highest power density, respectively (see the supplementary material for spec- trum calculation). The critical current dividing the multi-frequency and single frequency precessions were marked by avertical black dashed line. Current density values rangingfrom /C03.0 to /C09.0/C210 6A/cm2and bias electric ﬁelds E¼0, 60.1 V/nm were selected to study spectral variations. As shown in Fig. 2, patterns of spectra and critical curent density may be manipulated by electric ﬁeld. A positive Eassists the critical current moving to the low-density side [Fig. 2(g)], while a negative Efacilitates boundary shifting to the high- density side [Fig. 2(h)]. Such shift is mainly due to the IPMA change under electric ﬁeld. Details of uniform oscillation(region C) and the ﬁeld dependent shift are illustrated in thesupplementary material . For the uniform precession state of STO, we investigated the relationship among frequency ( f), current density ( J), and FIG. 2. Illustration of STO spectrum types altered by electric ﬁeld. (a) Self- oscillating spectrum regions of damp- ing (A), multi-frequency (B), single fre- quency (C) and ﬂipping (D) under different current density range. Points(b)–(e) correspond representing states in each region of (A)–(D) at current density /C01.2,/C03.0,/C06.0, and /C09.8 (/C210 6A/cm2), respectively. [Insets in (b)–(e) are ampliﬁed region for time range 44–49 ns.] (f)–(h) are spectra of normalized output power density ( P) versus current density at E¼0,60.1 V/ nm, respectively. The intensity of the power density was represented by a color bar. The critical current density dividing the multi-frequency and single frequency regions was marked by a black vertical dashed line.082406-2 Zhao et al. Appl. Phys. Lett. 111, 082406 (2017)electric ﬁeld ( E). It is shown that the frequency decreases with the increase in current density under different Evalues ranging from /C00.5 to 0.5 V/nm [Fig. 3(a)]. Meanwhile, fre- quency decreases with the increase in Eranging from /C00.3 to0.3 V/nm under certain current densities. The relationship between fandEis approximately linear [Fig. 3(b)]. The spec- tra under different electric ﬁeld for the current density value 8.0/C2106A/cm2are shown in Fig. 3(c). A slight shift in fre- quency peak to lower frequency side with the increase in electric ﬁeld was observed. The corresponding modulation rate for the peak deviation is estimated to be 31.2 MHz per0.1 V/nm Evalue. We calculated frequency using Larmor precession model x¼cM s½haniðEÞþhdþhstt/C138[see Eq. (S2) in the supplementary material ], which describes the relation- ship between precession frequency and angular moments, to verify our simulation and to explain the effect of electric ﬁeld on the rate of frequency modulation. The results showthat relative change of frequency Df=fis 3.7% and scale ofDfis 1–10 2MHz, estimated roughly by 3.7% of fscale 0.1–10 GHz, which concurs with our simulation result(31.2 MHz). From above f/h aniðEÞrelation, IPMA ﬁeld of free-layer are weakened by positive electric ﬁeld and subse- quently causes frequency red shift. As described earlier [Figs. 3(a)–3(c) ], the frequency can be modulated within the range of 10–100 MHz by apply- ing E(/C00.3–0.3 V/nm), meaning that electric ﬁeld may accelerate or decelerate spin precession. This offers a poten-tial approach of modulating the output phase by controlling duration of the electric ﬁeld pulse. The phase shift may be adjusted by changing phase velocity through applied electricﬁeld. The phase difference Dubetween the output signals with and without Epulse reads FIG. 3. Electric ﬁeld modulating STO frequency. (a) Frequency versus cur- rent density under different bias E(0,60.5,60.3). (b) Frequency versus bias Eunder different current density J¼5.0, 6.0, 7.0, 8.0( /C2106A/cm2). (c) Frequency peak of output power under different Evalues /C00.2–0.2 V/nm at a constant current density J¼/C08.0/C2106A/cm2, marked by the yellow line in (b). FIG. 4. Phase modulation by electric ﬁeld pulse. (a) 22 ns electric ﬁeld pulses applied during magnetization precession. (b), (c) Output resistantsignal after adding different negative or positive bias pulse. (d) Linear ﬁt of modulating phase versus electric ﬁeld under different eigen frequencies (1.86–1.99 GHz). (e) Phase lag under different pulse duration and electric ﬁelds.082406-3 Zhao et al. Appl. Phys. Lett. 111, 082406 (2017)Du¼ðtE/C0t0Þtpulse¼2pDftpulse ; (4) where t0is the initial phase velocity of eigen oscillation, tEis the phase velocity of oscillation with Epulse, and tpulseandDf represent pulse duration and frequency change, respectively. For example, tpulse¼22 ns, Df¼/C031:2M H z ( w h e n p u l s e amplitude E¼0:1V =nm and eigen current /C08.0/C2106A/ cm2), phase lag Duwas estimated to be /C01:372pwith Eq. (4). For demonstration of above assumption, we simulate oscilla-tions of magnetic resistance by applying a 22 ns Epulse under a stimulating current /C08:0/C210 6A=cm2,a ss h o w ni nF i g s . 4(a)–4(c) , illustrating phase shift appeared in both positive and negative E. According to Eq. (4), positive Ecauses a phase lag. The phase of wave peaks at Efrom 0 V/nm to 0.3 V/nm are labelled in the following order: u0!u1!u2!u3. Phase difference can be modulated by applying differentamplitude of Epulse. We compared the modulation rate in context of different eigen frequencies (1.86, 1.92, 1.95, and 1.99 GHz) under pulse duration 22 ns. Slope of ﬁtted resultsshow that the corresponding modulation rate are /C01:23p, /C01:38p,/C01:41p,a n d /C01:37pper 0.1 V/nm, respectively [Fig. 4(d)]. Phase difference can also be manipulated by regulating pulse duration. Effects of pulse time on phase modulation were investigated by setting different pulse duration of 22 ns, 44 ns, 66 ns, and 88 ns [Fig. 4(e)]. Current density /C08.0/C210 6A/cm2 was chosen to stimulate eigen precession. The linear results of phase variation Duversus the strength of Epulse were observed under different pulse w idths. Larger negative or posi- tive pulse amplitudes exhibit in creased positive or negative slopes, respectively. In summary, phase shift may be adjusted by both duration and strength of applied Epulse. This method can potentially be applied to STO a r r a yt oa d j u s ta n dl o c kt h e phase. In summary, the spin precession dynamics in a CoFeB/ MgO based STT nano-oscillator is simulated. By introducing a top oxide layer, the spin precession state can be manipu- lated by an electric ﬁeld between the two electrodes viainterfacial perpendicular magnetic anisotropy. The results can be summarized in following three points: (i) by applying appropriate electric ﬁeld and current, the spectra have multi-frequency modes and single frequency modes, and the range of the single frequency modes can be shifted by varying the amplitude of applied electric ﬁeld. (ii) The frequency of spinprecession may also be manipulated by electric ﬁeld. The frequency peak shifts to the low-frequency side with the increase in electric ﬁeld. (iii) by applying electric ﬁeld pulse,phase of the spin precession can be modulated by controlling the duration and amplitude of the pulses. A positive electric ﬁeld will cause phase lag, which is in proportion to theamplitude of the electric ﬁeld. 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1.4752260.pdf	Thermal stability of exchange-biased NiFe/FeMn multilayered thin films H. Y. Chen, Nguyen N. Phuoc, and C. K. Ong Citation: Journal of Applied Physics 112, 053920 (2012); doi: 10.1063/1.4752260 View online: http://dx.doi.org/10.1063/1.4752260 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/112/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of antiferromagnetic thickness on thermal stability of static and dynamic magnetization of NiFe/FeMn multilayers J. Appl. Phys. 113, 063913 (2013); 10.1063/1.4792223 Increased ferromagnetic resonance linewidth and exchange anisotropy in NiFe/FeMn bilayers J. Appl. Phys. 105, 063902 (2009); 10.1063/1.3086292 Combination of ultimate magnetization and ultrahigh uniaxial anisotropy in CoFe exchange-coupled multilayers J. Appl. Phys. 97, 10F910 (2005); 10.1063/1.1855171 Magnetic and thermal properties of IrMn/FeTaN films J. Appl. Phys. 87, 5867 (2000); 10.1063/1.372549 Brillouin light scattering investigations of exchange biased (110)-oriented NiFe/FeMn bilayers J. Appl. Phys. 83, 2863 (1998); 10.1063/1.367049 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 18:37:05Thermal stability of exchange-biased NiFe/FeMn multilayered thin films H. Y . Chen,1Nguyen N. Phuoc,2and C. K. Ong1,a) 1Center for Superconducting and Magnetic Materials, Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542 2Temasek Laboratories, National University of Singapore, 5A Engineering Drive 2, Singapore 117411 (Received 20 May 2012; accepted 9 August 2012; published online 11 September 2012) A systematic study of the effect of ferromagnetic thickness on magnetic and microwave properties of exchange-biased NiFe/FeMn multilayered thin ﬁlms was carried out with regards to thermal stability. The temperature-dependent microwave characteristics of the ﬁlms were obtained from thenear-ﬁeld microwave microscopy technique and analysed based on Landau-Lifshitz-Gilbert equation. The complex microwave permeability spectra of the magnetic thin ﬁlms up to 5 GHz in the temperature range from room temperature to 420 K were measured. It was found that thickerferromagnetic layers helped to reduce the dependence of the magnetic properties on temperature, leading to better thermal stability. The saturation magnetization M S, dynamic magnetic anisotropy ﬁeld H Kdyn, and ferromagnetic resonance frequency f FMRwere found to decrease with temperature, while the effective damping coefﬁcient aeffwas increased with temperature. We also investigate the rotational magnetic anisotropy ﬁeld H Krotwith temperature which gives a measure of the rotatable magnetization of the antiferromagnetic layers and its thermal stability. VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4752260 ] I. INTRODUCTION The exchange bias phenomenon between ferromagnet (FM) and antiferromagnet (AF)1,2has been widely used in many applications such as spin-valve sensors, magnetic re- cording in hard disk drives, and magnetic random accessmemory (MRAM). 2–5Most of these applications for the devi- ces are in the form of a thin ﬁlm in miniature devices2–5 employing exchange bias effect to providing an extra internal magnetic ﬁeld to pin a ferromagnetic layer.3Exchange-biased thin ﬁlms also enable the downsizing of communication devi- ces working in microwave frequencies.6–11It is therefore needful to characterize the magnetic properties of exchange- biased ﬁlms in the GHz frequency range.6–11In addition, ther- mal ﬂuctuations also become crucially important due to heattypically generated in these devices leading to degradation of magnetic properties. 12–16Yet, there have been few works in the literature focusing on the thermal stability study ofexchange-biased systems with regard to microwave character- istics at high temperature. 12In this paper, we thus performed a systematic investigation of an exchange-biased system con-sisting of alternating Ni 80Fe20layers being FM and Fe 50Mn50 layers being AF deposited on Si substrates with the FM thick-ness varied. This study investigates the temperature depend-ence of the magnetic properties up to 5 GHz, in the temperature range from room temperature to 420 K, with the inﬂuence of ferromagnetic layer thickness. II. EXPERIMENT Multilayered thin ﬁlms of [Ni 80Fe20(x nm)/Fe 50Mn50- (15 nm)] 10with the thickness of NiFe varied from 40 nm to120 nm were fabricated onto Si(100) substrates at an ambient temperature using a radio-frequency (RF) sputter-deposition system with the base pressure at 7 /C210/C07Torr. Both of the layers, NiFe and FeMn, are sputtered from alloy targets andthe ﬁrst layer deposited onto the substrate was a NiFe layer. A SiO 2layer with the thickness of 20 nm was coated on the thin ﬁlms to protect them from oxidation. The thickness of eachlayer was controlled both by the deposition time and by keep- ing the deposition rate constant, which was veriﬁed by a thick- ness proﬁle meter. A magnetic ﬁeld of approximately 200 Oewas applied in the plane of the ﬁlms to induce the unidirec- tional anisotropy. The argon pressure was kept at 2/C210 /C03Torr during the deposition process by introducing ar- gon gas at a ﬂow rate of 16 SCCM (SCCM denotes cubic cen- timeter per minute at STP). A M-H loop tracer and a vibrating sample magnetometer (VSM) were employed to measure thehysteresis loops of the samples from 300 K to 420 K. The per- meability spectra of the magnetic thin ﬁlms in the temperature range from 300 K to 420 K were measured using atemperature-dependent characterization system that has been fabricated in-house based on the near-ﬁeld microwave micros- copy (NFMM) technique. 17In the NFMM technique, the fer- romagnetic resonance (FMR) frequency is measured by a microwave probe that excites a small local area of the sample and picks up the electromagnetic response, which is then proc-essed by the vector network analyser (VNA). 17–20This tech- nique places no constraints on the sample and allows a local analysis of the sample and sample conditions (such as temper-ature) to be changed externally without damaging the trans- mission lines and connectors. 17–20Hence, this method is quite suitable for studying magnetic properties in temperature-dependent experiments. The microwave probe has an area of 2m m 2and the measurement frequency range used in the pres- ent study is from 0.05 GHz to 5 GHz.17a)Author to whom correspondence should be addressed. Electronic mail: phyongck@nus.edu.sg. Tel.: 65-65162816. Fax: 65-67776126. 0021-8979/2012/112(5)/053920/6/$30.00 VC2012 American Institute of Physics 112, 053920-1JOURNAL OF APPLIED PHYSICS 112, 053920 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 18:37:05III. RESULTS AND DISCUSSION Figure 1depicts typical magnetization hysteresis loops of [NiFe(40 nm)/FeMn(15 nm)] 10measured at T ¼330 K using VSM measurement along the easy and hard axes.Here, the direction of the ﬁeld applied during deposition is deﬁned as the easy axis, while the direction normal to the easy axis but still in the plane of the sample is deﬁned as thehard axis. 10,21,22Based on the magnetization curves as in Fig.1, we are able to derive several important static parame- ters such as the exchange bias ﬁeld H E, the coercivity H C and the static magnetic anisotropy ﬁeld H Ksta.8,21,22 The ﬁrst derived static parameter obtained from VSM measurement is the exchange bias ﬁeld H Ewhich is deﬁned as the hysteresis loop shift from the origin and occurs when the AF spins are coupled to the FM spins. As shown in Fig.2(a), the exchange bias decreases when the temperature increases because the FM-AF interface spins ﬂuctuate due to the thermal activation. 12–15This causes some of the AF spins to become unstable and do not couple fully to the FM spins. As a result, the exchange bias is reduced.12–15Another pa- rameter derived from VSM measurement is the coercivityﬁeld H C, which is deﬁned as half of the hysteresis loop width and is the ﬁeld required to switch the FM moment from one direction to the other. The enhancement of coercivity was aneffect already observed in early exchange biased experi- ments, 1but was not fully explained then due to many com- plications. The model proposed by Stiles and McMichael23 explained that when the ﬁeld is applied parallel or antiparal- lel to the bias direction, the AF grains apply torques to the FM magnetization that varies from grain to grain in magni-tude and direction. Hence, there will be torques on the FM magnetization in both directions, and some parts have clock- wise reversal while others have anticlockwise reversal. 23 These differences lead to a barrier for reversal, and are a source of enhanced coercivity for a exchange-biased sys- tem.23At higher temperatures, more energy is provided to overcome the reversal barrier and coercivity decreases,24 which is in agreement with our experimental results shownin Fig. 2(b). The magnitude of the coercivity is also related to the rotatable spin AF grains. 25,26When spins of the AF grains become rotatable with the FM magnetization and fol- low the rotation of FM moment, the coercivity would beenhanced due to the contribution of the AF reversal. Thethird derived parameter obtained from static magnetic mea- surement is the static magnetic anisotropy ﬁeld H Ksta extracted from the hard axis curve of the M-H loop.8,21,22As presented in Fig. 2(c), the static magnetic anisotropy ﬁeld HKstafor all the samples are observed to decrease with tem- perature. As is well-known, the static magnetic anisotropyﬁeld H Kstais the sum of the intrinsic uniaxial anisotropy ﬁeld HKof the FM layer and the exchange bias ﬁeld H E.21,22With the increase of temperature, the exchange bias ﬁeld isdecreased as discussed above and the intrinsic uniaxial ani- sotropy ﬁeld is also decreased owing to the thermal ﬂuctua- tions of the FM spins. 12As a result, H Ksta, which is the sum of these two contributions, decreases accordingly. We now turn to the discussion of the thermal stability of dynamic magnetic characteristics of our ﬁlms in conjunctionwith the static magnetic properties. Typical permeability spectra of [NiFe(40 nm)/FeMn(15 nm)] 10measured at vari- ous temperatures using the NFMM technique are presentedin Fig. 3. 17The shift of the peak of the imaginary permeabil- ity spectrum towards the lower frequency range with increas- ing temperature is clearly observed, which is indicative ofthe reduction of ferromagnetic resonance frequency when the temperature is increased. In order to have a more quanti- tative analysis, we employed the Landau-Lifshitz-Gilbert(LLG) equation 27,28as below to ﬁt the experimental dynamic permeability spectra. FIG. 1. Typical hysteresis loops along easy and hard axes of [NiFe(40 nm)/ FeMn(15 nm)] 10multilayer measured at T ¼330 K. FIG. 2. Temperature dependences of: (a) exchange bias ﬁeld H E, (b) coer- civity H C, and (c) static magnetic anisotropy ﬁeld H Kstafor [NiFe(t FMnm)/ FeMn(15 nm)] 10multilayers with various FM thicknesses as indicated in the legends.053920-2 Chen, Phuoc, and Ong J. Appl. Phys. 112, 053920 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 18:37:05d~M dt¼/C0cð~M/C2~HÞþaef f M~M/C2d~M dt: (1) Here, M represents the magnetization of the FM layers, H the magnetic ﬁeld, aeffthe dimensionless effective Gilbert damp- ing coefﬁcient ( aeffincludes both intrinsic and extrinsic damp- ings) and cthe gyromagnetic ratio. From the ﬁtting procedure, one can extract the dynamic magnetic anisotropy H Kdynas well as the effective Gilbert damping factor aeff.11,26 Figure 4(a) summarizes the temperature dependence of HKdynfor [NiFe(t FMnm)/FeMn(15 nm)] 10multilayers with various FM thicknesses. Unlike H Kstain Fig. 2(c) which shows a general decrease with temperature, H Kdynmanifests itself in a more complicated behaviour. For samples with thin NiFe layers, H Kdynis decreased with temperature while it is rather stable for the samples with thicker NiFe layers. We can also observe that there is a substantial discrepancy between the magnitudes of H Kdynand H Ksta. Both these behaviours can be explained in terms of the existence of rotational magnetic anisotropy.21,25,26,29The rotational mag- netic anisotropy is so-called because the direction of italways “follows” the magnetization. 30–32During the magnet- ization reversal as in the M-H loops, when the magnetization is switched, the anisotropy direction is also switched accord-ingly. Since the static magnetic measurement such as the VSM only “senses” the change of the magnetization with the applied ﬁeld, it cannot sense the rotational magnetic anisot-ropy which always rotates to follow the magnetization. Hence, the H Kstavalue obtained from VSM measurement does not include the rotational magnetic anisotropy.21,25,26,29 However, for the dynamic measurement such as our mea- surement of permeability spectra in Fig. 3, the small excited RF magnetic ﬁeld which changes directions at microwavefrequency is not large enough to make the magnetization reverse. Hence, this measurement can sense the existence ofthe rotational magnetic anisotropy while the static measure- ment cannot. 21,25,26,29Therefore, the magnetic anisotropy ﬁelds obtained from these two methods are quite different. In other words, the difference between the static and dynamicmagnetic anisotropy ﬁelds is due to the rotational magnetic anisotropy. We can therefore estimate the rotational mag- netic anisotropy ﬁeld H Krotby subtracting the H Kstafrom the HKdyn. This rotational anisotropy is deﬁned as an anisotropy that has an energy minimum that follows the FM magnetiza- tion direction.21,25,26,29Hence, this anisotropy can be “rotated” when the FM magnetization direction changes, which comes from changing the applied magnetic ﬁeld in terms of magnitude and direction.21,25,26,29–32It is also because of the contribution of this rotational anisotropy to dynamic magnetic anisotropy that makes the temperature behaviours of H Kstaand H Kdynquite different. The tempera- ture dependence of H Krot(which is deﬁned as H Krot¼HKdyn /C0HKsta) is presented in Fig. 4(b) showing a complicated behaviour depending on the FM thickness of the sample.This behaviour can be tentatively explained as follows. In the exchange-biased system, the AF grains can roughly be divided into three types as described in Fig. 5. The ﬁrst type, so-called random-spin grains (or disordered grains), is from those with very small grain sizes, of which the magnetic moments are thermally very unstable and fall into a para-magnetic state where their magnetic moments are random and do not have any contribution to the magnetic anisotropy. The second type of grains, which are frozen-spin grains (orfully ordered grains), has anisotropy K AFlarge enough to provide energy for a winding structure of a partial AF do- main wall when FM moment rotates. These AF grains willspring and unwind back to their original direction when the FM moment reverses back to the positive saturation. The FIG. 3. (a) Real and (b) imaginary permeability spectra of [NiFe(40 nm)/ FeMn(15 nm)] 10multilayer measured at various temperatures. Lines are LLG ﬁtting curves. FIG. 4. Temperature dependence of: (a) dynamic magnetic anisotropy H Kdyn and (b) rotational magnetic anisotropy H Krotfor [NiFe(t FMnm)/ FeMn(15 nm)] 10multilayers with various FM thicknesses as indicated in the legends.053920-3 Chen, Phuoc, and Ong J. Appl. Phys. 112, 053920 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 18:37:05grains remain in the original ﬁeld-cooling direction, continu- ing to contribute to the pinning of the FM magnetizationalong the ﬁeld-cooling direction 33,34and accounting for exchange bias anisotropy.33,34The third type of AF grains, which are rotatable-spin grains (or partially ordered grains),is in between the above two types, with weaker K AFsuch that the AF partial domain wall may form when the FM mag- netization is aligned to another direction. However, when theFM magnetization reverses back to positive saturation, the magnetization of these AF grains can be irreversibly “ﬂipped” to the FM layer direction. Hence, the magnetiza-tion of the AF grains becomes rotatable following the direc- tion of the FM magnetization which accounts for the rotational anisotropy. Figure 4(b) shows that the rotational anisotropy has large ﬂuctuations with temperature, but is rel- atively stable and increases for the samples with the thicker FM layers. In general, the rotational anisotropy should beincreased as the temperature increases, because the energy of spins to overcome the energy barrier increases. Essentially, the contributions to the rotational anisotropy depend on thedelicate balance among the three types of AF grains, namely random-spin grains, frozen-spin grains, and rotatable-spin grains. As temperature increases, some of the frozen-spin AFgrains may become rotatable-spin AF grains as described in Fig. 5, since these AF grains become more unstable and energetic. It is also possible for some weaker K AFrotatable- spin AF grains to become random-spin AF grains for the same reason. Hence, the number of rotatable-spin AF grains may be increased or decreased with rising temperature,depending on how many old rotatable-spin grains disappear (because temperature causes them to be more unstable andturn into random-spin grains) and how many new rotatable- spin grains are formed from the frozen-spin grains (also because temperature causes them to be more unstable andturn into rotatable-spin grains). For all the three samples with thicker FM layers, H Krotshows increasing trends as the temperature increases because the trend that frozen-spingrains are energetically unstable to become rotatable-spin grains dominates. Yet, for the ones with thinnest FM layers, H Krotis decreased with temperature because in these cases, the trend that rotatable-spin grains are energetically unstable to become random-spin grains dominates. As presented in Fig. 6(a), the FMR frequency is seen to decrease with increasing temperature. This behaviour is owing to the fact that the FMR frequency is dependent on both the saturation magnetization M Sand the dynamic mag- netic anisotropy ﬁeld H Kdynaccording to Kittel’s equation35 as follows: fFMR¼c 2pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ HKdynðHKdynþ4pMSÞq : (2) Since H Kdyndecreases with temperature as in Fig. 4(a),a n d MSobtained from the VSM is also decreased with tempera- ture, the FMR frequency shows the same decreasing trend. It is interesting to observe that the reduction of f FMR for the sample with thin FM layers is more drastic than that of thesample with thicker FM layers. This is also due to the temper- ature dependent behaviour of H Kdynand M S. For example, FIG. 5. Diagram describing how three types of AF grains change upon heat- ing. With increasing temperature, a fraction of rotatable-spin (partially or-dered) AF grains becomes random-spin (disordered) AF grains and another fraction of frozen-spin (fully ordered) AF grains become rotatable-spin (par- tially ordered) AF grains. FIG. 6. Temperature dependence of: (a) ferromagnetic resonance frequency fFMR, (b) frequency linewidth Df, and (c) effective Gilbert damping coefﬁ- cient aefffor [NiFe(t FMnm)/FeMn(15 nm)] 10multilayers with various FM thicknesses as indicated in the legends.053920-4 Chen, Phuoc, and Ong J. Appl. Phys. 112, 053920 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 18:37:05when temperature is increased from 300 K to 420 K, the reduction of H Kdynand M Sfor the sample with t FM¼120 nm is only 15% and 9%, while the reduction for the sample witht FM¼40 nm is 48% and 18%, respectively. As a result, the reduction of f FMRfor the sample with t FM¼120 nm is only 14% while the reduction of f FMR for t FM¼40 nm is 37%. This implies that in our exchange-biased system, the multi- layers with thicker FM layers have more thermal stability. The reason for this is possibly due to the fact that sampleswith thicker FM layers may have bigger FM grains leading to more thermal stability than that with smaller grains. 15,36At this moment, we should give a comment on the prospect ofemploying exchange-biased thin ﬁlms for high frequency applications at high temperatures. For the requirement from microwave applications, one needs magnetic thin ﬁlms withhigh resonance frequency and consequently exchange-biased ﬁlms with thinner FM layers are preferable as they offer higher resonance frequency. 6–10,22However, the ﬁlms with thinner FM layers may have problems when working at high temperatures as their thermal stability is worse than that of samples with thicker FM layers. This dilemma is a seriousissue that needs more research to tackle. The temperature dependence of the frequency linewidth Dfand the effective Gilbert damping coefﬁcient a effderived from LLG ﬁtting of the permeability spectra are shown in Figs. 6(b)and6(c), respectively. The frequency linewidth Df is determined by the following formula:11,26 Df¼caef fð4pMSþ2HKdynÞ 2p: (3) There are intrinsic and extrinsic contributions to the effective Gilbert damping.6,26,37The intrinsic part comes from the intrinsic Gilbert damping factor while the extrinsic part is likely to come from many other factors such as anisotropic dispersions, two-magnon scattering (TMS) processes,magnon-phonon scattering, and eddy current losses. 6,26,37,38 Increasing the temperature results in a higher damping due to the system becoming more thermally activated, and hencemore dispersions and scattering processes. 6,26,37In the sam- ples with thicker FM layers, the effective damping coefﬁ- cient is higher which is possibly due to more dispersion ofthe magnetic anisotropy. Another possible physical origin of the increasing of the effective damping coefﬁcient with FM thickness is the losses due to eddy current effect. As it iswell-known, the thicker the ﬁlm is the higher loss the eddy current effect produces 38and this higher loss causes a rise in the effective Gilbert damping with the ﬁlm thickness.Another possible mechanism that should not be ruled out is the TMS contribution. The TMS processes are known to be critically related to the ﬁlm microstructures, such as defects,grain size, and surface roughness. 6,37Hence, an increase of FM thickness may lead to a change to the ﬁlm microstruc- tures and consequently bring about a change in the effectiveGilbert damping due to the TMS contribution. However, an investigation of the detailed microstructures of the ﬁlm to shed a light on the correlation between the TMS processesand the ﬁlm microstructures in the present samples is rather complicated and beyond the scope of this paper.IV. SUMMARYAND CONCLUSION In summary, we report our systematic investigation of the temperature dependence of the magnetic properties at the gigahertz frequency range in the temperature range fromroom temperature to 420 K with the inﬂuence of ferromag- netic thickness. The NFMM technique was used to obtain the permeability spectra and the LLG theory was used to ﬁtthe spectra to extract dynamic magnetic properties of the multilayered NiFe/FeMn system. The static magnetic prop- erties were obtained using the vibrating sample magnetome-ter. For the samples with varying ferromagnetic thickness and ﬁxed antiferromagnetic thickness, the dynamic magnetic anisotropy ﬁeld, ferromagnetic resonance frequency, exchange bias ﬁeld, coercivity ﬁeld, saturation magnetiza- tion, and static magnetic anisotropy ﬁeld are found todecrease with temperature, while the effective damping coefﬁcient increases with temperature. The rotational mag- netic anisotropy ﬁeld becomes relatively stable at thickerferromagnetic layers. Finally, for this set of samples, this study concludes that thicker ferromagnetic layers result in more thermal stability of the magnetic properties at highertemperatures. ACKNOWLEDGMENTS The ﬁnancial support from the Defence Research and Technology Ofﬁce (DRTech) of Singapore is gratefullyacknowledged. 1W. H. Meiklejohn and C. P. Bean, Phys. Rev. 102, 1413 (1956). 2J. Nogu /C19es, J. Sort, V. Langlais, V. Skumryev, S. Suri ~nach, J. S. Mu ~noz, and M. D. Bar /C19o,Phys. Rep. 422, 65 (2005). 3B. Dieny, V. S. Speriosu, S. S. P. Parkin, B. A. Gurney, D. R. Wilhoit, and D. Mauri, Phys. Rev. B 43, 1297 (1991). 4C. Chappert, A. Fert, and F. Nguyen Van Dau, Nature Mater. 6, 813 (2007). 5C. Y. You, H. S. Goripati, T. Furubayashi, Y. K. Takahashi, and K. Hono,Appl. Phys. Lett. 93, 012501 (2008). 6B. K. Kuanr, R. E. Camley, and Z. Celinski, J. Appl. Phys. 93, 7723 (2003). 7S. Queste, S. 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1.5010948.pdf	Coupled breathing modes in one-dimensional Skyrmion lattices Junhoe Kim , Jaehak Yang , Young-Jun Cho , Bosung Kim , and Sang-Koog Kim Citation: Journal of Applied Physics 123, 053903 (2018); doi: 10.1063/1.5010948 View online: https://doi.org/10.1063/1.5010948 View Table of Contents: http://aip.scitation.org/toc/jap/123/5 Published by the American Institute of Physics Articles you may be interested in Motion of a skyrmionium driven by spin wave Applied Physics Letters 112, 062403 (2018); 10.1063/1.5010605 Optical properties of electrically connected plasmonic nanoantenna dimer arrays Journal of Applied Physics 123, 063101 (2018); 10.1063/1.5008511 Multiphase nanodomains in a strained BaTiO 3 film on a GdScO 3 substrate Journal of Applied Physics 123, 064102 (2018); 10.1063/1.5012545 Topological trajectories of a magnetic skyrmion with an in-plane microwave magnetic field Journal of Applied Physics 122, 223901 (2017); 10.1063/1.4998269 Pulsed coherent population trapping with repeated queries for producing single-peaked high contrast Ramsey interference Journal of Applied Physics 123, 053101 (2018); 10.1063/1.5008402 Theory of electromagnetic wave propagation in ferromagnetic Rashba conductor Journal of Applied Physics 123, 063902 (2018); 10.1063/1.5011130Coupled breathing modes in one-dimensional Skyrmion lattices Junhoe Kim, Jaehak Y ang, Y oung-Jun Cho, Bosung Kim, and Sang-Koog Kima) National Creative Research Initiative Center for Spin Dynamics and Spin-Wave Devices, Nanospinics Laboratory, Research Institute of Advanced Materials, Department of Materials Science and Engineering, Seoul National University, Seoul 151-744, South Korea (Received 27 October 2017; accepted 24 January 2018; published online 7 February 2018) We explored strong coupling of dynamic breathing modes in one-dimensional (1D) skyrmion latti- ces periodically arranged in thin-ﬁlm nanostrips. The coupled breathing modes exhibit characteris- tic concave-down dispersions that represent the in-phase high-energy mode at zero wavenumber (k¼0) and the anti-phase low-energy mode at the Brillouin zone boundary ( k¼kBZ). The band width of the allowed modes increases with decreasing inter-distance between nearest-neighboring skyrmions. Furthermore, the collective breathing modes propagate very well through the thin-ﬁlm nanostrips, as fast as 200–700 m/s, which propagation is controllable by the strength of magneticﬁelds applied perpendicularly to the ﬁlm plane. The breathing modes in 1D skyrmion lattices potentially formed in such nanostrips possibly can be used as information carriers in information processing devices. Published by AIP Publishing. https://doi.org/10.1063/1.5010948 I. INTRODUCTION Magnetic skyrmions1,2are topologically stable spin textures found in bulk crystals3,4and ultrathin-ﬁlm layered structures5–7owing to a broken inversion symmetry and a strong spin-orbit coupling. Dzyaloshinskii-Moriya interaction (DMI)8in these systems plays a key role in the formation of magnetic skyrmions.3–7The exotic topological spin texture of skyrmions and their manipulation, as well as their dynamic modes in the high-frequency sub-GHz-to-several-tens-of-GHz range, are of increasing interest from both fundamentaland technological perspectives. 6,9–23For example, reliable control of skyrmion motions in narrow-width thin-ﬁlm nano- strips by spin-polarized currents or magnetic ﬁelds allows forimplementation of skyrmions in potential information-storageand -processing devices. 6,9–14Moreover, single skyrmions in geometrically restricted magnetic dots exhibit unique GHz-range dynamic motions such as lower-energy gyrationmodes and higher-energy breathing modes. 15–17Furthermore, the breathing modes of dynamically stabilized skyrmions have been reported.18Therefore, microwave generators and detectors have been proposed based on their inherentdynamic modes. 19–22Additionally, spin-wave propagation and their dispersion characteristics in one-dimensional (1D) periodic arrays of skyrmions have been studied in the aspectof dynamic magnonic crystals. 23Very recently, the collective gyration modes in coupled individual skyrmions have been explored as information carriers,24,25as analogous to propa- gating spin waves in magnonic crystals26–30and to propagat- ing gyration modes in physically connected or separated vortex-state disks.31–33 In this work, we explore the coupled breathing modes between the nearest-neighboring skyrmions in thin-ﬁlm nano- strips showing characteristic concave-down dispersions. The collective motions of the individual skyrmions’ breathingmodes propagate well through the continuous nanochannels owing to the strong coupling characteristics of the neighboringskyrmions. Their propagation speed, in fact, is much higherthan those of skyrmion arrays’ coupled gyration modes, 25and also can be controlled by applying perpendicular magneticﬁelds. II. MODELING AND SIMULATIONS In the present study, we used the Mumax3 code34that incorporates the Landau-Lifshitz-Gilbert (LLG) equation35to numerically calculate the dynamic motions of individual sky-rmions periodically arranged in magnetic nanostrips withan open boundary condition at all of the edges. Co thin ﬁlmnanostrips of 1 nm thickness and 40 nm width interfaced withPt were modeled as follows: 9saturation magnetization Ms ¼580 kA/m, exchange stiffness Aex¼15 pJ/m, perpendicular anisotropy constant Ku¼0.8 MJ/m3, and DMI constant D¼3.0 mJ/m2. Different values of nanostrip length, l,w e r eu s e df o ra single skyrmion ( l¼40 nm), two skyrmions (64 nm), ﬁve sky- rmions (160 nm), and more skyrmions (800 nm).36,37The unit- cell size used in the micromagnetic simulations was 1 :0/C21:0 /C21:0n m3. An initial magnetization state was assumed to be that of the Neel-type skyrmion,38,39which was then relaxed for 100 ns using the damping constant a¼0.3, until its ground state could be obtained. Once we had obtained the ground stateof a single skyrmion or skyrmion lattices, we used a smallerdamping constant of a¼0.0001 for better spectral resolution of the excited dynamic modes. For relatively large damping con- stants, the frequency peaks of skyrmions’ coupled modesbecome broadened and the propagating signal of those modesis quickly attenuated (Refs. 25and33). However, the peak positions, dispersion curves, and propagation speeds showalmost the same trends for both smaller and larger dampingconstants. Although our results here were obtained with amuch smaller value of a¼0.0001 than those of real cases, such as Ta/CoFeB/MgO of a lower damping constant /C240.015, 40,41a)Author to whom all correspondence should be addressed: sangkoog@ snu.ac.kr 0021-8979/2018/123(5)/053903/6/$30.00 Published by AIP Publishing. 123, 053903-1JOURNAL OF APPLIED PHYSICS 123, 053903 (2018) the concept of the coupled modes of skyrmions can remain valid for real samples. In all of the simulations, we also used zero temperature without consideration of the thermal effect. Although thermalnoises were not included, our observation of breathing-modecoupling between neighboring skyrmions can be applicable toreal samples (Pt/Co/Ta and Pt/CoFeB/MgO multilayers 13,41)a t room temperature in cases where further optimizations ofmaterial parameters and structural design are made. III. RESULTS AND DISCUSSION First, in order to excite the breathing mode of a single skrymion in a square dot as depicted in Fig. 1(a), we applied the sinc-function ﬁeld Hz¼H0sin½xHðt/C0t0Þ/C138=½xHðt/C0t0Þ/C138 for 100 ns with H0¼10Oe;xH¼2p/C250GHz,a n d t0¼1n s over the entire area shown in Fig. 1(a). The temporal oscillation ofhmzi,mzaveraged over the entire region and its fast Fourier transformation (FFT) spectrum are plotted in Figs. 1(b) and 1(c), respectively. A single peak that appeared at 22.32 GHz corresponds to the breathing-mode frequency. Figure 1(d) shows the temporal oscillation hDmzðtÞið¼ h mzðtÞi /C0 h mzðt ¼0ÞiÞ(left column) along with snapshot images of the local Dmz(right column), as obtained by the inverse FFT of the FFTpower of hmziand the local mzin the frequency regions of Df¼22.30–22.34 GHz, respectively. The mzoscillation in the core region with its maintained radial symmetry corresponds toa characteristic breathing-mode behavior representative of peri- odic expansion and contraction of the core. 15–17 Next, we examined two skyrmions placed in a recta ngular dot as indicated in Fig. 2(a). To excite all of the possible breathing modes of the two skyrmions, we applied, in the – z direction, a 10 ns-width pulse ﬁeld of 500 Oe locally to onlyone of the two skyrmions. After the local ﬁeld was turned off, we monitored, under free relaxation for 100 ns, the hm ziin two different regions surrounding each core (i.e ., x¼0–32 nm for the left skyrmion and x¼33–64 nm for the right skyrmion), as s h o w ni nF i g . 3(b). The FFTs of the hmzioscillations in the two regions are plotted in Fig. 3(c). Two distinct peaks denoted as xlandxhare shown at 2 p/C220.55 and 2 p/C224.66 GHz, respectively, in the given frequency range.42The frequency splitting for the two skyrmions was the result of the symmetry FIG. 1. (a) Ground-state single skyrmion in a square dot. (b) Temporal varia- tion of the average mzcomponent hmzi, over an entire square dot by excitation of breathing mode using sinc-function ﬁeld Hz¼H0sin½xHðt/C0t0Þ/C138=½xHðt /C0t0Þ/C138, with H0¼10Oe ;xH¼2p/C250GHz, and t0¼1 ns. (c) FFT power spectrum, as obtained from fast Fourier Transform (FFT) of hmzioscillations. (d)hDmzioscillations obtained from inverse FFTs of peak with frequency band Df¼22.30–22.34 GHz. On the right are perspective-view snapshot images of the local Dmzof the breathing mode at the indicated two representa- tive times. FIG. 2. (a) Two skyrmions in a rectangular dot. (b) Temporal variation of average mzcomponents hmziover each skyrmion region separated by a verti- cal dotted line ( x¼0–32 nm for the left skyrmion and x¼33–64 nm for the right one). (c) FFT power spectra, as obtained from FFTs of hmzioscillations for both left and right skyrmion regions. (d) Perspective-view snapshot images of higher- and lower-frequency modes at indicated times for one cycle of each mode.053903-2 Kim et al. J. Appl. Phys. 123, 053903 (2018)breaking of the potential energy proﬁle of the isolated skyrmion due to its coupling, similarly to the coupled vortex disks described in Refs. 43and44. From the inverse FFTs of the local mz, we obtained, in the perspective view, snapshot images of both the lower xland higher xhmodes. For the xlmode, the left-side and right-side skyrmion cores oscillate as a breath- ing mode; that is, each core expands and contracts periodically, but both cores oscillate in anti-phase with each other. On theother hand, for the x hmode, both cores oscillate in-phase, as shown in Fig. 2(d). The anti-phase breathing motion of the two skyrmions shows a lower energy than the in-phase breathing mode, because the breathing modes represent the expansion and contraction of skyrmion cores. Therefore, the expansion of both cores in the same z(or – z) direction incurs higher energy cost of dynamic periodic motion. This effect is oppositeto those of the coupled gyration modes of skyrmions or mag- netic vortices. 25,31In-phase gyration motion between neighbor- ing skyrmions or magnetic vortices shows a lower energy than anti-phase gyration motion. This can be explained by the dynamic dipolar interaction between the gyrations of neigh- boring disks in vortex-state disks,31while the exchange, DMI, and magnetic anisotropy as well as the dipolar energy togethercontribute to the coupled gyration modes of neighboringskyrmions. 25Coupled breathing motions also are affected by the above-noted magnetic interactions, although gyration cou- pling between neighboring vortices in 1D arrays of vortex-statedisks is governed by dynamic dipolar interaction between the neighboring vortex-state disks. 31 Additionally, we studied with a coupling of ﬁve skyrmions in a nanostrip, as shown in Fig. 3(a). To excite all of the cou- pled modes in this ﬁve-skyrmion system, we applied a pulse ﬁeld only to the left region marked by the rectangular box (i.e.,sky1). The FFTs of the temporal oscillations of the hm ziof the individual skyrmions showed characteristic spectra for the indi- vidual skyrmions, as shown in Fig. 3(c). Five peaks denoted as xi(i¼1, 2, 3, 4, 5) were found, the relative FFT powers of which differed by each skyrmion. For all of the skyrmions, the ﬁve different peaks were located at 19.34, 20.31, 21.66,23.09, and 24.24 GHz. The ﬁrst and ﬁfth skyrmions showed all ﬁve peaks; the second and fourth skyrmions had no third peak, and the third skyrmion had no second or fourth peak.In order to understand the collective breathing modes, we per- formed, for all of the skyrmions, inverse FFTs of each peak. Figure 3(d) shows the spatial proﬁles of the hDm zivariation of the ﬁve individual cores for the different coupled modes. The hDmzivariation proﬁles represent ﬁve different standing- wave modes with ﬁxed boundaries at both ends (the imaginary0th and 6th skyrmion positions). The shapes of all of the standing-wave modes were either symmetric or anti-symmetric with respect to the center. For the x 1mode, all of the skyrmion cores oscillated in anti-phase between the nearest-neighboring skyrmions. For the x2,x3,a n d x4modes, the phase difference between the neighboring skyrmions decreased as the wave-length of the standing waves increased, thus resulting in a smaller number of nodes in the standing waves. For the x 5 mode, all of the cores oscillated in-phase without standing- wave nodes. Accordingly, on the basis of the ﬁxed boundary condition,31the wave vector of the allowed modes is expressed ask¼ðNþ1/C0mÞp=½ðNþ1ÞdintÞ/C138, with Nthe number of skyrmions, dintthe skyrmion inter-distance, and ma positive integer number that satisﬁes m/C20N. Therefore, the discrete ﬁve-modes’ k-values of collective skyrmion core breathing are given by km¼ð6/C0mÞp=6dint,w h e r e m¼1, 2, 3, 4, 5, as indicative of each mode. The nodes can thus be found at the nth skyrmion according to the condition sin ððð6/C0mÞp=6dintÞ^k /C1ndint^xÞ¼0. As noted earlier, the excited breathing mode from the left skyrmion can propagate through the neighboring skyrmionarrays. Thus, coupled breathing modes in skyrmion lattices can be used as information carriers, owing particularly to the strong coupling between the nearest-neighboring skyrmions.Therefore, we further examined a more general system: a one-dimensional (1D) skyrmion lattice strongly coupled in narrow-width strips as shown in Fig. 4(a).W eu s e d2 5s k y - rmions with d int¼32 nm average inter-distance between nearest-neighboring skyrmions in a given nanostrip. The cou- pled modes were excited by applying a pulse ﬁeld only to theﬁrst skyrmion at the left end. After the local ﬁeld was turned off, we monitored the dynamic oscillations of m zaveraged over each skyrmion region under free relaxation. From theFFTs of the temporal oscillations of the hm zifor each of the 25 skyrmion core regions, we obtained a dispersion relation FIG. 3. (a) Five skyrmions in the nanostrip of indicated dimensions. The black dotted rectangular box (i.e., sky1) indicates the region wherein a pulseﬁeld was applied for mode excitation. (b) Temporal evolution of hm zioscil- lations of each skyrmion and (c) their FFT spectra. (d) Spatial proﬁles of coupled breathing modes for each mode, as represented by hDmziof each skyrmion.053903-3 Kim et al. J. Appl. Phys. 123, 053903 (2018)within the corresponding reduced zone, as shown in Fig. 4(b). The overall shape of the dispersion curve was concave down;the frequency was highest at k¼0 and lowest at the Brillouin zone, k¼k BZ.A t k¼0, all of the skyrmion cores moved coherently, while at k¼kBZ, they showed anti-phase motions between the nearest-neighboring skyrmions (i.e., the nodes ofthe standing waves were between the nearest-neighboring skyrmions), as shown in Fig. 4(c). Such concave-down disper- sion is characteristic of the coupled breathing modes of sky-rmion motions. This dispersion shape is opposite to theconcave-up shape of the coupled gyration modes of sky-rmions and the vortex gyration modes. The breathing modesof skyrmions exhibit periodic expansion and contraction ofthe core in its motion. Next, we studied how the band structure can be con- trolled with d intin a 1D skyrmion lattice model. Figure 5(a) shows the variation of the dispersion curve with dint.B o t h band width Dxand angular frequency xk¼0atk¼0 increased with decreasing dint. This behavior can be explained by the variation of the interaction energy between the neighboringcores in dynamic motions for different d intvalues. In Ref. 45, it was reported that the repulsive force between the skrymioncores increases as d intdecreases. The decrease of dintleads to the increase of magnetic interaction energies between the neighboring skyrmion cores, resulting in a larger frequency splitting.43On the other hand, the angular frequency xk¼BZat k¼kBZdoes not signiﬁcantly vary with dint, because the neighboring skyrmion cores expand or contract in anti-phaseatk¼k BZ, resulting in rather less interaction between the neighboring cores. The external ﬁeld also can change the dispersion of sky- rmion lattices. Here, we performed further simulations byapplying perpendicular magnetic ﬁelds H z¼þ2,þ1,/C01, and /C02 kOe. Figure 6(a) plots the variation of the dispersion- curve with Hz.A sHzincreases from negative to positive val- ues, the bandwidth decreases, but xk¼0andxk¼BZincrease, as shown in Fig. 6(b). This band structure change is caused bythe change in the skyrmion core proﬁle along the ﬁeld direc- tion with ﬁeld strength. For skyrmions with the downwardcore, the size of the core expands under a negative magneticﬁeld, whereas it shrinks under a positive magnetic ﬁeld. As a result, the eigenfrequency of the skyrmion’s breathing mode increases linearly with the perpendicular ﬁeld strength H z, as similar to those found in single skyrmions17or their 1D arrays. Furthermore, we conﬁrmed by micromagnetic simula-tions that the eigenfrequency of the breathing mode of a sin- gle skyrmion of the downward core in a square dot changes linearly with the ﬁeld strength [see the red line in Fig. 6(b)]. Note that the breathing frequency monotonically increaseswith H zdue to the strong conﬁnement in nanostrips of small width ( w¼40 nm). Therefore, the linear dependences of xBZ onHzare mainly related to the change in the eigenfrequency of the single skyrmions.17,46,47Also, the size of the skyrmion core decreases with increasing Hz, resulting in a decrease in the interaction energy between the cores and in Dxas well [see Fig. 7(c)]. Finally, we calculated a technologically important parameter of the propagation speed of coupled breathing-mode signals through a given nanostrip wherein skyrmions are periodically arranged. Figure 7(a) plots the temporal FIG. 4. (a) 1D skyrmion lattice in 40 nm wide, 800 nm long strip. The col- ors, as indicated by the color bar, correspond to the out-of-plane components of local magnetizations mz¼Mz/Ms. (b) Dispersion curve of the 1D sky- rmion lattice and (c) Spatial proﬁles of Dmzcomponents of individual sky- rmions for coupled breathing modes at k¼0 and k¼p/a where a ¼32 nm. FIG. 5. (a) Dispersion curves of 1D skyrmion chains and (b) angular fre- quency at k¼k0andk¼kBZ.for different inter-distances dint¼27, 29, 32, 34, and 38 nm.053903-4 Kim et al. J. Appl. Phys. 123, 053903 (2018)oscillations of the hDmz0i¼hmz(t)i/C0hmz(ground state) icom- ponents of the individual skyrmion cores for the 1D skyrmion array ( N¼25) after applying a pulse ﬁeld only to the ﬁrst skyrmion. The excited signal from the left end propagates well through the neighboring skyrmions along the nanostrip. The speed of the collective-breathing-mode signal was esti-mated, using the 1st wave packet’s movement along the sky- rmion lattice (denoted by the black line), to be about 340 m/s. This value is faster than the group velocity ( /C24260 m/s) of spin-wave propagation in 2D skyrmion lattices with a MnSi material. 48Furthermore, it is also about three times faster than that of the gyration signal in the same system25and more than ﬁve times faster than that of the gyration signal invortex disk arrays. 32It has been known that the propagating speed in a two-coupled-vortex system is inversely propor- tional to the energy transfer time sex¼p jDxj¼pj x0jðgx/C0p1p2gyÞj, where x0is the eigenfrequency of the vortex gyration, jis the stiffness coefﬁcient, pis the vortex polarization, and g is the interaction strength.43Accordingly, the collective breathing modes also can offer an advantage in their propaga- tion speed as an information carrier, since the eigenfrequency of the breathing mode (tens of GHz) is much higher thanthose of other modes such as the skyrmion gyration mode (/C241 GHz) and vortex gyration mode (hundreds of MHz). Furthermore, the speed of the breathing-mode signal was also estimated for different values of dintandHz, as shown in Figs. 7(b) and7(c), respectively. The resultant propagation speeds generally followed the dependence of Dxondintand Hz, respectively. The most important point here is that the speed of the breathing-mode signal was found to be reliably controllable by the system, speciﬁcally by the skyrmion core inter-distance as well as external parameters such as perpen- dicular magnetic ﬁeld strength and direction. For example,for further reduction of the skyrmion inter-distance, the speed was increased to /C24540 m/s for the case of d int¼27 nm, and the speed also was readily increased to /C24700 m/s by applying a perpendicular ﬁeld of /C02 kOe in the þzdirection. Such controllability of the signal speed as well as the dispersion curve of collective breathing modes in 1D skyrmion lattices can be implemented in multifunctional microwave and logicdevices operating within a frequency range as wide as a few tens of GHz. 49 IV. SUMMARY In summary, we studied coupled breathing modes in 1D skyrmion lattices in thin ﬁlm nanostrips. The coupled modes and their characteristic concave-down dispersions were foundand explained by a standing-wave model with a ﬁxed bound- ary condition. The dispersion curve of the collective breath- ing modes of coupled skyrmions was controllable by the skyrmion inter-distance and externally applied perpendicular FIG. 6. (a) Dispersion curves of 1D skyrmion chains for different perpendic- ular ﬁelds Hz¼/C02,/C01, 0, 1, and 2 kOe. (b) Angular frequencies xk¼BZ, xk¼0, and eigenfrequency x0of breathing mode for isolated skyrmion ver- susHz. FIG. 7. (a) Contour plot of hDmz0iof individual cores’ oscillations with respect to time and distance in the whole chain. The bracket indicates the average value of mzin each skyrmion region and hDmz(t)0i¼hmz (t)i/C0hmz(ground state) i. (b) Propagation speed and Dxof coupled breathing mode versus (b) dintand (c) Hz.053903-5 Kim et al. J. Appl. Phys. 123, 053903 (2018)magnetic ﬁelds. Moreover, the coupled breathing modes of skyrmions propagated well through neighboring skyrmions, as fast as 200–700 m/s, whose functionality makes thempotential information carriers in future information process- ing devices. ACKNOWLEDGMENTS This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and Future Planning (NRF-2015R1A2A1A10056286). 1T. H. R. Skyrme, Nucl. Phys. 31, 556 (1962). 2A. A. Belavin and A. M. Polyakov, JETP Lett. 22, 245 (1975). 3S. M €uhlbauer, B. Binz, F. Jonietz, C. Pﬂeiderer, A. 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Serga, M. P. Kostylev, R. L. Stamps, H. Schultheiss, K. Vogt, S. J. Hermsdoerfer, B. Laegel, P. A. Beck, and B.Hillebrands, Appl. Phys. Lett. 95, 262508 (2009). 30A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, Nat. Commun. 1, 141 (2010). 31D.-S. Han, A. Vogel, H. Jung, K.-S. Lee, M. Weigand, H. Stoll, G. Sch €utz, P. Fischer, G. Meier, and S.-K. Kim, Sci. Rep. 3, 2262 (2013). 32D.-S. Han, H.-B. Jeong, and S.-K. Kim, Appl. Phys. Lett. 103, 112406 (2013). 33H.-B. Jeong and S.-K. Kim, Appl. Phys. Lett. 105, 222410 (2014). 34A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. V. Waeyenberge, AIP Adv. 4, 107133 (2014). 35L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935); T. L. Gilbert, Phys. Rev. 100, 1243 (1955). 36The thickness, width and length of nanostrips can affect the interdistance of neighboring skyrmions and the size of each skyrmion’s core. For exam-ple, the interdistance of the neighboring skyrmions in equilibrium statesincreases with the width and length of nanostrips, resulting in a weakercoupling of those skyrmions. Also, the skyrmion size can be increased byincreasing the thickness of nanostrips, as reported in Ref. 37. 37O. Boulle, J. Vogel, H. Yang, S. Pizzini, D. S. Chaves, A. Locatelli, T. O. Mentes, A. Sala, L. D. Buda-Prejbeanu, O. Klein, M. Belmeguenai, Y.Roussign /C19e, A. Stashkevich, S. M. Ch /C19erif, L. Aballe, M. Foerster, M. Chshiev, S. Auffret, I. M. Miron, and G. Gaudin, Nat. Nanotechnol. 11, 449 (2016). 38We performed additional simulations for coupling between two Bloch-type skyrmions with material parameters of FeGe thin ﬁlms. 39The cou- pled breathing motions of the skyrmions also showed almost the sametrends as those of the Neel-type skyrmions, except for the magnitude oftheDm zproﬁles around each core center. 39M. Beg, R. Carey, W. Wang, D. Cort /C19es-Ortu ~no, M. Vousden, M.-A. Bisotti, M. Albert, D. Chernyshenko, O. Hovorka, R. L. Stamps, and H.Fangohr, Sci. Rep. 5, 17137 (2015). 40T. Devolder, P.-H. Ducrot, J.-P. Adam, I. Barisic, N. Vernier, J.-V. Kim, B. Ockert, and D. Ravelosona, Appl. Phys. Lett. 102, 022407 (2013). 41H. Du, R. Che, L. Kong, X. Zhao, C. Jin, C. Wang, J. Yang, W. Ning, R. Li, C. Jin, X. Chen, J. Zang, Y. Zhang, and M. Tian, Nat. Commun. 6, 8504 (2015). 42We also found two other coupled gyration mode peaks at 2 p/C20.88 and 2p/C21.48 GHz, due to the neighboring core shift during the ﬁeld applica- tion, as described in Ref. 25. However, in the present study, we will cover only the high-frequency breathing modes. We performed additional simu-lations using different ﬁeld polarizations for more mode excitations aswell as different areas of magnetic ﬁeld application. No additionalbranches except for modes reported here were found in a given frequencyrange. 43H. Jung, K.-S. Kim, D.-E. Jeong, Y.-S. Choi, Y.-S. Yu, D.-S. Han, A.Vogel, L. Bocklage, G. Meier, M.-Y. Im, P. Fischer, and S.-K. Kim, Sci. Rep. 1, 59 (2011). 44K.-S. Lee, H. Jung, D.-S. Han, and S.-K. Kim, J. Appl. Phys. 110, 113903 (2011). 45X. Zhang, G. P. Zhao, H. Fangohr, J. P. Liu, W. X. Xia, J. Xia, and F. J. Morvan, Sci. Rep. 5, 7643 (2015). 46M.-W. Yoo, K.-S. Lee, D.-S. Han, and S.-K. Kim, J. Appl. Phys. 109, 063903 (2011). 47G. de Loubens, A. Riegler, B. Pigeau, F. Lochner, F. Boust, K. Y.Guslienko, H. Hurdequint, L. W. Molenkamp, G. Schmidt, A. N. Slavin,V. S. Tiberkevich, N. Vukadinovic, and O. Klein, Phys. Rev. Lett. 102, 177602 (2009). 48M. Garst, J. Waizner, and D. Grundler, J. Phys. D: Appl. Phys. 50, 293002 (2017). 49M. Krawczyk and D. Grundler, J. Phys.: Condens. Matter 26, 123202 (2014).053903-6 Kim et al. J. Appl. Phys. 123, 053903 (2018)
1.1662444.pdf	Theory of domainwall motion in magnetic films and platelets J. C. Slonczewski Citation: J. Appl. Phys. 44, 1759 (1973); doi: 10.1063/1.1662444 View online: http://dx.doi.org/10.1063/1.1662444 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v44/i4 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsTheory of domain-wall motion in magnetic films and platelets J. C. Slonczewski IBM Thomas J Watson Research Center, Yorktown Heights, New York 10598 (Received 15 August 1972) We calculate dynamical properties of plane and cylindrical magnetic domain walls in a uniaxial film or platelet whose plane is perpendicular to the easy axis. First, we calculate the internal structure of a freely moving wall to determine the nonlinear velocity-momentum relation. Using a Bloch-line approximation to the kinetic wall energy, we find that stray magnetic fields emanating from surface poles destabilize the structure at a critical velocity Vp' above which uniform motion is not possible. For a plane wall, we have Vp =24yA /hK 112, where y is the gyromagnetic ratio, A is the exchange stiffness, h is the film thickness, and K is the anisotropy. This velocity is usually much less than the critical velocity for bulk wall motion derived by Walker. Combination of the velocity-momentum relation with the conservation of momentum provides a simple method of calculating the time-dependent velocity in the presence of a small drive field and small viscous damping. For very small drive, the terminal velocity equals that given by the conventional linear-mobility theory. However, for small drives exceeding a critical value corresponding to Vp' a periodic cycle of Bloch-line generation, motion, and annihilation takes place at a frequency approaching the Larmor frequency corresponding to the driving field. This is accompanied by a synchronous nonsinusoidal oscillation of the wall, with a mean forward velocity tending to an asymptote Vo. Our relation Vo= 7.1 yA / hK 112 agrees within a factor of 3 with the onset of nonlinear behavior observed by means of bubble collapse in certain low-loss garnet specimens. Partial support is also given by data for hexaferrite platelets. For radial motion of cylinders with finite diameter D, numerically computed values of Vo are larger but never exceed this expression by more than 17%. The critical velocity r;, also varies little for D / h > 2, but varies rapidly and attains large values for small D / h . I. INTRODUCTION We have recently derived nonlinear equations of motion, based on the Landau-Lifshitz equation, for a generally curved surface representing the boundary between op positely oriented magnetic domains. 1,2 The material was assumed to be uniaxial in first-order approximation, with anisotropy constant K. The principal approximation made was to neglect the velocity dependence of the wall thickness .:l. This approximation is particularly appro priate to materials satisfying the strong inequality K»27TM2, which ensures the constancy of.:l. Moreover, the current interest in "bubble" devices spurs the inves tigation and perfection of materials satisfying the weak er condition K>27TM2. 3 Thus, on the one hand, we have a condition which facilitates the theory of domain-wall motion without restriction to cases of uniform motion, one-dimensional treatment, or small amplitude. On the other hand, we have available for experimentation mixed garnet and magnetoplumbite single-crystal films and platelets to which the theory applies as a limiting case. The first application considered was a medium without boundaries. 1,2 However, the preferred experimental geometry is a film or plate whose plane is normal to the easy axis. The stray fields arising' from the surface have a strong influence on the motion, and cannot be ig nored. The object of this paper is to propose a workable approach to this case and present results for simple limiting cases. A brief report of some of this work, to gether with experimental work of co-authors is already available. 4 SchlOmann has also studied the motion of domain walls in films. 5 Although his theory resembles ours, as de tailed below, his results cannot be compared with ours because he has considered the opposite case of easy axis parallel to film surface, applicable to Permalloy films. 1759 J. Appl. Phys., Vol. 44, No.4, April 1973 In Sec. II, we review the general equations of motion and derive a momentum conservation relation. In Sec. III, we reduce the problem of wall-surface motion in the presence of a drive field and damping to that of a single coordinate and its conjugate momentum, without sacri ficing the most important three-dimensional considera tions. In Sec. IV, we discuss the kinetic structure and resulting velocity-momentum relationShip for a plane wall in a film. In Sec. V, we derive the nonlinear velo city-field relation based on this kinetic structure. In Sec. VI, we adapt the plane-wall theory to the radial motion of a cylinder domain of finite diameter. In Sec. VII, we compare the theory with experiment. The dis cussion in Sec. VIII touches on the question of bubble device data rate. II. GENERAL RELATIONS The theory rests on wall-motion equations derived pre viously.1 The total energy W of the system is assumed to have the conventional form W=J J J [AM"2(VM)2+Ksin2e+H2/87T]dxdydz, (2.1) where M(x, y, z) is the spontaneous magnetization vector, A is the exchange stiffness constant, e(x, y, z) is the polar angle defined by e=arccos(M.lM), and K(>O) is the uniaxial anisotropy constant. The magnetic field H satisfies the magneto static equations v. (H+ 47TM) = 0, V XH= 0 (2.2) and the corresponding boundary conditions that (H + 47TM) • nand n XH are continuous, where n is a vec tor normal to the surface in question. Suppose e(x, y, z) to be continuous and to satisfy 0.,,; e.,,; 7T everywhere and that there exist two domains with e -0 and e -7T, respectively. Then we may define a time-de pendent geometric surface, the Bloch surface, by the Copyright © 1973 American Institute of Physics 1759 Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1760 J.e. Slonczewski: Domain-wall motion in films locus of points y == q(x, z, t) at which () == t1T. In first-or der approximation, the transition layer (wall) between domains has the conventional Bloch structure ()= 2 arctan{exp[(y -q)/ ~]}, (2.3) where ~ '" (A/K)1/2 is the conventional thickness para meter. This formula is accurate when I aq / ax I and I aq/ az I are small compared to unity, and ~ is small compared to domain size or other dimensions describing the instantaneous domain structure. In his solution of the one-dimensional uniform- motion problem, Walker found the azimuthal angle 1/!, defined by M,,= Msin() cos1/!, to be constant. 6,7 In our three-di mensional problem, we allow 1/!(x, z, t) to vary over the Bloch surface, and we have justified the neglect of its variation across the thickness of the wall. 1 This angle describes the orientation of a Bloch moment J.L per unit area of the Bloch surface. The moment J.L lies in the xy plane and its magnitude is /l = 1.: M sin() dy = 1T ~M, where () is given by Eq. (2.3). In our previous paper, 1 the equations of motion ~== _ (y/2M)(o%q) _ a~-iq, q== (y/2M)(oa/olf;) + a~~, (2.4) (2.5) (2.6) were derived from the Landau-Lifshitz equation. Here oa(x, z) is the surface variation of energy defined by OW== f f oadxdz, (2.7) and discussed in detail previously. 1 The quantities oa/oq and oa/o1/! are the usual functional derivatives oa", aa -V.~ oq aq avq' (2.8a) (2.8b) where V is now and henceforth the gradient in xz space. The constant a is Gilbert's dimensionless viscous damp ing coefficient. The formula for oa in the limit I Vq I «1, ~IV1/!1 «1, is oa= {iaO(VQ)2+ 2A~(V1/!)2+ 41T~M2 sin2(1/! -:~) ] + 1T~M(H"sin1/!-Hycos1/!)o1/!- 2MH.Oq (ao"'4A1I2K1I2). (2.9) Equation (2 _ 5) states that the precession rate of the Bloch moment at any point is proportional to the sum of conservative and dissipative pressures on the wall. Equation (2. 6) states that the velOCity of any point on the Bloch surface is proportional to the sum of conservative and dissipative torques on the Bloch moment. Aside from dissipative terms, these equations are of Hamil ton's form with 2M1f;/y the canonical momentum conju gate to q. Their principal limitation is to neglect the velocity-dependent contraction of the thickness~, im plying the condition sin21/!«Q=K/21TM2. (2.10) If we want 1/! to be arbitrary, then we are limited by Q J. Appl. Phys_, Vol. 44, No.4, April 1973 »1. Thus our equations represent a limiting case of "magnetic bubble" materials, which are required to satisfy Q:; 1.3 1760 With respect to the boundary conditions for Eqs. (2.5) and (2. 6), we note the absence of surface terms in the dynamic reaction of a magnetic medium at a magnetic nonmagnetic boundary. In other words, there are no surface concentrations of magnetic moment. In addition, we neglect surface terms, such as surface anisotropy, which might be present in the energy. We assume also that the function (3(x, z) = 0 which defines the magnetic nonmagnetic boundary does not depend on y. Then by the usual procedure of calculus of variations, we find the boundary conditions aa aa a(aq/an) = a(a1/!/an) = 0, (2.11) where n is a coordinate normal to the boundary. We note that, because of the term 41T~M20 sin2(1/! -aq/ ax) in Eq. (2.9), this condition is not generally equivalent to the limit ~ -0 of the condition aMI an = 0 of the Landau Lifshitz equation. Such an equivalence should not be ex pected because the order of a limit and derivative is not generally interchangeable. In addition to previously derived properties of Eqs. (2.5) and (2.6), 1 such as the energy-conservation relation (2. 12) we may write a useful momentum-conservation relation. Let us integrate both sides of Eq. (2.5) over the entire wall surface: f f dxdz[$+ (y/2M)(oa/oq) + a~ -lq]= O. (2.13) By integrating the second term of Eq. (2.8a) and substi tuting the boundary conditions (2.11) and Eq. (2.9), we find f f dxdz ~a = f f dxdz aa == -2M f f H.dxdz. (2.14) vq aq Thus Eq. (2.13) becomes the momentum-conservation relation expressed by ~= yii. -a~-i~, (2.15) where the bar over any quantity Signifies an average over x and z. According to this equation, the Bloch wall in its entirety constitutes a "compound body" whose to tal momentum is impelled by the total external force minus the viscous retarding force. In the limit a= 0, it represents a curious generalization of the Larmor rela tion n= yH., requiring the wall as a whole to satisfy it but not any local portion of it. It is clear that the self-field Hw that is, the field ema nating from the divergence of magnetization, contributes nothing to H". For, a virtual uniform translation, q -q + E: where E: is constant, involves virtual work in the amount 2ME: J J dxdz H., and only the external field can do work on our translationally invariant system. There fore, only the external field is involved in Eq. (2.15). III. VELOCITY-FIELD RELATION Independent of our work, SchlOmann has recently ex- Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1761 J.e. Slonczewski: Domain·wall motion in films (0) ENERGY (b) VELOCITY 2.". FIG. 1. Schematic energy Wi and velocity ui versus momentum ljj for stable kinetic domain-wall structures in a magnetic film or platelet. The values Iii = Iii! and lP2 are critical points at which the structure becomes unstable. amined the motion of free domain walls in infinite media 7 and thin films with the easy axis parallel to the film plane. 5 Neglecting both the applied field and dissipation, he calculates the internal structure of a freely moving domain wall consistent with an assumed uniform velo city u. Although the stray-field energy of the film case was calculated in two dimensions, the theory is one dimensional in the sense that the wall structure is as sumed constant over the wall surface. SchlOmann finds a velocity bound V crlt (which we denote Vp), depending on material parameters and film thickness, which can not be exceeded if the motion is to remain uniform. 5 Given the structure for any u less than Vp, it is simple to calculate the dependence of terminal velocity on ap plied field H. for light damping. It is only necessary to calculate the power dissipated by the structure corres ponding to the given velocity and equate it with the ex ternally supplied power 2MH.u. 8 Palmer and Willoughby have carried out this procedure for a bulk case involving cubic anisotropy and magnetostriction.9 They worked out numerically a nonlinear mobility relation, taking into account the velocity dependence of the wall struc ture. In the bulk uniaxial case, this approximate proce dure is not necessary because Walker's exact solution is correct for arbitrarily large damping. 6 Remaining in all work to date is the question of what happens when H. ex ceeds the field Hp corresponding to VI>' The theory beginning in this section differs in important respects from SchlOmann's film theory5 and all pre vious theories. First, we allow the wall structure rep resented by the functions q(x, z) and ljJ(x, z) to 7ary over the wall surface, but at the cost of neglecting the velo city-dependent wall contraction, which is unimportant for 21TM2 «K. Second, we go one step further to calcu late the nonuniform motion in the range H. >Hp. Since our application (in Sec. IV) is to the case of easy axis normal to the film plane, our results cannot be com- J. Appl. Phys., Vol. 44, No.4, April 1973 1761 pared with SchlOmann's. We assume again that the sample surfaces are generated by lines parallel to y, as in a plate of uniform thickness. Then, if we neglect at first the applied field (Ii.== 0) and ... viscous damping (a== 0), ljJ vanishes according to Eq. (2.15), We can then seek uniform solutions to Eqs. (2.5) and (2.6), that is, with constant q==u and vanishing $. These equations become 15(J/l5ljJ==2MuYl, 15(J/l5q==O. (3.1) Since the variable q' '" q -ut does not depend on time, and Wis translationally invariant by assumption, it fol lows that these equations represent static equilibrium in a coordinate system moving with velocity u. They may be written in the variational form I5(W- 2uMy-1~) == 0, (3.2) where 15 represents variation with respect to the func tions q'(x, z) and ljJ(x, z), and where we now and hence forth normalize W [Eq. (2.7)] to unit projected wall area. Equation (3. 2) in the limit of constant q' and ljJ re duces essentially to the variational principle of Doring, 10 which is expressed by Eqs. (1) and (2) of SchlOmann, 5 in the limit M2/K-0. Since the constant 2uM/y plays a role of a Lagrange multiplier, Eq. (3.2) may be written in the equivalent constrained form I5W== 0, ~== const, (3.3) with u to be determined as explained below. For any given ~ there exists at least one solution of this pro blem, namely, the absolute minimum of W, since W is a positive definite functional [Eq. (2.1)]. In general, there exist n solutions ljJ== ljJi(X, z, ~), q==u/t+ q/(x, z, ~ with kinetic energy W== W/(~, i== 1, 2, ... , n. Now let us evaluate the derivative dwliP) ==ff(l5(J 8ljJ/ + o(J 8q 1\ d d lliP" I5ljJ 8'iJi I5q 8lJi) x z. Upon substitution of Eqs. (3.1) this becomes dWI_ 2MUlff~d d (if -Y 8ijj x z. Since the last integral is unity, the result is _ 'Y dWI UI-2M d~ , i==1,2, •.. ,n. (3.4) (3.5) (3.6) (3.7) This equation expresses the expected relation between velocity, kinetic energy WI, and total cononical momen tum 2M~h, considering the wall as a compound body. Since all of the foregoing relations are unchanged by the transformation 1/J-ljJ+ 21T, the periodic pattern of UI(lJi) resembles the band structure of quantized electrons moving in a periodic crystalline potential. We describe the functions ljJl, q I' and UI as aspects of "kinetic wall structure" . We consider physically meaningful only those kinetic structures which are dynamically stable. For a general state of motion under the condition a== 0, dW/dt vanish es according to Eq. (2.15). Then, in view of the as- Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1762 J.e. Slonczewski: Domain·wall motion in films FIG. 2. Schematic mean velocity Ii versus t in the presence of a step-driving field Ha initiated at t~ 0, for two values of Ha. Also shown is the approximate velocity Uj(t) for the case Ha:> Hp' according to the present theory. sumed translational invariance of W, the question of dynamical stability in the laboratory frame reduces to the question of static stability in the frame of the moving domain wall. Thus, any given ldnetic structure specified by Zjj and i is stable if W is a positive-definite functional of 1/J(x, z), q(x, z) in a small neighborhood of 1/Jj(x, z, 1Jj), qj(x, z, ~), holding q and ~ constant, and not otherwise. In general, a stable structure for a given i will depend continuously on~, perhaps terminating at a certain point where the solution of Eq. (3.2) becomes either nonexistent or unstable. Therefore a plot of Wj(~) or Ui(~) for stable structures with all i generally consists of a family of curves, some having endpoints and others not. Since the equations of motion are invariant under the transformation 1/J -1/J+ 21T, such plots also have the period 21T. Schematic plots for the case of a film or plate (details in Secs. IV and VI) are shown in Fig. 1. Suppose the set of stable multivalued velocity functions UI(~) are known. Then, the motion of the wall for finite external driving field H.~(t) and ex may be found with re lative ease under certain assumptions. First, we as sume that Hu and ex are sufficiently small that the in stantaneous velocity lj(t) is given by one or another branch of the relation UI(ZP). This relation may be sub stituted for ~ in the momentum conservation relation (2.15) to give (3.8) This first-order differential equation is easily inte grated numerically in practical cases. Th~ solution Zjj(t) may in turn be used to provide q(t) = f~UI[1/J(t')]dt' by direct integration. The qualitative prediction of Eq. (3.8) for a step field li •• = 0 (t < 0), li •• = Ha (t :> 0), where Ha is constant, is illustrated in Fig. 2. Suppose the initial (at t= 0) condi tions q= 0 and $= 0, with the wall structure on branch i= 1 in Fig. 1. If Ha is small, then u, and ~ increase with t along the branch i = 1 in accordance with Eq. (3.8), approaching exponentially en asymptotic "termi nal" velocity q .. found by setting $= o. It is given by ~~=Uj(t=oo)=~'YHa/ex. (3.9) The velocity of this motion is sketched in Fig. 2. This equation, of course, is the conventional linear-mobility relation, which is unaffected by the structural details described by $1(X, z) and q1(X, z). It holds under the con dition J. Appl. Phvs., Vol. 44, No.4, April 1973 1762 (3.10) where Hp is the field corresponding to the peak velocity for stable motion on the appropriate branch, as indi cated in Fig. l(b) for branch 1. Now suppose Ha >Hp. Then, according to Eq. (3.8) Ifj and q = U1 respond to the step drive by increaSing with time (see Fig. 2) until q(t1) = Vp at which point (1jj= ~ in Fig. 1) the structure on branch i = 1 becomes unstable. Be yond this point the now unsteady motion does not obey q= u1 nor Eq. (3.8) and recourse must be made to the full partial differential equations (2.5) and (2.6). These equations describe rapid fluctuations Owing to internal forces not vanishing with Ha and ex. We assume that with the passage of time the nonlinearities in the equations cause these fluctuations to increase rapidly in spatial and temporal frequency so that at some point in time the fluctuations may be considered thermal and therefore negligible. Since W is considered to exclude thermal energy, we adjust W by subtracting the amount of ther malized energy and note that the attendant temperature rise is, in practical cases, a very small fraction of 10K. Moreover, given sufficient time, the thermal en ergy diffuses away from the wall via lattice vibrations and bulk spin waves. At room temperature, then, the "renormalized" equations of motion (2.5) and (2.6), ne glecting thermal fluctuations, are practically the same as before. It follows that the system tends to some new stable branch, say i= 2 with diminished W= W2(~) and a new velocity U2(~) (see Fig. 1). The velocity q is peri odic in time as illustrated in Fig. 2, with reversals of sign possible. To diSCUSS this case Ha :>Hp quantitatively, let us write q(t) = Uj[Zjj(t)] + q(t-tl_1), (3.11) where i is the branch index toward which the instanta neous wall structure tends at time t, following an in stability on branch i -1 at t= tl_1• Thus ~ represents, by definition, the correction to q caused by the unstable episode in the motion. Also let N be the smallest integer required to satisfy the condition 1jJCt + T) = 1jj(t) + 21TN. (3.12) (In simple cases N= 1, but other values are possible.) Then, if H.~ and ex are proportional to a common infini tesimal expansion parameter A, it is clear from Eq. (3.8) that the time period T is proportional to A-1• But, in the limit A-0 the dependence of ~ on t-t, should tend to a limiting function with a limiting characteristic time constant of decay. Therefore, the limiting displacement correction f~ dt during this finite period becomes ne gligible compared to the displacement based on Eq. (3.8), since T tends to infinity. It follows that Eq. (3.8), together with the set U1(iP), describes the motion in the limit of small Ha and ex. We derive a closed expression for the mean velocity in the case Ha >Hp as follows. Integration of Eq. (2.15) with respect to t from t= 0 to T gives (3.13) (3.14) where V is the time average velocity. Equation (2.15) Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1763 J.e. Slonczewski: Domain·wall motion in films h + + + o~------~-L~~+~+-y FIG. 3. Schematic illustration of domain wall with y-coor dinate q in a film of thickness h. Surface poles giving rise to stray-field component Hy acting on wall are indicated. q may also be written dt [H A -1( + l. ) ]-1 dlj/= Y a-a .... u, q/ • (3.15) Integrating this equation, taking care to use the value of i appropriate to the instantaneous~, one finds !o2rN dt -1iil T= -=d1/!=E [YHa-a~-l(ul+~I)]-ld~, o d1/! j ~j-l (3.16) where !PI is the unstable terminus of the stable branch u/i$). The summation is carried over one cycle in time ordered sequence, so that (3.17) From the argument given above it is clear that, in the limit A -0, the range of ~ over which ~ I is appreciable tends to zero. Neglecting ~ I in Eq. (3.16) and eliminat ing T with Eq. (3. 13), one has V=A:Hq{1_2WN[~ ~j~f1d~ (1-~;~!rrl} (3.18) under the condition that ~YHa/ a exceeds the maximum u, occurring in the cycle, i. eo, Ha > Hp. Now, consider the special case a/Ha= O. Retaining ~ in Eq. (3.11) and integrating with respect to t, one has V= r-1 E ftl [u,(t)+ ~(t-t'_l)]dt. , t'_l (3.19) Substituting the exact relation (3.13) with a= 0, and in tegrating with the help of Eqs. (2.15) and (3.7), one finds (3.20) where (3.21) The term Wf(~') -W'+l(~') is the energy disSipated at the ith instability in the cycle. The form of Eq. (3.20) shows that the neglected velocity correction ~ I contri butes to the first-order dependence of V on HQ• This observation illustrates the fact that our Eq. (3.18) is correct only to zero order in a and H •. It is, however, formally exact in the ratio H./ a. We remark in passing that Eq. (3.21) is a direct conse quence of conservation of energy and the Larmor pre cession principle. It falls out of equating the input power 2MH" Vo to the dissipated energy Ej [W,(iP,) -W1+1(iP1)] divided by the time 2wN /yH" required for N Larmor cycles. IV. KINETIC WALL STRUCTURE IN A THIN FILM According to Sec. TIl, the velocity-momentum relation J. Appl. Phys., Vol. 44, No.4, April 1973 1763 u,(~) of a freely moving wall is the key to deriving the velocity-field relation in the limit of low losses. We discuss it here for the simplest problem with one finite dimension-an infinite plate of constant thickness h. We assume its faces to be given by z = 0, h, and that it con tains a single infinite domain wall with q and 1/! indepen dent of x, as indicated in Fig. 3. Interpreted literally, this model is somewhat artificial because Hagedorn has shown that a single wall is stati cally unstable with respect to Sinusoidal deformation, in the absence of a gradient dHeei dy in the external field. 11 If a gradient sufficient for stability were provided, then servoing of He with respect to q(t) would be required to maintain the free-wall condition H"6[Y= q(t)]= O. None theless we believe this model takes into account the physical mechanisms essential to the velocity-field rela tion for stable structures such as cylinder and stripe domains. It should have some quantitative validity for a cylinder of diameter D in the limit D/h- oo• The case of general D/h is treated in Sec. VI. The assumption ~ «h, made in the remainder of this paper is usually well satisfied in experiments, Since, in garnet films, ~ "'0.1 IJ,m and h is several IJ,m typi cally. The more serious approximation of neglecting the Bloch-line thickness in comparison with h is discussed at the end of this section and in Sec. VII. Figure 3 indicates a stray-field distribution emanating from magnetic poles on the film surfaces. The presence of the nonuniform component H y of this field at the wall surface ensures that fJ1/!/ fJz"* O. Moreover, under .our free uniform-motion assumptions (H. = 0, a= 0, !P= 0, q=u), the differential equations (3.1) for 1/! and q-ut are coupled through the magnetostatic interaction of the surface monopole density -2M fJq / fJz with the surface di pole density w~M sin1/!. However, we assume, in spite of this, that the wall is flat (fJq/ fJz = 0). To support this assumption, we consider first that the energy terms de pending only on q(z), namely, the surface energy f~dz ~(]o( aq/ fJZ)2 and the magneto static self-energy of the pole density -2M fJq / fJz, are quadratic functionals of fJq/ fJz and are each minimized by aq/ fJz = O. Second, one can see from symmetry that the potential due to w~M sin1/!, for arbitrary 1/!(z), is odd to zero order in y -q. It follows that the magneto static interaction be tween -2MfJq/ fJz and w~M sin1/! vanishes to first order in q(z) -q. Therefore, the energy is stationary at the flat wall state fJq/fJz= O. Since the surface energy dominates in the limit of large Q, the flat wall provides a minimum of the energy. Therefore we may safely let W be a func tional of 1/!(z) alone. Given the flatness of the wall, the only remaining non local interaction is the magneto static self-energy of the surface dipole distribution w~M Sin1/!. This energy is small compared to the local demagnetizing energy 4w~M2 sin21/!, a point discussed fully in the Appendix. With the nonlocal interaction neglected, W reduces (ac cording to Eqs. (2.7) and (2. 9)J to a surface integral of a local energy density, W= h-1 JtO'dz, (4.1) 0'= 2A~ (~!y + 4w~M2 siull1/! -w~MHy sin1/!. (4.2) Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1764 J.e. Slonczewski: Domain·wall motion in films FIG. 4. Kinetic structure of a plane wall in a film. (a) indicates moment orientation in a plane perpendicular to both wall plane y = 0 and film plane z = O. Stray fields from surface poles twist wall moment as shown. (b) shows calculated contours of con stant magneto static energy urn -u"., and schematic func tions I/J (z) for three different velocities. Curve 2 with a Bloch line (BL) at Z= z, corre sponds to the moment config uration in (a). [The sign con vention for M and H in this figure is opposite that of Eq. (2.3) and Fig. 3.1 A simple magneto static calculation of the stray field yields the formula R ... =-4M[lnz-In(h-z)] (4.3) in the limit Il/h= O. (Here the Signs of M and H differ from Fig. 3). Let us now examine the variational problem (3.3) for any given~. First consider the magnetostatic terms O'rn = 4rr/lM2 sin21/! -rrllMH y(z) Sin1/! (4.4) of Eq. (4.2) alone. Let O'mo(z) be the minimum of O'm(z, 1/!) with respect to !/J at 1/!= trr ± ¢(z), 0 < ¢ < rr. Contours of O'm -O'mo in the space 1/!, z are shown by the faint (con tinuous and dashed) curves in Fig. 4(b). The dashed curves represent the minimum contour O'm = O'mo' (Note that subtracting O'mo from 0' has a trivial effect on the variational problem.) Suppose tge exchange coefficient All is small. Then the solution 1/!(z) of Eq. (3.3) should lie near a segment of the minimum contour. Beginning with ~= 0, it may lie along curve 1 of Fig. 4(b). If we now increase ~ con tinuously to some value ~(2)' then 1/!(z) may change con tinuously to a curve such as curve 2 in Fig. 4(b), which follows minimum contours except in a narrow transition region surrounding the point z = z" where 1/! = h. Fig ure 4(a) depicts the moment configuration represented by ~(2)' It is reasonable to expect most of the energy to be concentrated in this transition region, and we esti mate it using the concept of a Bloch line, analogous to a Bloch wall. To do this, we neglect the z dependence of Hy, setting it equal to the constant Hy(z ,). We write the Bloch line energy (b) 1764 (4.5) under the condition that 1/! approaches the magneto static minima h ± ¢(z ,) (0" ¢ ., rr) for z -z ,-'F 00, respective ly. The extremal can be found from the usual Euler equation. (See the Appendix for details.) There are three cases, pertaining to three ranges of Hy: I. Hy~ 8M, ¢= 0, WL= 0; IT. IHy I., 8M, ¢= Cos-1H/8M, (4.6) WL = BAMh-1(2rr/K)1/2(sin¢ -¢ cos¢); (4.7) ITI. Hy" -8M, ¢ = rr, W L = 41l(rrAM)1I2h-1 xl: [ -2M sin21/!+ t I H y 1(1 + cos1/!)]1/2 dl/!, (4.8) where ¢ and H yare understood to be evaluated at the point z=z,. By definition, the momentum is given by ~= h-1 1: 1/!dz. (4.9) In Fig. 4(b) it is evident that h1/!(2l> say, is equal to the area between curves 1 and 2. In the Bloch-line limit we consider l/! to lie exactly on minimum contours except near z = z ,. Therefore ~ becomes ¢=2h-1 f'¢(z)dz, a (4.10) where z=z. and zb-;;h-za are the critical points, satis fying ¢ (z a) = 0, ¢ (Z b) = rr, which separate the three re gions of Eqs. (4.6)-(4.8). The velocity is found from Eqs. (3.7, (4.7), and (4.10) to be U1 = _ yA (2..)1/2 BHy(z I) 2M 2K 8z, (4. 11) for IHyl., 8M, that is to say, for z. <z, <Zb' Substituting the stray-field expression (4.3), we find U1 = 2yA( rr/2K)1I2[z;1 + (h -z ,)"1] (4. 12) in this range, with z. = hl(1 + e2) now. The Bloch-line relations given above describe W1 = WL(~) and U1 (~) through the parameter ¢ (z I), in view of the in tegral (4.10), which requires numerical evaluation. We obtain thus the branch i= 1 of the relations W(~) and u/iP) plotted on a reduced scale in Figs. 5(a) and 5(b), respectively. The curves labelled i = 1 for ~ < 0 arise from a Bloch line which nucleates at z, = Z b' A second branch i= 2 may be constructed by beginning at 1fj= rr with a 1/!(z) which follows the minimum contour ABC in FIG. 5. Kinetic energy Wi (a) and velocity Uj (b) versus momentum ip -1T" n~ O-In~ +. 1T" U, u2 -2 -Vp -Vp for a stable kinetic wall structure, cal culated in the Bloch-line limit. WI is in units of SAM(2il IK)! IZh-! and u, (s in units of 4yA(27TIK)1/2h-1. J. Appl. Phvs., Vol. 44, No.4, April 1973 Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1765 J.e. Slonczewski: Domain·wall motion in films ." 2' z. z FIG. 6. Illustration comparable to Fig. 4(b) showing nature of instability for a Bloch line AB whose position lies between a critical point and the film surface (z, > z,). A second Bloch line may nucleate at a point c (z = zb) and move toward the left to any position, such as DE with a decrease in energy at constant iP. Fig. 4(b), having odd symmetry about 1fi== rr. Increasing (or decreasing) ijj from this point nucleates a Bloch line at z = Z b (or z.). It is clear that 1fi2(ijj, z) == 1fil (ijj+ rr, h -z) +rr, and U2==ul(ijj+rr). Although the general equations (Sec. III) require the period 2rr, the higher symmetry of this particular problem yields the period rr for u(lP) if we eradicate the subscript. Figure 5 shows the kinetic structure terminating abrupt ly at the point lP== lPh and similar points. The reason for this is that, for H y" -8M, corresponding to z > Z band lP > lPl' the wall structure with a single Bloch line (AB in Fig. 6) becomes unstable. A second Bloch line (DE in Fig. 6) may form. Rather than deal with the question of multiple Bloch lines we make the model assumption that the points lP= $1 and 1P== $2 (for i== 2) are critical insta bility pOints of the sort discussed in Sec. III. We note that the Bloch-line limit gives rise to a non phYSical discontinuity for Ul at ijj== O. This anomaly can be traced to the fact that the width of the Bloch line tends to infinity at z -z.. It follows that a better approxima tion than the Bloch line is required to describe the ef fective mass near rest (ijj== 0). Indeed direct numerical integration of the differential equation (3.1), using the energy density (4.2), eliminates the discontinuities at lP== rrn (n= integer), showing that UI is a smooth function of $ at such points. 12 In other respects, the exact re sults increasingly approach those of the Bloch-line ap proximation as h increases. In particular, the exact procedure reveals critical 1P1 of instability, for h in the range of existing experimental data; V. VELOCITY·FIELD RELATION FOR STRAIGHT WALL Here we use the kinetic structure of a straight wall in the Bloch-line limit, determined in Sec. IV, to deter- J. Appl. Phys., Vol. 44, No.4, April 1973 1765 mine the V(H.) relation in accordance with Sec. III. As remarked before, this relation always has the conven tionallinear form V== AyH. / Of under the condition H. < H p = Of V / Ay. In the opposite case, we may substitute the kinetic structure relation UI(lP), obtained numerically in Sec. IV, into Eq. (3.8), which may be integrated with respect to t. Alternatively, the function set UI(lP) is sub stituted directly into the closed expression (3.18), which is evaluated numerically. The numerical work of either procedure is trivial and yields the results plotted in Fig. 7. The behavior V(H.) is well characterized by the critical parameters Hp, Vp, and Va indicated in Fig. 7. The peak velocity V p is obtained by substituting z, == Z b' corres ponding to ¢,== rr, into Eq. (4.12). One finds Vp== 4yAh-1(2rr/ K)1/2 cosh21 = 24. yA/hKl/2. (5.1) The asymptote Va is obtained from Eq. (3.21). The re quired initial critical energy "'1 (~1) is obtained by sub stituting ¢ = rr in Eq. (4. 7) to find WI (iPl) = 8rrAMh-1 (2 rr/K)1I2 . (5.2) The corresponding 1Pl = O. 762rr is determined numerical ly from Eqs. (4.10), (4.6), and (4.3). By symmetry, W2(~I)== W1(rr-~1)' and we find numerically c= W2(ijjl)/ W1(lPl)=O.29. Thus, Eq. (3.21) reduces to Va= 4(1-c) yAh-l (2rr/K)1/2 = 7. 1yA/hK1I2. (5.3) VI. ADAPTATION TO CYLINDER Our theory for the ideal case of an infinite straight wall in a film is not susceptible to direct experimental veri fication because this configuration is not stable even statically. 11 (We note in passing that this difficulty does not arise in the usual Permalloy configuration, consid ered by SchlOmann, 5 in which the easy direction is par allel to the film plane.) However, a bias field along the z axis stabilizes a cylindrical domain; hence many "bub ble-collapse" studies in pulsed fields have been made. The collapse method suffers from the defect that the radial driving pressure on the cylinder varies with time even if the applied field H. is a perfect step function. As discussed by Bobeck et al. 13 and Callen and Josephs, 14 H. must be corrected for an effective field of internal origin, which may be written FIG. 7. Reduced mean wall velocit~ (Vh/4Ay)(K/27r)1/2 versus reduced driving field H.h/4a(27r.A)1 2 for a plane wall. Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1766 J.e. Slonczewski: Domain-wall motion in films -0.5 -1.0 _I .5'------'----'---.L-.---'-----"--' 0.0 0.2 0.4 0.6 O.S 1.0 z/h FIG. 8. Reduced radial component of stray field acting on cylinder wall of diameter D versus depth in a film of thickness h. The curve D/h=10 is indistinguishable from D/h=oo_ H eff= ~; (47TrhM)-1, (6.1) where r(r) is the potential energy of the cylinder of ra dius r in the presence of the static bias. Since Heff de pends on r, which varies in the course of the collapse experiment, the desirable condition of constant drive is not easily met. In practice, the applied field is kept constant, and a Callen-Joseph type of analysis is re quired to relate collapse time to applied-field ampli tude. 14 Such an analysis, in our case of nonlinear re sponse, would be complex. However, we note that, in the limit Q/= 0, the critical field Hp vanishes and the theory predicts the steprela tion V= Vo sgnHa' Since V is then independent of drive field, as long as its sign is constant, the measured col lapse velocity is simply Vo. This assumes that the col lapse time is many times the Larmor period 27T/YHa so that the sawtooth behavior of instantaneous velocity, dis cussed in Sec. IV, is averaged. Hence, Vo should cor- 1.0r-----,----,.---.--ro-r--rT"l 0.2 0.4 0.6 0.8 1.0 lf/7r FIG. 9. Branch i = 1 of reduced kinetic energy versus momen tum for a cylinder wall of diameter D in a film of thickness h. All of the curves terminate at a reduced energy of unity. J. Appl. Phys., Vol. 44, No.4, April 1973 1766 ->.. ..<:1« :::Iv ~l7r FIG. 10. Branch i= 1 of reduced velocity versus momentum for a cylinder wall, for three ratios of diameter to film thickness. respond to a plateau in the collapse velocity or, if high er-order terms in the motion are important, to an ap prOXimate intercept at the collapse threshold. With respect to calculation of Vp and Vo, the principal difference between the cylinder and the plane wall is the radial stray-field component Hr(z) to be used in place of H y(z) in Sec. IV. The required H r(z) is evaluated nu merically by elementary methods, and the resulting field distribution is plotted in Fig. 8. This is substi tuted in the still valid relations (4.6), (4.7), (4.10), and (4.11), also evaluated numerically. The principal change in the kinetic structure for small diameter D = 2r is to introduce plateaus in W!(~) corresponding to the reduced variation of Hr for Bloch-line pOSition Z I near the midplane. (See Figs. 8 and 9.) Correspondingly, the minimum of Uj lies lower and its maximum at critical points lies higher (Fig. 10). The V(H.) relation for cylinders is similar to Fig. 7 and is not displayed. Straightforward numerical evaluation yields the dependences of Vp and Vo on D/h shown in Figs. 11 and 12, respectively. We note that the quantity W1('i/il) does not depend on diameter, since it is the Bloch-line energy at a critical value Hr= 8M, regard less of dependence on z. Since the relatively variable WZ('i/il) is considerably smaller than Wl(~l)' it follows 12rT--,---,---,---,----~ 10 OL-----L---~----~~--~----_:5 D/h FIG. 11. Reduced critical peak velocity of uniform motion versus ratio of diameter to film thickness. Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1767 J.e. Slonczewski: Domain·wall motion in films 0.85,------~--.---___,--___,--- 0.80 0.75 5 D/h FIG. 12. Asymptotic reduced mean velocity Vo (see Fig. 7) versus ratio of diameter to film thickness. that Va' which is proportional to the function 1-W2(~1)/W1(~1) plotted in Fig. 12, varies little. VII. COMPARISON WITH EXPERIMENT Recent collapse experiments have revealed the existence of garnet platelets, 15 garnet films, 4 and hexaferrite platelets13 with pronounced nonlinear characteristics. In the case of the garnet compositions, the nonlinear behavior correlates with the presence of only low-loss ions on the RE sites, namely, Y, Gd, or Eu. Of these, Y is nonmagnetic, Gd is an S-state, and Eu is a singlet- IO,-----------------------,~, 9 8 7 6 3 2 • SmO.l Gd2.24 Tbo.66 FeSOl2 x Y3G0I.SFe3.S012 . x.-----x-------x--- 30 40 50 60 70 H(Oel FIG. 13. Reciprocal collapse time versus pulsed field ampli tude for two garnet film compositions (Calhoun et al .• Ref. 15) J. Appl. Phys., Vol. 44, No.4, April 1973 1767 20 • S z 0 E 14 ... frllS I/) 5 ..... 12 ffi 12 1&.1 ~ ~ 1&.1 1&.1 ~ 10 • 2 -10 ~ c ~ 1&.1 c:; oJ 8 9 III III 1&.1 ~ > III 6 oJ S oJ ~ o , 0 FIG. 14. Mean collapse velocity versus pulse field amplitude for the garnet GdO.3Y2.7Gal.03Fe3.9PI2 with (111) axis normal to film plane (Argyle et al .• Ref. 4). Static diameter versus bias field are shown in inset. all special cases with little spin-lattice interaction. Many of the other rare-earth ions are more lossy than these. 16 The two types of behavior are illustrated by the platelet data shown in Fig. 13. The high-loss composition (SmO.1Gd2.24Tbo.66Fes012) behaves linearly. The low-loss composition (Y3Ga1.5Fes.5012) has a very steep initial slope, followed by a much reduced but varying slope. Indeed, the initial slope is so great that the true initial mobility is difficult to extract from the data; particular ly because the corrections for bubble potential alluded to in Sec. VI should be very important. However, a wall-resonance experiment indicates that Cl! < 10-2 17 in this specimen. A recent study of five low-loss garnet films grown by liquid phase epitaxy included some spec imens with similar variable-slope behaVior, as well as two in which the velocity apparently saturated. 4 One of these, GdO.3 Y2.7Ga1.03Fe3.97012, is shown in Fig. 14. In such cases, it seems natural to associate the saturation TU .; 0.4 CD Q I- ~ 0.2 0'----2.,.0-#1I3OB PbAJ4FeaOl9 S-B 14008 0.6mil 190 0.3 BIAS SET AT 160 Oe lu=0.2 mil FIG. 15. Reciprocal collapse time (and indicated mean velocity) versus pulse field amplitude for platelet of the hexaferrite PbAl4Fes019 (Bobeck et al .• Ref. 13). Static fields and diameter for strip to bubble and collapse are indicated. Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1768 'u ll.o co2 >C I...... -0.5 J.e. Slonczewski: Domain·wall'motion in films ., 2259 SrO.sCoO.5At4FeaOI9 S-B 380 08 0.48 mil B 400 0.31 440 0.15 PULSE FIELD AMPLITUDE (Oe) FIG. 16. Reciprocal collapse time (and mean velocity versus pulse-field amplitude for the hexaferrite Sro.5Cao.:;A14FesOI9 (Bobeck et al. , Ref. 13). velocity with Vo. However, in order to treat the garnet data uniformly, we take the "knee" velocity, defined as the first point of deviation from linearity, to compare with our Vo. The great uncertainties in some of the parameters entering into the theory make such distinc tions immaterial. In the case of the hexaferrite platelets, all but one ap pear definitely to saturate, 13 so we take the plateau ve locity. (See Fig. 15.) The remaining compound (Sro.sCao.sA4Fes019), however, has a maximum followed by a minimum V min' (See Fig. 16.) Since the maximum might correspond to the peak in Fig. 7, in this case we associate V min with Vo. Comparison of theory and experiment is made in Table I. The parameters Mand K of the garnet platelet were estimated from the behavior of cylinder domains in static applied fields. 15,17 In the case of the garnet films M was Similarly obtained,4 but K /M was determined more reliably by magnetic resonance. is The exchange A for all garnets was estimated from Curie tempera tures.4,15 For the films, resonance valueslS of g were used in the expression 1'= 8. 8 X 1Q6g Oe-1 sec-l; in other cases, the value g= 2 was assumed. 1768 Also shown are values of Q to indicate the a priori re liability of our general equations, valid for Q = 00, as well as P =Mh(21T/ A)1/2= h/ t.L (t.L is the Bloch-line thickness at z = ~h) to indicate the reliability of the Bloch-line approximation (P = 00). Also included in this table are calculated values of V w = 21T t. I'M, the upper bound on uniform velocity of a wall in infinite space, ac cording to Walker, 6,7 in the limit Q = 00. Our Vo values are based on the median ordinate of Fig. 12, Vo = 7. 7yA/hK1/2• Thus we neglect the modest (± < 8%) de pendence on diameter. The Vo values are smaller than V w by factors ranging from 3 to 30, because of our in clusion of stray-field and nonuniform motion effects. Finally, the characteristic experimental velocity Ve, whether determined at the knee or plateau, as explained above, is shown. In some cases the knee velocity is not well defined experimentally so that a range is given for Ve' The discrepancy between Vo and Ve for the garnet platelets amounts to a factor of 3, and for the other cases 2.2 or less. Only one hexaferrite, PbA4Fes019, is listed in Table I because not all of the parameters required to evaluate Vo in other cases are known. 13,19 However, it is rea sonable to suppose that Sro.sCao.sAI4Fes019 is magneti cally similar to the cited compound, with comparable values of y, A, and K. Since h is about 30 times greater in the case of Sro.sCao.sA4Fes0t9, 19 Vo should be 30 times smaller, or about 1 cm/sec. But this is 100 times smaller than the minimum in Fig. 16. Thus the present theory cannot explain this observation. VIII. DISCUSSION We have found (Sec. IV) that stray fields have a subtle ~ffect on the internal structure of a domain wall moving in a thin film or platelet. In the Bloch-line limit, the structure is described by a Bloch-line pOSition corre sponding to the velocity u. For some ranges of the ca nonical momentum~, the velocity and Bloch-line posi tion are single valued, for others double valued. This structural detail has no effect on the initial linear mobi lity, which is given correctly by the conventional theory. However, when the drive field H. is increased to a cri- TABLE I. Parameter values for garnet specimens of composition Gdy Y3_yGaxFe5-xOI2 or EUzY3 ... GaxFe5_xOI2, plus one hexaferrite. Theory EAllt x y z h 47TM KX10-3 A X107 g Q p Vw Va Ve S Ga Gd Eu (Mm) (G) (erg/em3) (erg/em) (m/see) (m/sec) (m/sec) (erg/em 3) Platelet 1.5 36 40 0.3 2. 1 2 5 62 94 4.6 12 -17 1.0 (Refs. 15 and 17) Fllm, r" 0.3 3.7 58 1.4 2.6 1. 86 10 8 65 24 17 16 (Refs. 1. 05 0.47 3.7 150 3.6 2.5 2.2 4 22 120 17 8 40 4 and 1.1 0.5 6.8 320 13 2.7 1. 75 3 83 110 4 4-6 50 18) 1. 1 0.6 10.8 210 10 2.5 1. 64 6 90 75 2.5 2-6 20 1.1 5.2 43 1.0 2.5 2.19 14 9 65 22 7 -10 8 Platelet PbAl4Fes019 2.5 580 1500 1. 08 2 110 90 14 0.5 1.1 150 (Refs. 13 and 19) J. Appl. Phys., Vol. 44, No.4, April 1973 Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1769 J.e. Slonczewski: Domain·wall motion in films tical value H P' U attains a value V fJ> corresponding to a Bloch-line position for which the structure is unstable. Here, the Bloch-line becomes annihilated, some kinetic energy is dissipated, and a new Bloch line appears at some new position corresponding to practically the same /p. For Ha >Hp, a periodic behavior ensues, in which 1P increases monotonically with time. In this range, Bloch lines are repeatedly created, displaced, and annihilated; while the velocity executes sawtooth behavior as in Fig. 2. The time-average velOCity V has a sharp downward break at Ha= Hp and approaches the asymptote Vo for Hi 01-00 (Fig. 7). The validity of the theory is limited to small values of 01 and Ha' Our main predictions are embodied in Eqs. (5.1) and (5.3) for Vp and Vo in the limit of infinite diameter D, and the computed plots (Figs. 11 and 12) of dependence on D/h (h=thickness of platelet). These quantities de pend only on static material parameters and cylinder dimensions. They do not involve any sort of damping co efficient, the energy dissipation implied by Vo being in trinsic to the ferromagnet. The inverse dependence of Vp and Vo on h for constant cylinder proportions is re markable because the usual mechanisms of damping, such as spin-lattice relaxation or surface-pit scattering, predict velocities which are independent of thickness or increasing with thickness. The peak velocity Vp varies moderately with D/h over most of the range but in creases sharply for small D/h. The asymptote Vo, how ever, varies only by less than 8% from the median in the range of 0 < D/h < 00. The range of h for which the inverse relations hold needs consideration. In the direction of small h, one error is that of the Bloch-line apprOximation, which is serious when the inequality P = h/ A L »1 is violated, and is cur rently under numerical investigation. 12 As h increases, our theory limits Ho to smaller values because neglected wave motion of the wall surface becomes important. However, at what values of hand Ho the tranSition to bulk behavior occurs has not been determined. Indeed a large gap in understanding bulk behavior remains. 1 Our theory has implications for bubble-device data rates Suppose we take Thiele's preferred geometry for a cy linder domain,3 given by D= 2h= 81= 8(KA)1I2/1TM2, and assume that the cylinder moves s diameters during each bit transfer. We may define the data rate R= V/sD, where V is the velocity. We might assume that 01 is op timized, so that V= VI> at the driving field Hp, as in Fig. 7. Alternatively, we might consider 01 vanishes so that V= Vo. Referring to Figs. 11 and 12, we have alterna tive forms R= Va = Cy(21TK)1/2 = Cy(21TA)1/2 = C1TyM sD 32sQ! 2sDQ 16sQ3/2' (8.1) according to whether K, D, or M is taken to be the in dependent variable. (Here Q = K/21TM 2.) The coefficient C is read off Fig. 11 and 12, and has the value 2.8 or 0.8, depending on whether 01 is optimized or set equal to zero. Suppose that we accept the premises of Eq. (8,1), and also assume that sand Q have the smallest practical J. Appl. Phys., Vol. 44, No.4, April 1973 1769 values. Then, since there is little scope for varying y or A, the optimum R is a fairly unique function of K or D or M. Suppose we let s = 3, Q= 3, Y= 2 X 107 sec-1 Oe-1, and assume that M == 2 x 103 G (iron) is an upper limit on the magnetization. Assuming (II to be optimized (i. e., C = 2. 8), the resulting R = 1.4 X 109 sec-1 is an "ultimate" upper bound on the data rate obtainable at low driving fields, assuming the requiSite K= 7 x 107 erg cm-3 were provided. The diameter would be less than 1000 A. In addition, a driving-field differential BH across the wall diameter must be provided to accelerate the wall to the required velocity. The time required to accele. rate to V p is roughly one-half of a Larmor period. This fact implies the condition BH:; 6. R/ Y in order to achieve the rate R. In the above illustration BH== 600 Oe. Several qualifying remarks are in order: (i) The theory applies strictly only to radial motion rather than trans lation of the cylinder as a whole, whose details will be different. In the case of translation, one expects at the very least sinuous Bloch curves rather than Bloch lines parallel to the film surface, so there must be some dif ference in the details. (ii) The mean velOCity V is given only in the limit 01== 0 and Ha-0, so that the theory does not consider the increase of V beyond the knee seen in some experimental cases such as Y3Ga1,sFe3.S012 in Fig. 13. (iii) The theory should not apply to weak ferromag nets such as orthoferrites in which the presence of Dzyaloshinsky fields makes the single-sublattice Landau-Lifshitz theory inapplicable. It is known that the orthoferrite YFe03 attains velocities in excess of 105 cm/ sec, 20 which approaches the Walker limit of 3. 6 X 105 cm/ sec. 16 Also, the large orthorhombic component of anisotropy makes the present theory inapplicable to or thoferrites, as explained below. As a final remark, we point out that the theory neglects all "basal" magnetic anisotropy (such as cubic crystal line, rhombic growth, or hexagonal crystalline) which would make different directions of M in the platelet plane energetically inequivalent. (The effect of a rhom bic component in the one-dimensional theory has already been discussed. 1,2,21) The effect of basal anisotropy gen erally is to make the two" easy" directions 1/1 = h ± ¢ of the wall moment inequivalent, thus contributing kinetic energy throughout the entire wall surface. In the above Bloch-line theory, however, the kinetic energy includes only the demagnetizing energy (Plus an equal amount of exchange) distributed within the Bloch-line regions of the wall surface. Thus the basal anisotropy need only be as large as the quantity S=21TALM2/h = 21TM2/P == (21TA)1I2M/h to have a significant effect. Since P = h/ A L ranges from 8 to 90 for the samples we have discussed, the basal anisotropy need not be large to be significant in these samples, as shown by the S values in Table I. A fortunate circumstance with respect to the garnet data cited above is that the film plane in every case is (111), which has magnetically sixfold rotational symmetry. Thus, no orthorhombic component of growth anistropy, which has twofold symmetry, is possible. Also, the cu bic anisotropy coefficient K1 contributes no sixfold ener gy variation in first order since it represents the am plitude of a fourth-degree spherical harmonic. However, Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1770 J.e. Slonczewski: Domain·wall motion in films in garnet films of (100) orientation, the effect of Kl( -6 x10S erg/cm3 for YIG), which is larger by far than the S values in Table I, should be significant. In other orientations [neither (100) nor (111)], the rhombic component of "growth anisotropy", which may be as large as -104 erg/ cm3, 22 should also be Significant. In such cases, extension of the theory is required. In the case of hexaferrites, the presence of a sixfold crystal axis diminishes the danger of basal anisotropy effects, because the sixfold coefficient is usually small. In the case of the hexaferrite Znl.SFeO.s Y, for example, the sixfold anisotropy is less than 6 erg/ cm3, 23 which is much less than the value S= 150 erg/cm3 for the hexaferrite in Table I. ACKNOWLEDGMENTS The author gratefully acknowledges stimulating discus sions with B. E. Argyle, access to computations of V. L. Moruzzi before publication, and private communications fromA.H. Bobeck, D.C. Cronemeyer, B.A. Calhoun, and F. B. Hagedorn. APPENDIX We define p = -1/J+ tIT and (A1) Then the variational problem (4.5) becomes equivalent to the Euler equation aJ _ ~~ = 0 J=pI2+ g(p) ap dz ap' ' with the boundary condition g -0 for z -z ! -l' 00. A first integral is (A2) g(p)=p,2, J=2g(p). (A3) Consequently, the Bloch-line energy becomes W L = 4A~h-l 1'" g(p) ddZ dp= 4A~h-l f' g 1/2 dp, (A4) -I/> P -q, where p= ±¢ minimizes g(p). Suppose H y ;, BM. Then ¢ = 0, W L = O. (A Bloch line of angle less than 27T does not exist because there is only one "easy" direction.) Suppose IHyl <BM. Then cos¢=H/BM, g= 27TM2A-l(cosp -COS¢)2, and (A4) integrates to give (4.7). Suppose Hy<-BM. Thencos¢= -1, ¢=7T, and (A5) g= (2A~)"1[47T~M2(cos2p -1) -7T~MHy(cosp+ 1)], (A6) and (A4) takes the form (4. B). Now we can compare WL with the magneto static self-en ergy Ws of the surface dipole distribution 7T~M sin1/J, J. Appl. Phys .• Vol. 44, No.4, April 1973 1770 which has been neglected. By means of Fourier analYSis one finds that for a plane wall this energy is given for mally by Ws= (7T~M)2(2h)"1 JOh J: (z -z,)"2[sin1/J(z') -sin1/J(z)]2 dz dz'. (A7) Therefore, in order of magnitude, Ws'" (7T~)2/h. From Eq. (4.7) we have WslW L '" 7T/16Q 1/2. (AB) (A9) Thus, in the limit of large Q =K/27TM2, a basic approxi mation of the general equations (2.5) and (2.6), Ws may be neglected. lJ. C. Slonczewski. Int. J. Magn. 2,85 (1972). 'J. C. Slonczewski, in AlP Conference Proceedings No.5. Magnetism and Magnetic Materials-1971 (American Institute of Physics, New York. 1972), p. 170. 3A. A. Thiele, J. App!. Phys. 41, 1139 (1970). 'B. E. Argyle, J. C. Slonczewski, and A. F. Mayadas, in AlP Conference Proceedings No.5, Magnetism and Magnetic Materials- 1971 (American Institute of Physics, New York, 1972). p. 175. Abo B. E. Argyle (private communication). sE. Schlomann, App!. Phys. Lett. 20, 190 (1972). 6L. R. Walker (unpublished). The calculation is reproduced by J. F. Dillon, Jr. in Treatise on Magnetism. edited by G. T. Rado and H. Suhl (Academic, New York, 1963), Vol III, p. 450. 7E. Schlomann, App!. Phys. Lett. 19, 274 (1971). SA. M. Clogston, Bell Syst. Tech. J. 34, 739 (1955). 9W. Palmer and R. A. Willoughby, IBM J. Res. Dev. 11,284 (1967). lOW. Doring, Z. Naturforsch. A 3, 373 (1948). "F. B. Hagedorn, J. App!. Phys. 41, 1161 (1970). "J. C. Slonczewski and V. L. Moruzzi (unpublished). 13 A. H. Bobeck, I. Danylchuk, J. P. Remeika, L. G. van Vitert, and E. M. Walters, Proceedings of the International Conference on Ferrites. 1970 (V. of Tokyo Press, 1971), p. 361. Also A. H. Bobeck (private communication). 14H. Callen and R. M. Josephs, J. App!. Phys. 42, 1977 (1971). 15B. A. Calhoun, E. A. Giess, and L. L. Rosier, App!. Phys. Lett. 18,287 (1971); L. L. Rosier and B. A. Calhoun, IEEE Trans. Magn. 7,747 (1971). 16See the review article by F. B. Hagedorn, in AlP Conference Proceedings No.5, Magnetism and Magnetic Materials-1971 (American Institute of Physics, New York, 1972), p. 72; O. P. Vella-Coleiro, D. H. Smith, and L. G. van Vitert, App!. Phys. Lett. 21, 36 (1972). 17B. A. Calhoun (private communication). ISO. C. Cronemeyer (private communication). 19In private communications, A. H. Bobeck has kindly supplied estimates of h, and F. B. Hagedorn and D. H. Smith have kindly provided values for M, A , and K. 'OF. C. Rossol, Phys. Rev. Lett. 24, 1021 (1970). 21A. A. Thiele (unpUblished); see F. B. Hagedorn, Ref. 16. 22A. Rosencwaig and W. J. Tabor, in AlP Conference Proceedings No. 5. Magnetism and Magnetic Materials-1971 (American Institute of Physics, New York, 1972), p. 57. 23J. Smit and H. P. J. Wijn, Ferrites (Wiley, New York, 1959), p. 210. Downloaded 28 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
1.4772613.pdf	Exploring the accessible frequency range of phase-resolved ferromagnetic resonance detected with x-rays P. Warnicke, R. Knut, E. Wahlström, O. Karis, W. E. Bailey, and D. A. Arena Citation: Journal of Applied Physics 113, 033904 (2013); doi: 10.1063/1.4772613 View online: http://dx.doi.org/10.1063/1.4772613 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/113/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Phase-resolved x-ray ferromagnetic resonance measurements in fluorescence yield J. Appl. Phys. 109, 07D353 (2011); 10.1063/1.3567143 A compact apparatus for studies of element and phase-resolved ferromagnetic resonance Rev. Sci. Instrum. 80, 083903 (2009); 10.1063/1.3190402 Combined time-resolved x-ray magnetic circular dichroism and ferromagnetic resonance studies of magnetic alloys and multilayers (invited) J. Appl. Phys. 101, 09C109 (2007); 10.1063/1.2712294 X-ray ferromagnetic resonance spectroscopy Appl. Phys. Lett. 87, 152503 (2005); 10.1063/1.2089180 Spatially resolved ferromagnetic resonance: Imaging of ferromagnetic eigenmodes J. Appl. Phys. 97, 10E704 (2005); 10.1063/1.1860971 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.129.164.186 On: Fri, 19 Dec 2014 17:35:07Exploring the accessible frequency range of phase-resolved ferromagnetic resonance detected with x-rays P . Warnicke,1,a)R. Knut,2E. Wahlstr €om,3O. Karis,2W. E. Bailey,4and D. A. Arena1 1National Synchrotron Light Source, Brookhaven National Laboratory, Upton, New York 11973, USA 2Department of Physics, Uppsala University, Uppsala, Sweden 3Department of Physics, Norwegian University of Science and Technology, Trondheim, Norway 4Materials Science and Engineering Program, Department of Applied Physics and Applied Mathematics, Columbia University, New York,, New York 10027, USA (Received 4 April 2012; accepted 29 November 2012; published online 16 January 2013) We present time- and element-resolved measurements of the magnetization dynamics in a ferromagnetic trilayer structure. A pump-probe scheme was utilized with a microwave magnetic excitation ﬁeld phase-locked to the photon bunches and x-ray magnetic circular dichroism intransmission geometry. Using a relatively large photon bunch length with a full width at half maximum of 650 ps, the precessional motion of the magnetization was resolved up to frequencies of 2.5 GHz, thereby enabling sampling at frequencies signiﬁcantly above the inverse bunch length. Bysimulating the experimental data with a numerical model based on a forced harmonic oscillator, we obtain good correlation between the two. The model, which includes timing jitter analysis, is used to predict the accessible frequency range of x-ray detected ferromagnetic resonance. VC2013 American Institute of Physics .[http://dx.doi.org/10.1063/1.4772613 ] I. INTRODUCTION A preferred technique for examining magnetization dy- namics at GHz frequencies is ferromagnetic resonance (FMR), which has been used extensively to examine issues such as damping mechanisms in elemental and compound fer-romagnets. 1–4FMR is particularly well suited for examining magnetization dynamics as the energy scale of the microwave excitation naturally matches with the precessional motion ofthe moments in many ferromagnetic materials. Recently, the technique has been expanded by combining it with another powerful spectroscopic technique: x-ray magnetic circulardichroism (XMCD). With XMCD, contributions from individ- ual elements can be isolated and examined, 5and the spin and orbital moment can be extracted via use of sum-rule analy-ses. 6These attributes make XMCD a particularly useful tech- nique for examining alloys and other complex magnetic materials such as oxides,7multilayers,8diluted magnetic semiconductors,9and molecular compounds.10 The combination of XMCD and FMR, often referred to as x-ray detected ferromagnetic resonance (XFMR), hasgained increasing popularity. 11,12,14–16Several different modes have been employed to record XFMR spectra. In one approach, time averaged and element-speciﬁc FMR spectrahave been recorded using microwave excitations that are asynchronous with the photon bunches. 11–13A main advant- age of this approach is that the full frequency space is accessi-ble as no phase relationship needs to be maintained between the microwave source and the x-ray bunches. An alternative approach takes advantage of the sub-ns x-ray pulses availableat most modern x-ray facilities to probe the orbit of precessing magnetic moments stroboscopically as a function of the phasebetween the excitation of the magnetization dynamics and the x-ray bunches. 15,16In this approach, a main distinction is the type of source used to excite magnetization dynamics. Pulsedcurrent sources with fast ( /C24100 ps) rise times can be used to provide a short Oersted ﬁeld, which couples to the magnetiza- tion in a thin ﬁlm sample. 14,17The resulting motion can be characterized by a relatively large angular deviation of the magnetization ( /C2420/C14or more) before the magnetization decays in an oscillatory fashion back to the equilibrium posi-tion. In principle, any resonance frequency of the system is accessible by tuning an external magnetic bias ﬁeld. With the pulsed excitation approach, the timing resolu- tion, which ultimately determines the ability to distinguish phase differences between dissimilar magnetic moments, depends primarily on factors such as the timing jitter ( t j) between the pulser and the bunch clock and the width of the x-ray bunches ( sc). To date, timing resolution in the order of 645 ps has been achieved, which is comparable to the bunch width used in the measurements.17,18 As mentioned, another method to implement XFMR is to use continuous microwave excitations that are synchronized(phase locked) with the photon bunches. The technique offers a number of advantages, including low jitter and wide range of excitation angles, enabling studies of the magnetic systemin the low angle ( <1 /C14) continuous precession mode up to the larger angle, non-linear regime. Moreover, XMFR with synchronized microwave excitations adds the ability to detectthephase of precessing magnetic moments relative to the driv- ing ﬁeld. 19The latter characteristic is of particular importance as it can be used to examine small differences in the phase ofprecessing elemental moments, which in turn can reveal weak coupling between dissimilar moments. 20Another advantage is the ability to measure the element-speciﬁc complex magneticsusceptibility ( v¼v 0þiv00) at arbitrary admixture between the in-phase ( v0) and out-of-phase ( v00)r e s p o n s e .15,16,21,22a)Present address: Swiss Light Source, Paul Scherrer Institut, 5232 Villigen - PSI, Switzerland. Electronic mail: peter.warnicke@psi.ch. 0021-8979/2013/113(3)/033904/6/$30.00 VC2013 American Institute of Physics 113, 033904-1JOURNAL OF APPLIED PHYSICS 113, 033904 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.129.164.186 On: Fri, 19 Dec 2014 17:35:07In conventional FMR, a wide frequency range for mea- surement of the resonant response is useful as it enables assessment of the frequency response at microwave bandsthat are of technological relevance (X-band and beyond). Also, conventional FMR at high frequencies can reveal new physics, such as the contribution of non-Gilbert damping inthin ferromagnetic ﬁlms. 23It is, therefore, useful to examine the potential limiting factors of XFMR such as the time struc- ture of the probe. Of course, additional factors can limit thepractical frequency limit of XFMR, such as noise in the detec- tion circuitry (e.g., voltage noise). However, this quantity is speciﬁc to the experimental apparatus used to measure theXFMR signal and hence in this paper we focus our investiga- tion on the more general consideration of the timing system, parameterized by t jandsc, and analyze how these parameters inﬂuence the upper accessible frequency of XFMR. We begin by reviewing the experimental methods used in the measure- ments and then develop a numerical model to explore theeffects of t jandsc. II. METHODS The measurements were carried out at the I1011 soft x-ray beam line, located on the MAX-II storage ring at MAX-lab, Lund University, Sweden. MAX-II operates with a fundamental bunch clock of 100 MHz. The x-ray source for I1011 beam line is an elliptically polarizing undulator,which permits selection of circularly or linearly polarized soft x-rays for experiments. The beam line is also equipped with an eight-pole vector magnet (octupole magnet). Thepoles of the magnet are oriented along the body centered directions of a conventional cubic geometry, and the maxi- mum gap between the poles is small (22 mm). The magneticarrangement provides on-sample magnetic ﬁelds ( H)u pt o /C240:5 T in an arbitrary direction ( ^H). The vector magnet is particularly useful for XFMR as it permits a simple changeof^Hfrom (i) partially parallel to the photon direction, which is necessary for conventional XMCD scans, to (ii) orthogo- nal to both the photon direction and the microwave excita-tion, suitable for phase-resolved XMFR measurements. A photodiode mounted in the octupole chamber is used to detect the photon transmitted through the thin-ﬁlm samples. Our microwave circuitry is similar to the one presented by Arena et al. in 2009. 16Here, we brieﬂy review the main components of the measurement system, which affect the sig-nal levels and timing measurements. The harmonic spectrum of the bunch clock from the synchrotron is generated by a low phase-noise comb generator and the desired frequency for theXFMR measurements is selected by a narrow band pass ﬁlter (BPF). This method restricts the available frequencies to the harmonics of the fundamental x-ray bunch clock, but resultsin a phase resolution which is considerably smaller than the bunch length of the x-rays. The relatively low frequency of the MAX-II storage ring (100 MHz) correspondingly resultsin a ﬁne spacing in the harmonic spectrum. Under normal operation, the full width at half maximum (FWHM) of the longitudinal photon bunch length amounts to s c¼650 ps. Timing control is obtained by a digital delay generator located on the high-frequency side of the circuit, which permitsvariation of the delay time between the microwave pump and the x-ray probe in steps of 5 ps. A ﬁnal ampliﬁcation stage in the microwave circuit increases the power of the single-frequency signal to about þ30 dBm before it is directed on to a co-planar waveguide (CPW) where the sample is mounted. Alternating magnetic ﬁelds inductively generated from the microwaves pass through the CPW and excite the magnet- ization precession in the sample. Under forced precession, the magnetization follows a highly elliptical orbit deﬁned by theeffective ﬁeld, which consists of contributions from the applied bias, anisotropy, and dipolar ﬁelds. Our samples have negligible anisotropy and the magnetic easy axis lies in-planedue to the dipolar ﬁelds; hence, the in-plane to out-of-plane axis ratio of the precessional orbit is /C291. As the projection of the XMCD is measured along the propagation direction ofthe x-ray photons, the alignment of the sample surface with respect to the incoming photons, therefore, determines the XMCD intensity. To increase the component along the photonpropagation direction from the magnetization precession, the sample was rotated 20 /C14away from normal incidence. As the photons arrive at the detector at a rate of 100 MHz, we strobo-scopically sample the instantaneous projection of the magnet- ization along the photon beam direction. XFMR can be used to examine complex structures com- prising one or more magnetic elements in several layers. In this study, we examine a trilayer sample consisting of two magnetic layers with in-plane easy axes separated by a non-magnetic layer. The sample with a layer structure of Ni 81Fe19ð15Þ=Cuð10Þ=Co93Zr7ð4Þ=Cuð2Þ(thickness in nm and Ni 81Fe19as the bottom layer) was deposited onto a 1m m/C21 mm, 100 nm thick, silicon nitride membrane under UHV base pressure using magnetron sputtering. The sample was mounted with the ﬁlm side facing a shorted CPW. Aperforated hole in the current-conducting central strip-line of the CPW enables transmission of x-rays. Two measurement protocols were used in the experi- ments. Field scans are performed by sweeping the magnetic bias ﬁeld at a constant delay time while recording the XMCD intensity. By selecting an energy corresponding to an elemen-tal core-level, this approach measures the element-speciﬁc complex susceptibility, containing linear combinations of v 0 andv00, with relative amplitudes chosen through the delay.16 Timing delay scans, on the other hand, are performed by scan- ning the delay time at a ﬁxed bias ﬁeld and photon energy and such scans can provide a direct mapping of the magnetizationprecession orbit in the sample. III. RESULTS Conventional XMCD spectra were measured in trans- mission mode using magnetic ﬁeld switching. Fig. 1shows x-ray absorption spectra (XAS) over the L3andL2edges of Fe recorded using a ﬁxed helicity of the x-rays. Below the XAS, the XMCD difference spectrum is presented. Understatic conditions, the magnetization makes a complete rever- sal in a single XMCD measurement. A different situation applies under continuous excitation where the precessionangle is small: typically, the magnetization cone angle is in the order of 1 /C14.20Consequently, the XMCD signal intensity033904-2 Warnicke et al. J. Appl. Phys. 113, 033904 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.129.164.186 On: Fri, 19 Dec 2014 17:35:07is about two orders of magnitude smaller in precessional mode compared to static full-switching mode. For the forced precession measurements acquired under microwave excitation, we tune the energy to the point ofmaximum spin asymmetry which maximizes the XMCD sig- nal (see arrow in Fig. 1). Field scans, obtained with a ﬁxed helicity, are presented in Fig. 2(a). In these scans, we mea- sure the amplitude of the magnetization projected along the beam direction as the external ﬁeld is swept through reso- nance. The line shape of the ﬁeld scan is determined by themixture between v 0andv00, and the relative contribution can be selected by tuning the phase delay between the micro- wave driving ﬁeld and the photon bunch clock. As wechange the frequency, and thereby the wavelength of the microwaves, we expect a shift in the relative phase. The am- plitude as well as the relative phase can be recovered from aﬁt of the ﬁeld scan with a complex Lorentzian. In Fig. 2(a), as the frequency is increased from 1.0 to 2.3 GHz, we observe a positive shift in the resonance ﬁeld in accordancewith conventional FMR. 2The recovered phases of the dis- played 1.0, 1.2, 1.3, 1.7, 2.0, and 2.3 GHz ﬁeld scans are 84/C14, 68/C14,6 7/C14,9 8/C14, 102/C14, and 111/C14, respectively, indicating that the scans are not purely imaginary but contain also some real part. The square of the resonance frequency is plotted against applied ﬁeld in Fig. 2(b). Assuming the only signiﬁcant con- tribution to the effective ﬁeld is the applied bias ﬁeld H(i.e., neglecting anisotropy and contributions from exchange cou- pling between the layers) our data are well described by theKittel relation 2 f2¼ðl0c=ð2pÞÞ2HresðHresþMsÞ (1) at resonance H¼Hres,w h e r e l0is the vacuum permeability andcthe gyromagnetic ratio. From the ﬁt, we extract a sat- uration magnetization l0Ms¼1:0 T, which is in line with expected values for Ni 81Fe19.24 As mentioned, an alternative way to probe the precessional motion is to stroboscopically sample the direction of themagnetization as a function of the delay time between the x-ray pulses and the microwave exc itation, and such timing delay scans are presented in Fig. 2(c). At a ﬁxed frequency and near the resonant ﬁeld, the timing scans have a maximum amplitude, which decays away as the ﬁeld deviates from the resonantvalue. As expected, the resonant amplitude is reduced as the frequency is increased, indicating that the opening angle of the precession cone decreases with increasing frequency at a con-stant power. At frequencies above 2.3 GHz, the noise level of the ﬁeld scans becomes comparable with the signal level and ﬁeld scans similar to the data presented in Fig. 2(a)reveal only noise. In contrast, timing scans performed at an extrapolated resonance ﬁeld value reveal disc ernible oscillations at a driving f r e q u e n c yo fu pt o2 . 5G H z ,a ss e e ni nF i g . 2(d). This is the cen- tral result of our paper. It is clear that the observed oscillation period (400 ps) is considerably shorter than the x-ray bunch length from the synchrotron (650 ps). To address the experimental factors on the timing side of the detection circuit, which limit the accessible frequency range, we construct a numerical model for the sampling pro-cess. We will assume that, in analogy with a forced harmonic oscillator, the magnetization is driven to forced precessionFIG. 1. X-ray spectroscopy for Fe recorded at reversed magnetic polarities S/C0andSþ. The XMCD is obtained from the difference of these spectra. Time-resolved XMCD scans were taken at the energy marked by the arrow. FIG. 2. XFMR measurements. (a) Field scans of the normalized XMCD in-tensity showing data as solid circles and complex Lorentzian ﬁts as solid lines. (b) Resonance frequency squared plotted against ﬁeld. (c) Time delay scans of the normalized XMCD intensity (solid circles) obtained at the reso- nance ﬁeld with sinusoidal ﬁts (solid line). (d) Time delay scan at 2.5 GHz averaged over 5 scans at a bias ﬁeld of 78 Oe. The vertical bars are standard deviations to the data.033904-3 Warnicke et al. J. Appl. Phys. 113, 033904 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.129.164.186 On: Fri, 19 Dec 2014 17:35:07and attains the same frequency fas the magnetic driving ﬁeld. The projection of the precessing magnetization onto the vector of the incoming x-rays for a given driving fre-quency fcan be represented by sðtÞ¼A 0sinð2pftþdÞwith a phase d. The coefﬁcient A0for a forced harmonic oscillator can be described by A0/1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4p2ðf2 0/C0f2Þ2þk2f2q ; (2) where f0is the resonance frequency and ka damping parame- ter.19At resonance, this expression reduces to A0/1=f assuming the damping is constant. The model further assumesthat the x-ray probes, p(t), are uniform, Gaussian pulses pðtÞ¼1 rcﬃﬃﬃﬃﬃﬃ 2pp e/C0t2=ð2r2 cÞ(3) with a FWHM of sc¼2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2l n2p rc. We simulate a delay scan at a ﬁxed frequency fnumerically by evaluating the inte- grated product of the precession orbit with a sampling pulse,accumulated over Nphoton bunchesSðs c;rj;td;fÞ¼1 NXN n¼1ð1 /C01sðtÞpðt/C0td/C0tjðnÞÞdt (4) for each delay time tdin the time range.25To take into account the jitter of the bunch clock, the timing of the probe for each sampling event nis shifted in time by a Gaussian dis- tributed random number tjwith a standard deviation rj.28In accordance with the experimentally measured amplitude, the numerically sampled amplitude Ais extracted from a ﬁt to the simulated delay scan. In Fig. 3, we compare the numerical calculations of the amplitude with experimental data. The inﬂuence of timing jitter and frequency on ampli- tude at a ﬁxed bunch length is investigated in Fig. 3(a).A t any given point in the parameter space, the amplitude drops monotonically with increasing forrj. In relative terms, the amplitude is more sensitive to a change of fcharacterized by a slope more than two orders of magnitude larger than for rj at the position of the solid circle. The shape of the amplitude landscape can be understood by noting that the jitter pro-duces a shift of each probe, which results in a net broadening and reduction of the sampled intensity. As seen in Eq. (2), there is an increase in the precessional orbit amplitude forlower frequencies due to the 1/ fdependence. FIG. 3. Numerical amplitude calculations compared with experiment. (a) For visual clarity, the square-root of the normalized amplitude Ais plotted versus rj andffor a ﬁxed bunch length of sc¼650 ps. The solid circle marks the experimental conditions of the highest attainable frequency at MAX-lab. (b) Experi- mental and numerical amplitudes of the timing scans, both following a decaying trend with increasing frequency. The standard deviation of the ﬁts to t he ex- perimental data is normalized with the amplitude (star symbols). (c) Square-root of the normalized amplitude plotted versus scandffor a ﬁxed jitter of rj¼40 ps. A dashed line marks the 1/ fbunch length for comparison. (d) Amplitude proﬁles for 40 to 650 ps bunch length at rj¼40 ps.033904-4 Warnicke et al. J. Appl. Phys. 113, 033904 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.129.164.186 On: Fri, 19 Dec 2014 17:35:07The experimental amplitude decays to a frequency region where the relative noise is about one order of magnitude larger. This is illustrated in Fig. 3(b) together with numerical data using parameters corresponding to the experimental con- ditions: sc¼650 ps and rj¼40 ps.26The numerical ampli- tude follows a non-linear decaying trend which is alsoobserved in the experimental data. Comparing the ﬁtting error of the measured timing scans with the numerical timing scans two regimes are present in the frequency spectrum. Forf/C202:3 GHz, the ﬁtting error of the experimental data (star symbols) increases linearly with a slope of /C240:14 GHz /C01fol- lowed by a sharp increase in slope to 1 :8G H z/C01for f>2:3 GHz. The numerical ﬁts follow a similar trend with a cut-off frequency at about 2.6 GHz (not shown). The rela- tively small discrepancy in the ﬁtting error for the amplitudebetween the experimental data and the simulated delay scans may be due to the voltage noise in the measurement, which is not included in our modeling. Keeping the jitter parameter constant at 40 ps, the inﬂu- ence of the photon bunch length s cis examined in Fig. 3(c). Again, a monotonic decay in amplitude is observed forincreasing fors c. At a ﬁxed frequency, the amplitude is sen- sitive to a relative change in sc. For example, a reduction of the bunch length from 650 ps to 550 ps leads to an increasein the sampled amplitude from 3 :3/C210 /C05to 4:9/C210/C04.A t the same position, the gradients have the same order of mag- nitude along fandscimplying that the amplitude is more sensitive to a change in scas compared to a change in rj. In Fig. 3(d), the amplitude decay is shown as a function of frequency for different choices of the bunch length and ata constant jitter of 40 ps. At a bunch length of 650 ps, the rel- ative amplitude crosses the 10 /C04limit at about 2.5 GHz, which coincides with the frequency where signals are stilldiscernible in the experiments (see Fig. 2). Using this value as an estimate for the detection limit, the upper detectable frequency increases at shorter bunch lengths. For 100 ps pho-ton bunches (available at several synchrotrons), the upper de- tectable frequency exceeds 10 GHz. Detection at higher frequencies can be a practical challenge due to prominentnoise. The signal-to-noise ratio scales with the square root of the number of probing events so to maintain the signal level at 10% of the amplitude one needs to increase the measure-ment time by a factor 100. The choice of a constant jitter in Fig.3(d) focuses the analysis of the XFMR frequency limit on the bunchlength. It should be noted that the jitter is alsoexpected to inﬂuence the upper detectable frequency. IV. DISCUSSION The experimental data and the numerical analysis imply that the accessible frequency range in phase-resolved XFMRextends well beyond 1 =s c(this ratio amounts to 1.54 GHz at the MAX-II storage ring). While the amplitude of the signal decays rapidly for increasing frequencies above this point, theﬁeld scans can be measured at c onsiderably highe r frequencies and the timing scans have a detectable signal at even higher fre- quencies. These results suggest that it is feasible to extend thefrequency range of phase-sensitive and x-ray detected FMR beyond the X-band, especially in consideration to the <80 psbunch lengths available at sever al of today’s third generation synchrotrons. Furthermore, experimental schemes to transfer- ring the XFMR technique to fourth generation synchrotrons,where photon bunch lengths in the sub-ps regime can be achieved, have recently been proposed. 27 In conclusion, experimental measurements of the upper accessible frequency for transmission mode XFMR exceed the inverse bunch length. The observed upper accessible fre- quency is supported by a numerical model, which accountsfor the sampled amplitude dependence on the pump fre- quency as well as the time structure of the probe, i.e., the bunch length and timing jitter. The model could be extendedfurther by including additional sources of jitter, for example, sampling noise 28originating from the detector side of the circuit. ACKNOWLEDGMENTS The authors thank Dr. Gunnar €Ohrwall at MAX-lab for beamline support. The support of the Swedish Foundation for International Cooperation in Research and Higher Educa-tion (STINT) is gratefully acknowledged. The support of the NSLS under DOE Contract No. DE-AC02-98CH10886 is also gratefully acknowledged. W.E.B. acknowledges supportfrom the U.S. NSF-ECCS-0925829. 1J. H. E. Grifﬁths, Nature 158, 670 (1946). 2C. Kittel, Phys. Rev. 73, 155 (1948). 3M. Sparks, Ferromagnetic Relaxation Theory (McGraw-Hill, New York, 1964). 4M. Farle, Rep. Prog. Phys. 61, 755 (1998). 5G. Sch €utz, W. Wagner, W. Wilhelm, P. Kienle, R. Zeller, R. Frahm, and G. Materlik, Phys. Rev. Lett. 58, 737 (1987). 6C. T. Chen, Y. U. Idzerda, H.-J. Lin, N. V. Smith, G. Meigs, E. Chaban, G. H. Ho, E. Pellegrin, and F. Sette, Phys. Rev. Lett. 75, 152 (1995). 7M. Besse, V. Cros, A. Barth /C19el/C19emy, H. Jaffre `s, J. Vogel, F. Petroff, A. Mir- one, A. Tagliaferri, P. Bencok, P. Decorse, P. Berthet, Z. 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Br €ussing, R. Abrudan, and H. Zabel, J. Phys. D 44, 165001 (2011). 15M. K. Marcham, P. S. Keatley, A. Neuder, R. J. Hicken, S. A. Cavill, L. R.Shelford, G. van der Laan, N. D. Telling, J. R. Childress, J. A. Katine, P. Shafer, and E. Arenholz, J. Appl. Phys. 109, 07D353 (2011). 16D. A. Arena, Y. Ding, E. Vescovo, S. Zohar, Y. Guan, and W. E. Bailey, Rev. Sci. Instrum. 80, 083903 (2009). 17W. E. Bailey, L. Cheng, D. J. Keavney, C.-C. Kao, E. Vescovo, and D. Arena, Phys. Rev. B 70, 172403 (2004). 18S. Buschhorn, F. Br €ussing, R. Abrudan, and Hartmut Zabel, J. Synchrotron Radiat. 18, 212 (2011). 19Y. Guan, W. E. Bailey, E. Vescovo, C. C. Kao, and D. A. Arena, J. Magn. Magn. Mater. 312, 374 (2007).033904-5 Warnicke et al. J. Appl. Phys. 113, 033904 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.129.164.186 On: Fri, 19 Dec 2014 17:35:0720D. A. Arena, E. Vescovo, C.-C. Kao, Y. Guan, and W. E. Bailey, Phys. Rev. B 74, 064409 (2006). 21T. Martin, G. Woltersdorf, C. Stamm, H. A. D €urr, R. Mattheis, C. H. Back, and G. Bayreuther, J. Appl. Phys. 105, 07D310 (2009). 22P. Klaer, F. Hoffmann, G. Woltersdorf, E. Arbelo Jorge, M. Jourdan, C. H. Back, and H. J. Elmers, J. Appl. Phys. 44, 425004 (2011). 23J. Lindner, K. Lenz, E. Kosubek, K. Baberschke, D. Spoddig, R. Mecken- stock, J. Pelzl, Z. Frait, and D. L. Mills, Phys. Rev. B 68, 060102 (2003). 24G. Counil, J.-V. Kim, T. Devolder, C. Chappert, K. Shigeto, and Y. Otani, J. Appl. Phys. 95, 5646 (2004). 25For the numerically calculated timing scans, a total range of 2 ns is used with a time delay interspacing of 10 ps. The number of probing events perdata point was kept at N¼1000, where the data showed good conver- gence. In the experimental delay scans, the number of photon bunches per data point was typically 108. 26The experimental jitter was measured using a digital sampling oscillo-scope and deﬁned as the rof approximately 10 000 excitation events trig- gered with respect to the bunch clock. It should be noted that the timing jitter depends on the components in the microwave circuit and may there- fore vary between similar setups at the same experimental station. 27A. Rogalev, J. Goulon, G. Goujon, F. Wilhelm, I. Ogawa, and T. Idehara, J. Infrared Millim. Terahertz Waves 33(7), 777–793 (2012). 28M. Shinagawa, Y. Akazawa, and T. Wakimoto, J. Solid-State Circuits 25, 220 (1990).033904-6 Warnicke et al. J. Appl. Phys. 113, 033904 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.129.164.186 On: Fri, 19 Dec 2014 17:35:07
1.4977490.pdf	Thickness dependence study of current-driven ferromagnetic resonance in Y 3Fe5O12/ heavy metal bilayers Z. Fang , A. Mitra , A. L. Westerman , M. Ali , C. Ciccarelli , O. Cespedes , B. J. Hickey , and A. J. Ferguson Citation: Appl. Phys. Lett. 110, 092403 (2017); doi: 10.1063/1.4977490 View online: http://dx.doi.org/10.1063/1.4977490 View Table of Contents: http://aip.scitation.org/toc/apl/110/9 Published by the American Institute of PhysicsThickness dependence study of current-driven ferromagnetic resonance in Y 3Fe5O12/heavy metal bilayers Z.Fang,1,a)A.Mitra,2A. L. Westerman,2M.Ali,2C.Ciccarelli,1O.Cespedes,2B. J. Hickey,2 and A. J. Ferguson1 1Microelectronics Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom 2School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom (Received 19 December 2016; accepted 14 February 2017; published online 28 February 2017) We use ferromagnetic resonance to study the current-induced torques in YIG/heavy metal bilayers. YIG samples with thickness varying from 14.8 nm to 80 nm, with the Pt or Ta thin ﬁlm on top, aremeasured by applying a microwave current into the heavy metals and measuring the longitudinal DC voltage generated by both spin rectiﬁcation and spin pumping. From a symmetry analysis of the FMR lineshape and its dependence on YIG thickness, we deduce that the Oersted ﬁeld domi-nates over spin-transfer torque in driving magnetization dynamics. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4977490 ] Insulating magnetic materials have recently played an important role in spintronics since they allow pure spin cur- rents to ﬂow without the associated charge transport. Withinthe family of ferromagnetic insulators, yttrium iron garnet(YIG) holds a special place owing to several favourable properties, including ultra-low damping, high Curie tempera- ture, and chemical stability. 1–3By growing an overlayer of heavy metal (HM), such as platinum or tantalum, severalimportant spintronic phenomena have been explored in the YIG/HM bilayer system, including the magnetic proximity effect, 4,5spin pumping,6,7spin Hall magnetoresistance (SMR),8,9spin Seebeck effect,10,11and so on. Furthermore, the spin Hall effect in HM can convert a charge current into a transverse pure spin current, making it possible to manipu- late the ferromagnetic insulator by spin transfer torque(STT). Recently, several groups have reported controllingthe damping in YIG by applying a DC charge current in a Pt capping layer, 12by which spin-Hall auto-oscillation can be realized.13,14Replacing the DC current with a microwave current, the electrical signal in Pt can also be transmitted viaspin waves in YIG. 3In order to further explore the applica- tion of the YIG/HM system, it is necessary to understand the torque on YIG induced by the charge current in HM. Current-induced ferromagnetic resonance (CI-FMR) is an effective method to characterize ferromagnetic samples atmicrometre-scale and quantify the current-induced magnetictorques in ferromagnetic/HM bilayer systems. 15As shown in Fig.1(a), an oscillating charge current in the HM layer gen- erates a perpendicular pure spin-current, oscillating at thesame frequency via the spin Hall effect. This oscillating spincurrent ﬂows into the ferromagnetic layer, exerting an oscil- lating STT, which can drive magnetization precession when the FMR condition is satisﬁed. 15–19Since no charge current is required to ﬂow inside the ferromagnetic layer in this pro-cess, it is possible to extend this method to ferromagnetic insulator/HM bilayers. Instead of penetrating into the ferro- magnetic layer, the electrons undergo spin-dependentscattering at the interface between the ferromagnetic insula- tor and the HM, transferring angular momentum to theferromagnetic insulator. The accompanied Oersted ﬁeld gen-erated by the charge current in Pt can also drive the magneti-zation precession. Although from the symmetry point ofview, the torques induced by the Oersted ﬁeld and the ﬁeld-like component of STT are indistinguishable from eachother, in our work, we conﬁrm that the driving ﬁeld is domi-nated by the Oersted contribution by repeating the measure-ment with Pt and Ta. Recently, several groups have studied the CI-FMR in YIG/Pt both theoretically 20,21and experimentally.22–24 Using the theoretical model built by Chiba et al.,21Schreier et al. did the ﬁrst experiment on in-plane CI-FMR in YIG/Pt and identiﬁed the current-induced torque by the symmetryand the lineshape of the signals. 22Sklenar et al . then repeated the experiment for an out-of-plane external mag-netic ﬁeld. 23Very recently, Jungﬂeisch et al . imaged the current-driven magnetization precession in YIG/Pt at the res-onance condition with Brillouin light scattering spectroscopyand argued that uniform precession is no longer applicable ata high microwave power. 24The behaviour of CI-FMR in YIG/HM, however, should depend on the thickness of theﬁlms, 20which is one aspect that remains under explored. Here, we study the current-induced torque in YIG/Pt (or Ta) bilayer structures with different YIG thicknesses usingCI-FMR. By applying a microwave current into the HM andsweeping the external magnetic ﬁeld in the plane of the devi-ces, a DC voltage is observed at the resonance condition.This DC voltage is generated simultaneously by two mecha-nisms: 20spin rectiﬁcation and spin pumping. The nature of the torque can be understood from the lineshape and the sym-metry of the DC voltage obtained from different samples. YIG ﬁlms with different thickness (listed in Table I)w e r e grown using RF sputtering on substrates of (111) gadoliniumgallium garnet at a pressure of 2.4 mTorr. Since the depositedYIG was nonmagnetic, the ﬁlm was annealed ex situ at 850 /C14C for 2 h. A layer of 4.2 60.1 nm Pt (or 5.0 60.1 nm Ta) was then deposited via DC magnetron sputtering. Both YIG and Pta)zf231@cam.ac.uk 0003-6951/2017/110(9)/092403/5/$30.00 Published by AIP Publishing. 110, 092403-1APPLIED PHYSICS LETTERS 110, 092403 (2017) thicknesses were measured by x-ray reﬂectivity. The sam- ples were patterned into 5 /C250lm2bars by using optical lithography and argon ion milling. After a second round of optical lithography, a layer of 5 nm Cr/50 nm Au was evap- orated as the contact electrodes. Each bar was mounted on a low-loss dielectric circuit board and connected to a micro- strip transmission line via wire bonding. By using a bias- tee, the DC voltage across the bar was measured at the same time as microwave power was applied. A magnetic ﬁeldHextwas swept in the ﬁlm plane at an angle hwith respect to the bar, as deﬁned in Fig. 1(a). The lineshape and the symmetry of the resonance in the DC voltage that we measure depend on both how the magne-tization precession is driven and how the DC voltage is gen-erated. As for the driving mechanism, when a microwavecurrent I 0ejxtﬂows though the HM, two types of torques are expected to act on YIG: a ﬁeld-like torque sOe¼M/C2hOe induced by the Oersted ﬁeld hOe//y, where yis the unit vector along the y axis, and an antidamping-like STTs ST¼M/C2hSTinduced by an effective ﬁeld hST//y/C2M.I f Mis in the x-y plane, both torques reach their maximum when Mis along the x-axis and become zero when Mis per- pendicular to the current direction. As for the generation of the longitudinal voltage, two mechanisms are mainly involved: spin rectiﬁcation and spin pumping. At the FMR condition, the oscillating magnetiza- tion leads to a time-dependent SMR in the HM at the samefrequency: R¼R 0þDRcos2h(t),8,25which rectiﬁes the microwave current, inducing a DC voltage along the bar. Wehave characterised the SMR for each YIG thickness byTABLE I. Summary of sample characteristics. The HM cap is Pt unless speciﬁed. tYIG (nm)SMR (10/C05)ceff/2p (GHz/T)Meff (kA/m)aeff (10/C03)K2? (kJ/m3) 14.8 4.8 60.6 30.0 60.1 69 63 1.41 60.02 12.6 61.3 22 5.9 61.0 29.9 60.1 81 64 1.15 60.03 11.2 61.1 36 2.9 60.3 29.8 60.1 82 63 1.00 60.02 11.1 61.0 49.5 5.1 60.1 29.8 60.1 82 63 0.97 60.02 11.1 61.0 62 3.1 60.5 29.9 60.1 77 64 0.93 60.04 11.6 61.2 80 (Ta) 1.2 60.5 28.1 60.9 90 67 1.36 60.31 10.2 60.9 (a) FIG. 1. (a) Scheme of CI-FMR in YIG/Pt and the experiment setup. (b) and (c) Results from a YIG(14.8)/Pt(4.2) sample: (b) spectra of CI-FMR at 4–8 GHz; (c) resonance frequency fas a function of the resonant ﬁeld l0Hres, ﬁtted with in-plane Kittel’s formula in dashed line. Inset: frequency dependence of the FMR linewidth l0DH. (d) Plot of the effective damping aeffas a function of tYIG-active . Red dashed line represents the ﬁtting result using EQ. (4).092403-2 Fang et al. Appl. Phys. Lett. 110, 092403 (2017)measuring the resistance of the Pt bar as an external in- plane magnetic ﬁeld is rotated. The results are reported in Table I. The spin-rectiﬁcation DC voltage consists of a symmetric ( Vsym-SR ) and an antisymmetric ( Vasy-SR ) Lorentzian components26,27 VDC¼Vsym-SRDH2 Hext/C0Hres ðÞ2þDH2 þVasy-SRDHH ext/C0Hres ðÞ Hext/C0Hres ðÞ2þDH2; (1) Vsym-SR¼I0DR 2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HresHresþMeff ðÞp DH2HresþMeff ðÞhSTsin 2h; (2) Vasy-SR¼I0DR 2HresþMeff ðÞ DH2HresþMeff ðÞhOesin 2hcosh; (3) where Hext,Hres, andDHare the external applied magnetic ﬁeld, the resonant ﬁeld, and the linewidth (half width at halfmaximum), respectively; M eff¼Ms–2K2?/l0Msis the effec- tive magnetization, where Ms,K2?, and l0refer to the satura- tion magnetization, the interface anisotropy energy density, and the vacuum permeability, respectively. Eqs. (2)and(3) show that Vsym-SR is induced by the out-of-plane ﬁeld hST while Vasy-SR is induced by the Oersted ﬁeld hOe. The other mechanism that could lead to a DC longitudi- nal voltage is spin pumping. The DC voltage from spin pumping, irrespective of the driving mechanism, is describedby a symmetric Lorentzian component V sym-SP , being inde- pendent of the phase between the microwave current in the HM and the precessing magnetization. Table IIsummarizes the lineshape and the angle dependence of the DC voltagefor each of the driving and detecting mechanisms discussedabove. Here, the effective damping factor a effis a function of tYIGand includes the spin pumping term aSP28–30 aefftYIGðÞ ¼a0þaSP¼a0þglB 4pMstYIGg"# eff; (4) where a0is the intrinsic Gilbert damping coefﬁcient of YIG without HM cap; gis the g-factor; lBis the Bohr magneton; andg"# effis the interface effective spin mixing conductance taking into account the backﬂow. Assuming that a0does not change with tYIG,g"# effcan be determined by measuring aeff for samples of different YIG thicknesses. Fig. 1(b) shows an example of CI-FMR signals mea- sured at f¼4–8 GHz and h¼45/C14for a sample YIG(14.8)/ Pt(4.2). The resonances are well described by a Lorentzian lineshape consisting of symmetric and antisymmetriccomponents. Fig. 1(c)plots the frequency dependence of the resonance ﬁeld and the linewidth (inset), which are well ﬁtted by the in-plane Kittel formula f¼(l0ceff/2p)[Hres(Hres þMeff)]1/2and the linear linewidth function l0DH¼l0DH0 þ2pfaeff/ceff, respectively. Here, ceffis the effective gyro- magnetic ratio and DH0is the frequency-independent inho- mogeneous linewidth broadening. From the ﬁtting, wecalculate the parameters, c eff,Meff, and aeff, as summarized in Table I, together with those obtained from each of the samples considered in this study. By using a vibrating sam- ple magnetometer (VSM), we measured the values of Msto be 180620 kA/m for all the samples at room temperature, andK2?can now be calculated (Table I). From VSM mea- surement, we also ﬁnd that there is a 5.1 60.1 nm-thick non- magnetic dead layer in our YIG ﬁlms. Therefore, we deﬁnethe thickness of active YIG layer as t YIG-active ¼tYIG– 5.1 in nm, and this value should be used in our calculation. As shown in Fig. 1(d), the value of aefffor different tYIG-active is well ﬁtted by Eq. (4), and we ﬁnd the values of a0andg"# effto be (8.1 60.1)/C210/C04and (7.1 60.2)/C21017m/C02for our YIG/Pt samples, respectively, in reasonably good agreementwith the literature. 31 To characterize the current-induced torque, we now ana- lyse the angle dependence of the symmetric and the antisym-metric components of the resonance signal. Fig. 2(a) shows the result obtained from the YIG(14.8)/Pt(4.2) sample. The V asyis ﬁtted well with a –sin2 hcoshfunction alone (red dash) in agreement with a resonance driven by the Oersted ﬁeld and detected by spin-rectiﬁcation (Table II). In contrast, Vsymis ﬁtted by the sum of a sin2 hcoshterm (orange dash) and a sin h (green dash) term noted as Vsym-sin2 hcoshandVsym-sin h,r e s p e c - tively. All components are linear in power (Fig. 2(b)), indicat- ing that the small-angle precession approximation is satisﬁed.By carrying a quantitative analysis of V asybased on Eq. (3), we extract the value of the effective ﬁeld that generates the torque for each sample (Fig. 2(c)), normalized to a unit cur- rent density of jc¼1010A/m2. This can be compared with the value of the Oersted ﬁeld calculated from Ampere’s law as l0hOe¼l0jctPt/2/C2526lT( r e dd a s hi nF i g . 2(c)), where tPtis the thickness of Pt. The good agreement between the two val- ues conﬁrms that the ﬁeld-like torque is mainly attributed to the Oersted ﬁeld. The analysis of the sin2 hcoshterm (orange dash) in symmetric component is richer since it contains three different terms as shown in Table II. Despite this, we can still identify the main driving mechanism by comparing Vsym-sin2 hcoshandVasy. Fig. 3plots the ratio Vsym-sin2 hcosh/Vasy in each Pt/YIG sample with respect to 1/ aeff, showing a lin- ear dependence. Referring to Table II, only jVOe-SP /VOe-SRj /1/aeff, while the ratios between other terms have a more complicated relation to aeffasaeffalso depends on tYIG-active (Eq. (4)). From this, we conclude that Vsym-sin2 hcoshcan be mainly attributed to the spin pumping driven by the Oersted ﬁeld. In addition to this, we carried out an experiment inwhich we replaced the Pt with Ta. Fig. 2(d) shows the angle dependence for a YIG(80)/Ta(5.0) sample. While V sym changes its sign compared with the YIG/Pt case, the sign of Vasystays the same. The change in the sign of Vsymis explained with the opposite sign of the spin-pumping, which results from the opposite value of the spin-Hall angle of TaTABLE II. Summary of resonance DC signal components involved, with their Lorentzian lineshape and dependence on h, spin Hall angle #SH,tYIG, andaeff.Ciare the positive coefﬁcients independent from the parameters listed above (see supplementary material for the deduction). Driving Detecting lineshape Dependence on h,#SH,tYIG, and aeff hST SR Symmetric /C0CST-SR½#3 SH=ðaefftYIGÞ/C138sin 2hcosh SP Symmetric CST-SP½#3 SH=ðaefftYIGÞ2/C138sin 2hcosh hOe SR Anti-symmetric /C0COe-SRð#2 SH=aeffÞsin 2hcosh SP Symmetric COe-SPð#SH=a2 effÞsin 2hcosh092403-3 Fang et al. Appl. Phys. Lett. 110, 092403 (2017)compared with Pt.15,16,32The fact that the sign of Vasydoes not depend on the sign of the spin-Hall angle of the metallayer further conﬁrms that the Oersted ﬁeld dominates overthe ﬁeld-like STT in driving the magnetization dynamics inour samples. 33Since Vsymis dominated by the spin pumping signal driven by the Oersted ﬁeld and we have already deter- mined the value of g"# eff, following the method in Ref. 36by assuming the spin diffusion length in Pt to be 1.5 nm37and the precession ellipticity factor to be 1, we can estimate thespin Hall angle of Pt to be 1.0 60.2%, agreeing well with the previous work. 7,28Our results from YIG/Pt are different from some ferromagnetic metals/HM bilayers, such as Co/Pt 27or Py/Pt,15,34,35where the rectiﬁcation signal driven by STT dominates over the spin pumping signal, and Vsymand Vasyare comparable with each other with the same sign in the CI-FMR measurement, which might be caused by thelow interface spin mixing conductance in our samples. We also brieﬂy comment on an additional sin h(green dash) term that appears in the ﬁtting of V sym. We note that when measuring other material systems in our setup, e.g.,Co/Pt 27or Py/Pt,38this sin hcomponent is absent, indicatingFIG. 3. Plot of the ratio Vsym-sin2 hcosh/Vasyas a function of 1/ aeff, measured from the YIG/Pt samples at 8 GHz. The dashed line represents the linear ﬁtting.FIG. 2. (a) Angle dependence of the symmetric part Vsym(blue) and anti-symmetric part Vasy(red) from YIG(14.8)/Pt(4.2) at 8 GHz. Dashed lines are ﬁtting results, where Vasyis ﬁtted by sin2 hcosh, while Vsymneeds a sin hterm (green) in addition to the sin2 hcoshterm (orange). (b) Power dependence of the three resonance components at 8 GHz. (c) Oersted ﬁeld l0hycalculated from Ampere’s law (red dashed line) and Vasyusing Eq. (3)(black dot) for each YIG/Pt sam- ple, normalized to jc¼1010A/m. (d) Angle dependence measurement from a YIG(80)/Ta(5.0) sample.092403-4 Fang et al. Appl. Phys. Lett. 110, 092403 (2017)its origin in the sample. This term was previously attributed to an on-resonance contribution from the longitudinal spin Seebeck effect.24However, neither STT nor Oersted ﬁeld can drive FMR at h¼90/C14, where this term is maximum. In conclusion, we have used CI-FMR to investigate the charge-current-induced torque on YIG magnetization in a series of YIG/HM samples with different YIG thickness between 14.8 nm and 80 nm. Our measurements show that the Oersted ﬁeld gives the dominant contribution to driving the magnetisation precession and should therefore be taken into account when carrying out CI-FMR studies in YIG/HM systems. See supplementary material for the deduction of the dependence of the d.c. voltage on the parameters h,#SH, tYIG, and aeffin Table II. We thank L. Abdurakhimov for the experiment help and T. Jungwirth, M. Jungﬂeisch and A. Hoffmann for the valuable discussions. Z.F. thanks the Cambridge Trusts forthe ﬁnancial support. B.J.H. thanks D. 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1.5129016.pdf	AIP Advances 10, 025120 (2020); https://doi.org/10.1063/1.5129016 10, 025120 © 2020 Author(s).Effect of asymmetric Pt thickness on the inverse spin Hall voltage in Pt/Co/Pt trilayers Cite as: AIP Advances 10, 025120 (2020); https://doi.org/10.1063/1.5129016 Submitted: 26 September 2019 . Accepted: 17 January 2020 . Published Online: 12 February 2020 Tzu-Hsiang Lo , Yi-Chien Weng , Chi-Feng Pai , and Jauyn Grace Lin ARTICLES YOU MAY BE INTERESTED IN Current-induced spin–orbit torque efficiencies in W/Pt/Co/Pt heterostructures Applied Physics Letters 116, 072405 (2020); https://doi.org/10.1063/1.5133792 Spin transfer torque devices utilizing the giant spin Hall effect of tungsten Applied Physics Letters 101, 122404 (2012); https://doi.org/10.1063/1.4753947 Spin-torque ferromagnetic resonance measurements utilizing spin Hall magnetoresistance in W/Co 40Fe40B20/MgO structures Applied Physics Letters 109, 202404 (2016); https://doi.org/10.1063/1.4967843AIP Advances ARTICLE scitation.org/journal/adv Effect of asymmetric Pt thickness on the inverse spin Hall voltage in Pt/Co/Pt trilayers Cite as: AIP Advances 10, 025120 (2020); doi: 10.1063/1.5129016 Submitted: 26 September 2019 •Accepted: 17 January 2020 • Published Online: 12 February 2020 Tzu-Hsiang Lo,1Yi-Chien Weng,2 Chi-Feng Pai,1,3 and Jauyn Grace Lin2,3,a) AFFILIATIONS 1Department of Materials Science and Engineering, National Taiwan University, Taipei 10617, Taiwan 2Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan 3Center of Atomic Initiative for New Materials, National Taiwan University, Taipei 10617, Taiwan a)Author to whom correspondence should be addressed: jglin@ntu.edu.tw ABSTRACT Ferromagnetic resonance (FMR) is an effective technique for probing the magnetization dynamics of magnetic thin films. In particular, bilayer systems composed of a paramagnetic layer and a ferromagnetic layer are commonly used for FMR-driven spin pumping experi- ments. Spin pump-and-probe models have been adopted to obtain the spin Hall angle ( θSHE) and spin diffusion length ( λN) for various single layer and bilayer systems. Trilayer systems, however, have rarely been studied with the same model. In this work, we study the structural asymmetry effect on Pt/Co/Pt trilayers and find that the different thicknesses of Pt on two sides of Co may change the spin current sign. Furthermore, we propose a method that allows analysis of Pt/Co/Pt trilayers using the spin pump-and-probe model. The obtained values of θPtandλPtin the Pt/Co/Pt system are 0.116 nm and 1.15 nm, respectively, which are consistent with the values obtained from other Pt-based bilayer systems. ©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5129016 .,s I. INTRODUCTION In 2004, Kato et al.1first observed the spin Hall effect (SHE) experimentally in semiconductors through optical meth- ods. Since then, the SHE has attracted considerable interest and led to several experimental studies, including the nonlocal detec- tion of the inverse spin Hall effect (ISHE),2,3spin Seebeck effect,4,5 spin transfer torque,6spin-orbit torque,7and spin pumping.8–11 The spin–orbit interaction plays an indispensable role in these effects. Among these studies, the spin pump-and-probe measure- ment in heavy transition metals involves the ISHE model due to the strong spin–orbit interaction therein, which converts spin cur- rent into charge current. In the ferromagnetic layer, spin cur- rent is generated via ferromagnetic resonance (FMR) and injected into the adjacent heavy metal layer, resulting in an electrical sig- nal which provides a quick way to acquire information of crit- ical parameters such as spin diffusion length, spin mixing con- ductance, and spin Hall angle. In both the SHE and ISHE, the efficiency of spin–charge conversion can be expressed as the spinHall angle θSHE, which is the ratio of spin current to charge current. In general, the spin pump-and-probe measurement requires a bilayer structure composed of one magnetic layer and one non- magnetic layer. Permalloy (Py) is usually used as the magnetic layer, and Pt is used as the non-magnetic heavy transition metal layer. The charge current detected in the Pt layer is due to the ISHE of the spin current generated in the Py layer. Recently, some groups have proposed a self-induced spin pumping effect in a single mag- netic layer.12–14The spin current flowing from the magnetic layer to the substrate may explain the observation of self-induced spin Hall voltage. To understand the “self-induced spin pumping” on the single layer system, a systematic study on the thickness depen- dent ISHE voltage was recently carried out on Co/Si and attributed to the combination of spin-orbit coupling and long spin diffusion length of Si.15In this study, we design a Pt/Co/Pt trilayer sys- tem with different Pt thicknesses on two sides of Co and com- pare the results with the Co/Pt bilayer system to investigate the asymmetry effect on the ISHE. AIP Advances 10, 025120 (2020); doi: 10.1063/1.5129016 10, 025120-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 1 . Schematic illustration of FMR and ISHE measurements. II. SAMPLE FABRICATION AND THE EXPERIMENT SETUP Magnetic heterostructures Pt(t)/Co(10)/Pt(2) were deposited onto Si(100) substrates with a dimension of 1.5 mm ×3 mm, where t = 0, 1, 2, 3, 4, 6, 8 and 10. The values indicated in parentheses are expressed as nanometers. Trilayer stacks were deposited using an ultra-high vacuum magnetron sputtering chamber with a base pressure of 3 ×10−8Torr and an Ar working pressure of 3 mTorr. Room temperature FMR spectra were obtained from a Bruker EMX system with the samples placed at the center of a TE 102microwave (MW) cavity, in which the magnetic field of the MW was max- imum and the electric field was minimum. The frequency of the MW was 9.78 GHz, and the power of the MW ( PMW) varied from 10 mW to 60 mW. Angle dependent FMR measurement was then performed. The DC voltage was detected with a Keithley 2182A nanometer. The schematic illustration of FMR and spin pump-and- probe measurements is shown in Fig. 1, with θHdefined as the angle between the applied field (H) and the sample plane, and w and ldefined as the respective width and length of the sample, respectively. III. RESULTS AND DISCUSSION The FMR spectra of Pt(t)/Co/Pt with PMW= 40 mW are shown in Fig. 2(a). In order to obtain the resonance field ( HR) and linewidth (ΔH), the differential Lorenz function is used to fit the FMR spectra as follows:9,15 dI dH=−16A⋅ΔH⋅(H−HR) π[4(H−HR)2+(ΔH)2]2, (1) where Ais the area of curve I(H) . The fitting results of HRand ΔHare shown in Table I. The significant linewidth change in the FIG. 2 . FMR spectra and DC voltages of Pt(t)/Co/Pt: (a) FMR spectra of Pt(t)/Co/Pt where the resonance field increases with Pt thickness and (b) DC voltages of Pt(t)/Co/Pt where the voltage decreases with increasing Pt thickness from t = 0 to t = 6; then, the voltage is saturated at t = 6 to t = 10. trilayers (∼17 Oe in average) compared with the bilayer is mainly due to the spin injection at the additional Pt/Co interface.8However, the linewidth of Pt(t)/Co/Pt oscillates with changing t, which cannot be simply explained by the modified bilayer model. This oscillation behavior may be due to the mirror reflection of the top Pt(2 nm), which requires further investigation. It is noted that H rincreases with increasing t, suggesting the change in magnetic anisotropy. In principle, effective magnetization increases with increasing mag- netic anisotropy; however, in most spin pumping models, the cal- culation of spin mixing conductance treats the anisotropy as the angle of magnetization with respect to the sample plane (later, this is denoted as θM]. Furthermore, we measured the magnetiza- tion of Co(10) and Co(10)/Pt(10) in our samples and found only 2% enhancement in magnetization with Pt capping. Thus, we do not consider the magnetic proximity effect of Pt in our analysis. TABLE I . Resonance field and linewidth of Pt(t)/Co/Pt. Materials Resonance field (Oe) Linewidth (Oe) Pt(0)/Co(10)/Pt(2) 663 185 Pt(1)/Co(10)/Pt(2) 703 204 Pt(2)/Co(10)/Pt(2) 794 191 Pt(3)/Co(10)/Pt(2) 742 213 Pt(4)/Co(10)/Pt(2) 780 197 Pt(6)/Co(10)/Pt(2) 795 197 Pt(8)/Co(10)/Pt(2) 810 202 Pt(10)/Co(10)/Pt(2) 812 205 AIP Advances 10, 025120 (2020); doi: 10.1063/1.5129016 10, 025120-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv The interface roughness of Co/Pt may induce magnetic inhomo- geneity, which is reflected in the linewidth modulation. In this work, we consider the interfacial roughness is the same for all trilayer sam- ples and included the change in linewidth in the calculation of spin mixing conductance. The DC voltage of the Pt(t)/Co/Pt samples induced by spin pumping is shown in Fig. 2(b). We define the direction of the spin current from the Co layer to the top Pt layer as positive sinceθSHE of Pt is positive. For the case of t >0, the spin cur- rent jswould also flow from the Co layer toward the bottom Pt layer during magnetization-precession, i.e., the precession generates two spin currents of opposite directions according to the following equation:10 jc=θSHE(js×σ), (2) where the spin polarization σis parallel to the external magnetic field. Therefore, for the case of Co/Pt, jcshould be positive. Based on this spin pump-and-probe model, ideally, the Pt(2)/Co/Pt(2) sample should have no signal at all since the spin currents on the two Pt layers should be the same in magnitude but opposite in direction. However, our experimental result indicates that the voltage of Pt(2)/Co/Pt(2) still shows a small positive voltage, as seen in Fig. 2(b), which could be due to the differences in voltage at the top and bottom Pt/Co interfaces. Conversely, Pt(3)/Co/Pt(2) demonstrates a negative voltage, indicating that down-direction spin current dominates the system. FIG. 3 .α,γ, and the g-factor of Pt(t)/Co/Pt: (a) αas a function of Pt thickness; αis constant at different Pt thicknesses as t = 2, 4, 6, 8, and 10 and (b) γand the g-factor as a function of Pt thickness; γand the g-factor show the same trend; they are constant at different Pt thicknesses at t = 2, 4, 6, 8, and 10.For the trilayer structure, the bottom Pt layer leads to an additional spin current flowing from the Co downward to the Pt layer. The spin current depriving the magnetization-precession of the Co, therefore, enhances the linewidth which is proportional to the Gilbert damping constant α. This leads to the relaxation of magnetization-precession, and we can obtain αby measuring the res- onance field and linewidth at different θHfrom 0○to 85○14while simultaneously obtaining the g-factor and the gyromagnetic ratio γ. Figures 3(a) and 3(b) show the thickness dependence of αand the thickness dependence of γand the g-factor, respectively. The verifi- cation of the mechanism of the ISHE by measuring the DC voltage at 0○and 180○is shown in Fig. 4(a). The opposite sign of the voltage is due to the opposite direction of the spin current. The obtained DC voltage, as in Fig. 2(b), is composed of a symmetric part VISHE and an antisymmetric part VAHE shown as follows:16 Voltage=VISHEΔH2 (H−HR)2+ΔH2+VAHE−2ΔH(H−HR) (H−HR)2+ΔH2. (3) Figure 4(b) shows the decomposition of the DC voltage into VISHE (green line) and VAHE(pink line). Figure 5 shows the value of VISHE as a function of PMW. The linear dependence corresponds to the spin pump-and-probe theory.8 FIG. 4 . (a)VISHEof Pt(4)/Co/Pt: the opposite voltage at 0○and 180○demonstrates the net opposite spin current flowing from Co to Pt and (b) a decomposition of VISHEandVAHEin Pt(2)/Co/Pt: the green curve is the contribution of VISHE, and the pink curve is the contribution of VAHE. AIP Advances 10, 025120 (2020); doi: 10.1063/1.5129016 10, 025120-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 5 .VISHEof Pt(t)/Co/Pt as a function of PMW; linear dependence indicates that the inverse spin Hall effect is the main contributor of VISHE. VISHE is converted to Icby Ohm’s law and then normalized to the width of the sample.Ic wvstPtis plotted as shown in Fig. 6. For the convenience of data analysis, we treat the trilayer system as a bilayer system, by considering the top Co/Pt(2 nm) as the spin pumping layer and the bottom Pt(t) as the spin current detection layer. Based on the spin pump-and-probe model for the bilayer system,10,17,18 Ic=θPtw(2e ̵h)λPttanh(tPt 2λPt)j0 s, (4) whereθPtis the spin Hall angle of Pt, w = 1.5 mm, and λPtis the spin diffusion length of Pt. j0 sis the spin current density, which can be expressed as j0 s=g↑↓ effγ2PMW̵h[4πMsγsin2θM+√ (4πMs)2γ2+ 4ω2] 8πα2√ (4πMs)2γ2+ 4ω2, (5) where Msis the saturation magnetization of Co (1414 Oe/cc obtained from VSM), ωis the angular frequency of MW, and g↑↓ eff is the spin mixing conductance,8,19 FIG. 6 . Charge current as a function of Pt thickness at PMW= 40 mW;λPtandθPt can be obtained using the red fitting curve.TABLE II . Summary of λPt,θPt, and g↑↓ eff. λPt θPt g↑↓ eff Materials (nm) (%) (1019m−2)Backflow Reference Py/Pt 7.7 ±0.7 1.3 ±0.1 3.02 Y 10 3.7±0.2 4.0 ±1.0 2.4 Y 22 1.2 8.6 ±0.5 3.0 N 20 8.3±0.9 1.2 ±0.2 2.5 ±0.2 N 23 YIG/Pt 1.5 11 0.97 Y 21 7.3 10 ±1 0.69 ±0.06 N 24 LSMO/Pt 5.9 ±0.5 1.2 ±0.1 1.8 ±0.4 N 9 Pt/Co/Pt 1.15 11.6 2.85 N This work g↑↓ eff=4πMsdF γ̵h(ΔHF/N−ΔHF), (6) where ΔHF/Nis the linewidth of Co/Pt, and ΔHFis the linewidth of the single Co layer in FMR spectra. For the convenience of analysis, we propose a modified bilayer model. We treat the Pt(t)/Co(10)/Pt(2) as a bilayer system, where the Pt(t) is considered the spin-detection layer and the Co(10)/Pt(2) is considered the effec- tive spin injection layer. With this assumption, we only take into account the interfacial effect between the bottom Pt(t) layer and the Co(10). By fitting the data of IcvstPtwith Eq. (4), which can be seen as the solid line in Fig. 6, we obtain g↑↓ eff=2.85×1019m−2 and j0 s=2.04×10−10A m2.λPtandθPtare estimated to be 1.15 nm and 0.116, respectively, which are comparable to previously reported values.20,21 We summarize the results of λPt,θPt, and g↑↓ efffrom various research groups and have listed them in Table II.9,10,20–24It is inter- esting to note that a wide range of θPthas been obtained by differ- ent groups using different measurement techniques. However, the productλPtθPtis almost constant.25,26Our result, the product of λPtθPt∼0.13, is equivalent to that obtained by Wang et al.26There- fore, we suggest that the spin pump-and-probe model is also feasible in fitting the trilayer system via a bilayer approach. IV. CONCLUSIONS In Pt(t)/Co/Pt samples, FMR and spin pumping measurements were performed to analyze the effect of the bottom Pt layer on VISHE. The linewidth, α, the g-factor, and γwere obtained. VISHE of a series of Pt(t)/Co/Pt samples demonstrates a clear asymme- try effect by changing the thickness of the bottom Pt layer. The Pt(3)/Co/Pt(2) has a negative voltage in contrast with the positive voltage in Pt(2)/Co/Pt(2). This result infers that spin currents with opposite directions are driven by the precession of Co magnetiza- tion, resulting in an imbalance of spin currents in asymmetric struc- tures. Based on the spin pump-and-probe model, λPtandθPtare 1.15 nm and 0.116, respectively. The comparable values of λPtand θPtin bilayer and trilayer systems suggest that spin pump-and-probe models can also apply to a trilayer system. However, the oscillation AIP Advances 10, 025120 (2020); doi: 10.1063/1.5129016 10, 025120-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv behavior in the FMR linewidth of Pt(t)/Co/Pt with changing t cannot be simply explained by the modified bilayer model, which requires further investigation. ACKNOWLEDGMENTS This work was financially supported in part by the Min- istry of Science and Technology of the Republic of China and National Taiwan University under the projects of Grant Nos. MOST 108-2112-M-002-022 and NTU-107L900803, respectively. REFERENCES 1Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004). 2S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006). 3T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007). 4C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nat. Mater. 9, 898 (2010). 5K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). 6C. F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 101, 122404 (2012). 7K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Blügel, S. Auffret, O. Boulle, G. Gaudin, and P. Gambardella, Nat. 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1.3675614.pdf	Tuning of magnetization relaxation in ferromagnetic thin films through seed layers Lei Lu, Jared Young, Mingzhong Wu, Christoph Mathieu, Matthew Hadley, Pavol Krivosik, and Nan Mo Citation: Applied Physics Letters 100, 022403 (2012); doi: 10.1063/1.3675614 View online: http://dx.doi.org/10.1063/1.3675614 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/100/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Influence of the interface on the magnetic properties of ferromagnetic ultrathin films with various adjacent copper thicknesses J. Appl. Phys. 115, 17C108 (2014); 10.1063/1.4861555 Ferromagnetic resonance and damping properties of CoFeB thin films as free layers in MgO-based magnetic tunnel junctions J. Appl. Phys. 110, 033910 (2011); 10.1063/1.3615961 Magnetization relaxation and structure of CoFeGe alloys Appl. Phys. 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Downloaded to IP: 129.21.35.191 On: Sun, 21 Dec 2014 09:34:10Tuning of magnetization relaxation in ferromagnetic thin films through seed layers Lei Lu,1Jared Y oung,1Mingzhong Wu,1,a)Christoph Mathieu,2Matthew Hadley,2 Pavol Krivosik,3and Nan Mo4 1Department of Physics, Colorado State University, Fort Collins, Colorado 80523, USA 2Seagate Technology, Recording Head Operations, Bloomington, Minnesota 55435, USA 3Department of Physics, University of Colorado at Colorado Springs, Colorado Springs, Colorado 80933, USA 4Southwest Institute of Applied Magnetics, Mianyang, Sichuan 621000, China (Received 11 November 2011; accepted 16 December 2011; published online 9 January 2012) Tuning of the magnetization relaxation in Fe 65Co35thin ﬁlms via seed layers was demonstrated. Through the use of different types of seed layers, one can tune substantially both the magnitude and frequency dependence of the relaxation rate gof the ﬁlm. This tuning relies on the change of the ﬁlm grain properties with the seed layer and the correlation between grain properties andtwo-magnon scattering processes. In spite of a signiﬁcant change of gwith the seed layer, the ﬁlm static magnetic properties remain relatively constant. VC2012 American Institute of Physics . [doi:10.1063/1.3675614 ] The magnetization in a magnetic material can precess around the direction of a static magnetic ﬁeld, and such pre-cession typically has a frequency in the microwave range. One can excite and maintain a uniform magnetization pre- cession with an external microwave magnetic ﬁeld. Once themicrowave ﬁeld is turned off, however, the magnetization will tend to relax back to the static ﬁeld direction. Such mag- netization relaxation can be realized through energy redis-tribution within the magnetic subsystem, energy transfer out of the magnetic subsystem to non-magnetic subsystems such as phonons and electrons, or energy transfer out of the mate-rial to external systems. 1,2 The tailoring of the magnetization relaxation rate gin ferromagnetic thin ﬁlms is of great fundamental and practicalsigniﬁcance. In practical terms, for example, the relaxation rates in thin ﬁlm materials in magnetic recording heads and media set a natural limit to the data recording rate; 3the band- width, insertion loss, and response time of a magnetic thin ﬁlm-based microwave device are critically associated with g in the ﬁlm.4 Previous work has demonstrated three approaches for the tuning of gin ferromagnetic thin ﬁlms: (1) control of ﬁlm thickness, (2) addition of non-magnetic elements, and (3)doping of rare earth elements. Regarding (1), the tuning of g relies on the sensitivity of two-magnon scattering (TMS) and eddy current effects on the ﬁlm thickness. 5,6Regarding (2), one makes use of the addition of non-magnetic elements to control the microstructural properties of the ﬁlms and, thereby, control the TMS processes.7Regarding (3), the relaxation rates are enhanced through the slow relaxing im- purity mechanism.8These approaches, however, are not practically desirable. Approach (1) sets a limit to ﬁlm thick-ness for a speciﬁc relaxation rate. For (2) and (3), the change ingis always accompanied by signiﬁcant changes in other ﬁlm properties, such as saturation induction 4 pM s. It shouldbe noted that the TMS processes in thin ﬁlms are critically associated with ﬁlm microstructures, such as defects, grainsize, and surface roughness. 9,10The processes manifest themselves in a broadening in the ferromagnetic resonance (FMR) linewidth and nonlinear behavior in the linewidth vs.frequency response, rather than linear behavior expected by the Gilbert model. It is also possible that the processes give rise to a saturation response or even a decrease in the line-width as one moves to higher frequencies. 11 This letter reports on the tuning of gin ferromagnetic thin ﬁlms through the use of different types of seed layers.Speciﬁcally, the letter presents experimental and numerical results that demonstrate the tuning of g, both the magnitude and frequency dependence, in 100-nm-thick Fe 65Co35ﬁlms through seed layers. It is found that the use of different types of seed layers leads to ﬁlms with different grain sizes and grain-size distributions. The difference in the ﬁlm grainproperties results in a difference in the levels of both grain- grain two-magnon scattering (GG-TMS) 11and grain- boundary two-magnon scattering (GB-TMS)12processes. As a result, the ﬁlms grown on different seed layers show differ- ent relaxation properties, which manifest themselves as dif- ferent FMR linewidth properties. It is also found that theﬁlms on different types of seed layers show similar static magnetic properties, although they differ signiﬁcantly in g. These results clearly demonstrate a simple and practicalapproach for the control of relaxation properties in ferromag- netic thin ﬁlms. The Fe 65Co35ﬁlms were deposited at room temperature by dc magnetron sputtering. The substrates were (100) Si wafers with a 300-nm-thick SiO 2capping layer. Prior to the growth of each ﬁlm, a thin seed layer was deposited. Duringﬁlm deposition, a ﬁeld of 80 Oe was applied to induce an in- plane uniaxial anisotropy in the ﬁlm. The nominal thick- nesses of the Fe 65Co35ﬁlms are 100 nm. The grain size dand grain-size distribution rof each ﬁlm were determined by transmission electron microscopy. The static magnetic prop- erties were measured by vibrating sample magnetometry.a)Author to whom correspondence should be addressed. Electronic mail: mwu@lamar.colostate.edu. 0003-6951/2012/100(2)/022403/3/$30.00 VC2012 American Institute of Physics 100, 022403-1APPLIED PHYSICS LETTERS 100, 022403 (2012) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.21.35.191 On: Sun, 21 Dec 2014 09:34:10The FMR measurements were carried out by shorted rectan- gular waveguides over a frequency range of 8.2-18.0 GHz. The peak-to-peak ﬁeld separation in each power absorptionderivative proﬁle was taken as the FMR linewidth DH. Table Iprovides the details of the samples. Column 2 gives the material, nominal structure, and nominal thickness(nm) of each seed layer. Column 3 gives the values of d (left) and r/d(right). These values vary signiﬁcantly with the seed layer, but are relatively independent of the seed layerthickness (except for sample 9). Column 4 lists the 4 pM sval- ues, which are all close to each other. The average value is 23.1 kG, which matches that reported in Ref. 13and was used in the numerical analyses. The small variation in 4 pMs is probably due to the deviation of the ﬁlm thickness from the nominal value, which is in the 5% range. Column 5 givesthe gyromagnetic ratio jcjvalues, which are close to each other. Column 6 gives the anisotropy ﬁeld H uvalues, all of which are smaller than 40 Oe. The jcjandHuvalues were obtained through ﬁtting the measured FMR ﬁeld vs. fre- quency responses with the Kittel equation. Figure 1shows the linewidth DHvs. frequency responses. Graph (a) shows the data for ﬁlms deposited on different seed layers, as indicated. Graph (b) shows the data for ﬁlms deposited on Ru seed layers of different thick-nesses, as indicated. Four important results are evident in Fig.1. (1) By using different seed layers, one can tune the magnitude of DHover a rather wide range of 80-490 Oe. (2) Films on different seed layers also show signiﬁcantly different DH-frequency responses. (3) None of those responses shows linear behavior with a zero DHintercept at zero frequency, as expected by the Gilbert model. (4) For a given type of seed layer, a change in the seed layer thick- ness leads to a notable change in the magnitude of DH, but produces negligible effects on the frequency dependence of DH. These results clearly indicate the feasibility of tuning the FMR linewidth properties of the Fe 65Co35ﬁlms via the use of different seed layers. They, however, provide no details on the effects of the seed layer on physicalrelaxation processes in the ﬁlms. To understand such effects, numerical analyses were carried out as explained below. The experimental linewidth DHusually takes the form DH¼DH rþDH0; (1)where DHroriginates from the magnetization relaxation and DH0takes into account the sample inhomogeneity-caused FMR line broadening. The term DHrcan be related to gas DHr¼2g @xFMR =@H; (2) where xFMRis the FMR frequency. In this work, one consid- ers three contributions to DHr(andg): (1) Gilbert damping,1,2 (2) GG-TMS relaxation,11and (3) GB-TMS relaxation.12The Gilbert damping results mainly from magnon-electron scatter- ing, and contributions from magnon-phonon scattering, eddy current, and spin pumping effects are relatively weak.2,11,12 The term DH0in Eq. (1)is not a loss. Rather, it arises from the simple superposition of several local FMR proﬁles for dif- ferent regions of the ﬁlm. If the inhomogeneity is strong andDH 0is comparable to DHr,E q . (1)is inappropriate and the combined linewidth DHshould take the form11,14 DH¼DH2 rþ1:97DHrDH0þ2:16DH2 0 DHrþ2:16DH0: (3) The discussions below were based on Eq. (3). Figures 2and3show the results from the ﬁtting of DH data with four linewidth contributions described above. The ﬁtting used a Gilbert damping constant a¼0.003,15an exchange constant A¼1.25/C210/C06erg/cm, which was 30% lower than that reported in Ref. 13, and a magneto-crystallineTABLE I. Summary of sample properties. # Seed layer d(nm) and r/d4pMs(kG) jcj(MHz/Oe) Hu(Oe) 1 Cr/bcc/2.5 40.0/0.355 23.5 2.93 37.8 2 Ta/bcc/2.5 26.0/0.354 23.1 2.93 28.03 Pt/fcc/2.5 14.7/0.299 22.4 2.90 12.84 Cu/fcc/2.5 16.8/0.506 23.4 2.92 17.95 Ti/hcp/2.5 20.6/0.291 22.8 2.90 9.86 Ru/hcp/2.5 12.5/0.280 23.7 2.93 23.47 Ru/hcp/0.5 12.6/0.270 23.1 2.93 24.28 Ru/hcp/2.0 12.5/0.280 23.1 2.93 20.59 Ru/hcp/5.0 10.5/0.429 23.4 2.93 29.010 Ru/hcp/10 13.3/0.293 22.7 2.93 25.3 FIG. 1. (Color online) FMR linewidth vs. frequency responses for ﬁlms grown on (a) different types of seed layers and (b) Ru seed layers of differ-ent thicknesses. FIG. 2. (Color online) Theoretical ﬁts of FMR linewidth vs. frequencyresponses and relaxation rates for ﬁlms grown on different seed layers. (a) Film on Ru seed layer. (b) Film on Cr seed layer. (c) and (d) Six ﬁlms.022403-2 Lu et al. Appl. Phys. Lett. 100, 022403 (2012) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.21.35.191 On: Sun, 21 Dec 2014 09:34:10anisotropy ﬁeld Ha¼960 Oe, which was close to that reported in Ref. 16. The values of DH0,d,r/d, and grain boundary surface anisotropy constant Ksused in the ﬁtting are given in Table II. The use of higher avalues resulted in poor ﬁts, which are not shown in the ﬁgures. In Fig. 2, graphs (a) and (b) show the total ﬁts of the DH data and the four components for the ﬁlms on Ru (2.5 nm) and Cr seed layers, respectively. Graph (c) shows the theoreti-cal ﬁts (curves) to all the DHdata shown in Fig. 1(a).G r a p h (d) shows the gvalues obtained with a two-step procedure: (1) calculation of DH rusing Eq. (3)with the experimental DH values and the DH0values from the ﬁtting and (2) calculation ofgusing Eq. (2). The data in Fig. 2indicate three important results. First, through the use of different seed layers, one cantune gover a rather wide range from 0.5 GHz to 4 GHz as well as its frequency dependence, as shown in (d). Second, this tuning relies on the changes of the GG-TMS and GB-TMS processes with the seed layer, as shown representatively in (a) and (b). Third, the dominant contributions to the relaxa- tion are from the TMS processes, whereas the contributionfrom Gilbert damping is relatively small. In Fig. 3, graphs (a) and (b) show the total ﬁts of the DH data and the four components for the ﬁlms on 0.5 nm and10 nm thick Ru seed layers, respectively. Graph (c) shows the ﬁts to all the DHdata shown in Fig. 1(b). Graph (d) shows the corresponding gvalues obtained with the proce- dure described above. Two important results are evident in Fig.3. First, a change in the Ru seed layer thickness results in negligible effects on the TMS processes, as shown in (a)and (b), and thereby gives rise to insigniﬁcant changes in the relaxation properties, as shown in (d). Second, the seed layer thickness change results in a notable change in DH 0and a corresponding change in DH. There are two important points to be emphasized. (1) Thedvalues used in the ﬁtting were all close to the experi- mental values. Most of the ﬁtting r/dvalues were smaller than the experimental values, and this is probably due to the relatively small numbers of grains (about 30) used in the sta-tistical analyses of the grain properties. Nevertheless, the rel- ative differences in r/dbetween the samples are consistent with those from the measurements. These facts strongly sup-port the interpretation of the mechanism of the presentedrelaxation tuning. (2) The tuning relies on the fact that the GG-TMS and GB-TMS processes are the dominant relaxa- tion processes. For ﬁlms much thinner than the ﬁlms in thiswork, spin pumping is also an important damping source so that one can vary both the material and thickness of the seed layer to tailor the ﬁlm relaxation. 2 In summary, this letter reported the effects of seed layers on the relaxation and FMR responses of 100-nm-thick Fe65Co35ﬁlms. It was found that the use of different types of seed layers results in ﬁlms with different relaxation rates, both in magnitude and frequency dependence, but similar static magnetic properties. No signiﬁcant effects on therelaxation rate were observed when one varied the thickness of the Ru seed layer. These results can be interpreted in terms of the effects of the seed layers on the ﬁlm grain prop-erties and the correlation between the grain properties and the GG-TMS and GB-TMS processes. This work was supported in part by the U. S. National Science Foundation, the U. S. National Institute of Standardsand Technology, and Seagate Technology. 1M. Sparks, Ferromagnetic-relaxation Theory (McGraw-Hill, New York, 1964). 2B. Heinrich and J. A. C. Bland, Ultrathin Magnetic Structures: Fundamen- tals of Nanomagnetism (Springer, Berlin, 2005). 3J. G. Zhu and D. Z. Bai, J. Appl. Phys. 93, 6447 (2003). 4R. E. Camley, Z. Celinski, T. Fal, A. V. Glushchenko, A. J. Hutchison, Y. Khivintsev, B. Kuanr, I. R. Harward, V. Veerakumar, and V. V. Zagorodnii, J. Magn. Magn. Mater. 321, 2048 (2009). 5B. K. Kuanr, R. E. Camley, and Z. Celinski, J. Appl. Phys. 75, 6610 (2004). 6C. Scheck, L. Cheng, and W. E. Bailey, Appl. Phys. Lett. 88, 252510 (2006). 7J. Lou, R. E. Insignares, Z. Cai, K. S. Ziemer, M. Liu, and N. X. Sun, Appl. Phys. Lett. 91, 182504 (2007). 8G. Woltersdorf, M. Kiessling, G. Meyer, J.-U. Thiele, and C. H. Back, Phys. Rev. Lett. 102, 257602 (2009). 9R. D. McMichael and P. Krivosik, IEEE Trans. Magn. 40, 1 (2004). 10P. Krivosik, N. Mo, S. S. Kalarickal, and C. E. Patton, J. Appl. Phys. 101, 083901 (2007). 11S. S. Kalarickal, P. Krivosik, J. Das, K. S. Kim, and C. E. Patton, Phys. Rev. B. 77, 054427 (2008). 12N. Mo, J. Hohlfeld, M. ul Islam, C. S. Brown, E. Girt, P. Krivosik, W. Tong, A. Rebei, and C. E. Patton, Appl. Phys. Lett. 92, 022506 (2008). 13J. O. Rantschler, C. Alexander, Jr., and H.-S. Jung, J. Magn. Magn. Mater. 286, 262 (2005). 14A. M. Stoneharm, J. Phys. D 5, 670 (1972). 15K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007). 16R. C. O’Handley, Modern Magnetic Materials: Principles and Applica- tion, (Wiley, New York, 2000).FIG. 3. (Color online) Theoretical ﬁts of FMR linewidth vs. frequency responses and relaxation rates for ﬁlms grown on Ru seed layers of differentthicknesses, as indicated.TABLE II. Summary of ﬁtting parameters. # d(nm) r/dK s(erg/cm2) DH0(Oe) 1 36.0 0.278 0.438 100 2 30.0 0.233 0.324 803 14.0 0.118 0.438 704 15.0 0.100 0.450 805 20.0 0.400 0.258 706 12.5 0.145 0.438 307 12.3 0.133 0.438 258 12.2 0.144 0.438 389 12.5 0.136 0.438 4510 12.7 0.134 0.438 50022403-3 Lu et al. Appl. Phys. Lett. 100, 022403 (2012) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.21.35.191 On: Sun, 21 Dec 2014 09:34:10
1.4817076.pdf	Spin-relaxation modulation and spin-pumping control by transverse spin- wave spin current in Y3Fe5O12 Y. Kajiwara, K. Uchida, D. Kikuchi, T. An, Y. Fujikawa et al. Citation: Appl. Phys. Lett. 103, 052404 (2013); doi: 10.1063/1.4817076 View online: http://dx.doi.org/10.1063/1.4817076 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v103/i5 Published by the AIP Publishing LLC. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 03 Aug 2013 to 129.93.16.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsSpin-relaxation modulation and spin-pumping control by transverse spin-wave spin current in Y 3Fe5O12 Y . Kajiwara,1,a)K. Uchida,1,2,b)D. Kikuchi,1,3T. An,1,4,c)Y . Fujikawa,1and E. Saitoh1,3,4,5 1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan 3WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4CREST, Japan Science and Technology Agency, Tokyo 102-0076, Japan 5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan (Received 13 June 2013; accepted 16 July 2013; published online 30 July 2013) Heat-current-induced manipulation of spin relaxation in Y 3Fe5O12under an in-plane temperature gradient is investigated. We show that the linewidth of the ferromagnetic resonance spectrum, i.e., the spin relaxation, in an Y 3Fe5O12ﬁlm increases or decreases dependi ng on the temperature-gradient direction and that this modulation is attributed to the spin-transfer torque caused by a thermally induced transverse spin-wave spin current in the Y 3Fe5O12ﬁlm. The experimental results also show that the spin-current magnitude generated by spin pumping in an attached Pt ﬁlm is inversely proportional tothe square of the modulated Gilbert damping cons tant, consistent with a phenomenological spin- pumping model. VC2013 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4817076 ] The spin-transfer torque (STT) is a phenomenon that exerts torque on magnetization via a spin current, i.e., the angular momentum transferred from conduction-electronspins to magnetic moments in a ferromagnet. 1,2Since the STT enables modulation of spin relaxation and manipulation of a magnetization direction, it plays an essential role in spin-tronics. Recent studies in spintronics have revealed that a spin current induces the STT not only in metallic magnets 3,4but also in insulating magnets,5–13such as Y 3Fe5O12. The STT acting on Y 3Fe5O12has been investigated by using electrically and thermally generated spin currents in Pt/Y 3Fe5O12 junctions with the aid of the spin Hall effect14–21and the spin Seebeck effect (SSE).22–35 The thermally induced STT has been observed in Pt/Y 3Fe5O12junctions in two different device conﬁgurations. One is the longitudinal conﬁguration, where a temperature gradient is applied perpendicular to the Pt/Y 3Fe5O12inter- face. Recently, several research groups7–11observed the modulation of the spin relaxation in Y 3Fe5O12in the longitu- dinal conﬁguration, and showed that the thermally induced STT in this conﬁguration originates from the spin transferbetween conduction electrons in Pt and localized magnetic moments in Y 3Fe5O12via the s-dexchange interaction across the Pt/Y 3Fe5O12interface. The other is the transverse conﬁguration, where a temperature gradient is applied along the Pt/Y 3Fe5O12interface. In the transverse conﬁguration, a spin-wave spin current5ﬂowing along a temperature gradient plays an important role in the STT. Using this conﬁguration, da Silva et al.12,13demonstrated that the inverse spin Hall effect (ISHE) induced by the spin pumping36–39is modulated by a transverse temperature gradient. However, the direct measurement of the spin-relaxation modulation associated with the thermally induced STT in the transverseconﬁguration is yet to be reported. In this paper, we report on the observation of the spin relaxation that is modulated by the thermally induced STT mediated by a transverse spin-wave spin current in a Pt/Y 3Fe5O12structure. Furthermore, we also show that, by making use of the thermally induced STT, the spin pumping efﬁciency can be enhanced or sup-pressed depending on the direction of the spin-wave spin current in Y 3Fe5O12. Figure 1(a) shows schematic illustrations of the experi- mental conﬁguration used in the present study. The sample system consists of a 2.7- lm-thick ferrimagnetic insulator Y3Fe5O12(111) ﬁlm and a 15-nm-thick Pt wire attached on the center of the Y 3Fe5O12ﬁlm. The Y 3Fe5O12ﬁlm was grown on a single-crystalline Gd 3Ga5O12(111) substrate by liquid phase epitaxy, and then the Pt wire was sputtered onthe top of the ﬁlm in an Ar atmosphere. Here, the surface of the Y 3Fe5O12ﬁlm has a 3 /C29m m2rectangular shape. The length and width of the Pt wire are 3 mm and 0.1 mm,respectively. We put two thermoelectric Peltier modules under both the edges of the Pt-wire/Y 3Fe5O12-ﬁlm sample and a heat sink under the center of the sample [see Figs. 1(b) and1(c)]. When electric currents are applied to the Peltier modules, in-plane heat currents ﬂowing towards the Pt wire or the edges of the Y 3Fe5O12ﬁlm are generated [see the tem- perature proﬁles shown in Figs. 1(b) and1(c)]. To measure the spin-wave-resonance (SWR) spectra and the spin pump- ing, a static in-plane magnetic ﬁeld Hand a microwave with the frequency fof 3.8 GHz and the power of 20 mW are applied to the Pt-wire/Y 3Fe5O12-ﬁlm sample. The thermally induced STT in the present Pt-wire/ Y3Fe5O12-ﬁlm structure is measured by using the ISHE com- bined with the spin pumping. In the conﬁguration shown in Fig.1(a), the transverse temperature gradients generate spin- wave spin currents in the Y 3Fe5O12layer along the gradients. The excited spin-wave spin currents ﬂow from the edges to the center (from the center to the edges) of the Y 3Fe5O12in the Pt-Lower T (LT) (Pt-Higher T (HT)) setup. At the center of the Pt-wire/Y 3Fe5O12-ﬁlm sample, the thermally induceda)Present address: Toshiba Corporation, R&D Center, Kanagawa 212-8583, Japan. b)Electronic mail: kuchida@imr.tohoku.ac.jp c)Present address: RIKEN, Saitama 351-0198, Japan. 0003-6951/2013/103(5)/052404/4/$30.00 VC2013 AIP Publishing LLC 103, 052404-1APPLIED PHYSICS LETTERS 103, 052404 (2013) Downloaded 03 Aug 2013 to 129.93.16.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsspin-wave spin currents modulate the spin relaxation of the Y3Fe5O12[see Figs. 1(d) and1(e)]. Since the spin-wave dy- namics near the Pt/Y 3Fe5O12interface induces a spin current in the Pt layer under the SWR conditions via the spin pump-ing, we can detect the modulation of the spin relaxation by measuring the spectrum of an electric voltage induced by the ISHE in the Pt. In Figs. 2(a)and2(b), we, respectively, show the spectra of the microwave absorption Iand the DC electric voltage V for the Pt-wire/Y 3Fe5O12-ﬁlm sample under 20 mW micro- wave excitation, measured when Hwas applied perpendicu- lar to the inter-electrode direction. In the Vspectrum, the clear voltage signals with peak structures, VSP, appear around the SWR ﬁelds ( /C2460:75 kOe); the VSPsignals are attributed to the ISHE induced by the spin pumping. The shapes of the VSPpeak structures are well reproduced by multiple Lorentz functions and, from the ﬁtting result, the spectrum linewidth DHat the ferromagnetic resonance (FMR) mode ðk¼0Þat the center of the Y 3Fe5O12ﬁlm is estimated to be DHk¼0¼3:9 Oe. From the value of DHk¼0, the Gilbert damping constant ak¼0¼ðc=4pfÞDHk¼0at the FMR mode is calculated as 1 :5/C210/C03, where cdenotes the gyromagnetic ratio. In addition to VSP, when a ﬁnite temper- ature difference was applied to the sample, a clear voltage step originating from the SSE,22–35VSSE, was found to appear around zero magnetic ﬁeld [Fig. 2(b)]. We conﬁrmed thatVSSEin the Pt-LT setup is opposite in sign to that in the Pt-HT setup [Figs. 2(c)and2(d)] and that the magnitude of VSSEin both the setups is proportional to the temperature dif- ference DT[Fig. 2(e)], where DT/C17Tedge/C0Tcenter with Tedge andTcenter being the absolute temperature at the edges andthe center of the Y 3Fe5O12ﬁlm, respectively. These results are consistent with the feature of the SSE.24,40 Now, we focus on the DTdependence of the Vspectra to investigate the modulation of the spin relaxation ofY 3Fe5O12by the transverse temperature gradient. Figure 3(a) shows the comparison of the normalized Vspectra at DT¼ 14:5 K (red curve) and DT¼0 K (black dotted curve) around the FMR ﬁeld. We found that the spectrum linewidth of VSP atDT¼14:5 K is clearly narrower than that at DT¼0K . By ﬁtting the observed VSPstructure using multiple Lorentz functions, DHk¼0andak¼0atDT¼14:5 K are estimated to be 2.3 Oe and 0 :9/C210/C03, respectively. These values are smaller than DHk¼0and ak¼0atDT¼0 K, showing that the spin relaxation of Y 3Fe5O12is modulated by the FIG. 1. (a) Schematic illustrations of the Pt-wire/Y 3Fe5O12-ﬁlm sample and the experimental conﬁguration. The Pt-wire/Y 3Fe5O12-ﬁlm sample was bridged between two Peltier modules. A microwave with the frequency of 3.8 GHz and the power of 20 mW was applied to the sample by using a microstrip line connected to a vector-network analyzer, where the microstrip line was placed on the top of the sample. [(b), (c)] Temperature Tproﬁles in the Pt-wire/Y 3Fe5O12-ﬁlm sample, measured when the edges of the Y3Fe5O12ﬁlm are heated [(b): Pt-LT setup] and cooled [(c): Pt-HT setup]. TPtdenotes the temperature of the Pt wire. The Tproﬁles were measured with an infrared camera (NEC-Avio TH9100MR). (d) A schematic illustra- tion of the spin-wave spin current generated by the in-plane temperature gra- dientrTin a ferromagnet. (e) A schematic illustration of the STT induced by the spin-wave spin current and the damping torque (DT) of the localized spins. FIG. 2. [(a), (b)] The magnetic ﬁeld Hdependence of the microwave absorp- tionI(a) and the electric voltage V(b) for the Pt-wire/Y 3Fe5O12-ﬁlm sample at the temperature difference DT¼14:5 K in the Pt-LT setup. [(c), (d)] H dependence of Vfor various values of DTin the Pt-LT setup (c) and the Pt- HT setup (d) in the low Hregion. (e) jDTjdependence of the SSE voltage VSSEatH¼0:1 kOe in the Pt-LT and Pt-HT setups. FIG. 3. (a) H/C0HFMRdependence of V=Vmaxin the Pt-wire/Y 3Fe5O12-ﬁlm sample at DT¼14:5 K (red curve) and DT¼0 K (black dotted curve) in the Pt-LT setup. HFMRandVmaxdenote the FMR ﬁeld and the maximum value of V,r e s p e c t i v e l y .( b ) H/C0HFMRdependence of V=VmaxatDT¼/C04:7K ( b l u e curve) and DT¼0 K (black dotted curve) in the Pt-HT setup. (c) DTdepend- ence of ak¼0ðDTÞ=ak¼0ðDT¼0KÞin two different Pt-wire/Y 3Fe5O12-ﬁlm samples. ak¼0ðDTÞdenotes the Gilbert damping constant at the FMR mode.052404-2 Kajiwara et al. Appl. Phys. Lett. 103, 052404 (2013) Downloaded 03 Aug 2013 to 129.93.16.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsspin-wave spin current induced by the transverse temperature gradient. We also found that, by reversing the direction of the spin-wave spin current and temperature gradient, thespectrum linewidth of V SPbecomes wider than that at DT¼ 0 K [see Fig. 3(b)], consistent with the characteristic of the STT. As shown in Fig. 3(c), the value of ak¼0monotonically decreases with increasing DT. We conﬁrmed that, when the whole sample is uniformly heated, in which the transverse spin-wave spin current is absent, the value of ak¼0does not change. These results indicate that this spin-relaxation modu- lation is attributed to the STT from the thermally induced transverse spin-wave spin current. Finally, we demonstrate that, by making use of the ther- mally induced STT and the transverse spin-wave spin cur- rent, the spin pumping efﬁciency can be thermallycontrolled. In the phenomenological spin-pumping model, 38 a spin current generated by the spin pumping under the FMRcondition is inversely proportional to the square of theGilbert damping constant, suggesting that the above spin- relaxation modulation by the transverse spin-wave spin cur- rent also can tune the magnitude of spin currents induced bythe spin pumping. In Fig. 4(a), we show the Hdependence of Vunder 20 mW microwave excitation for various values of DTin the Pt-wire/Y 3Fe5O12-ﬁlm sample in the Pt-LT setup. In this setup ( DT>0), the magnitude of the VSPsignal generated by the spin pumping was observed to monotonically increasewith increasing DT[see Figs. 4(a),4(c), and 4(e) and note that the shift of the resonant peak structure is attributed to the increase in the base temperature]. In contrast, Figs. 4(b), 4(d), and 4(e) show that the V SPsignal is suppressed in the Pt-HT setup ( DT<0), indicating that the spin pumping is modulated by the thermally induced STT. As shown in Fig.4(f), the magnitude of V SPis almost proportional to 1 =~a2, where ~a/C17ak¼0ðDTÞ=ak¼0ðDT¼0KÞ, consistent with the phenomenological spin-pumping model.38We conﬁrmedthat this spin-pumping modulation does not appear when the Pt-wire/Y 3Fe5O12-ﬁlm sample is uniformly heated [see Fig. 4(g)]. This modulation also disappears in the Pt/Y 3Fe5O12 bilayer wire on a paramagnetic Gd 3Ga5O12substrate, where the width of the Y 3Fe5O12layer is the same as that of the Pt wire. Since the transverse spin-wave spin current does notexist in the Pt/Y 3Fe5O12bilayer wire, this result becomes strong evidence that the spin-pumping modulation observed here is due to the transverse spin-wave spin current gener-ated by the in-plane temperature gradient. In summary, using Pt/Y 3Fe5O12junctions, we have investigated the magnetization dynamics coupled with theSTT and the transverse spin-wave spin current generated by an in-plane temperature gradient. The experimental results show that the heat-current-induced STT modulates the spinrelaxation of Y 3Fe5O12, which in turn changes the magnitude of the inverse spin Hall voltage induced by the spin pumping in the Pt layer. Since the heat-current-induced STT enablesthermal manipulation of spin relaxation and spin-pumping efﬁciency, it will be one of the basic principles for driving future spin-caloritronic devices. The authors thank S. Maekawa, H. Adachi, J. Ohe, B. Hillebrands, and S. M. Rezende for valuable discussions. 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1.3067757.pdf	Transverse wall dynamics in a spin valve nanostrip J. M. B. Ndjaka, A. Thiaville, and J. Miltat Citation: Journal of Applied Physics 105, 023905 (2009); doi: 10.1063/1.3067757 View online: http://dx.doi.org/10.1063/1.3067757 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/105/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Current-induced resonant depinning of a transverse magnetic domain wall in a spin valve nanostrip Appl. Phys. Lett. 97, 182506 (2010); 10.1063/1.3507895 Investigating the exchange bias in multilayer triangular nanorings J. Appl. Phys. 105, 123916 (2009); 10.1063/1.3153274 Direct visualization of three-step magnetization reversal of nanopatterned spin-valve elements using off-axis electron holography Appl. Phys. Lett. 94, 172503 (2009); 10.1063/1.3123290 Off-axis electron holography of pseudo-spin-valve thin-film magnetic elements J. Appl. Phys. 98, 013903 (2005); 10.1063/1.1943511 Exploring spin valve magnetization reversal dynamics with temporal, spatial and layer resolution: Influence of domain-wall energy Appl. Phys. Lett. 85, 440 (2004); 10.1063/1.1772520 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Thu, 02 Oct 2014 07:41:27Transverse wall dynamics in a spin valve nanostrip J. M. B. Ndjaka,a/H20850A. Thiaville,b/H20850and J. Miltat Laboratoire de Physique des Solides, CNRS, Université Paris-Sud, Bât. 510, 91405 Orsay Cedex, France /H20849Received 26 August 2008; accepted 3 December 2008; published online 22 January 2009 /H20850 The magnetism of a Fe 20Ni80/Cu /Co spin valve, in which a layer of FeNi containing a head-to-head transverse domain wall is coupled to a uniformly magnetized Co layer, via a nonmagnetic Cu layer,was investigated by micromagnetics /H20849mainly numerical simulations /H20850. In equilibrium, due to the magnetostatic coupling between the layers, a quasiwall is created in the Co layer, which affects thedomain wall proﬁle in the FeNi layer. The dynamics of the domain wall under an applied ﬁeld is alsomodiﬁed, and two opposite effects due to the spin valve geometry have been found, resulting, on theone hand, from the variation in the width of the domain wall and, on the other hand, from theadditional damping of magnetization dynamics due to the cobalt layer. © 2009 American Institute of Physics ./H20851DOI: 10.1063/1.3067757 /H20852 I. INTRODUCTION Domain wall motion in magnetic nanowires and nanos- trips under the inﬂuence of an applied magnetic ﬁeld or spin-polarized current is currently the subject of worldwide inten-sive theoretical and experimental research. 1–4In a previous theoretical study,5the displacement, under an applied mag- netic ﬁeld, of a transverse domain wall within a Permalloynanostrip revealed the existence of laminar and turbulent re-gimes, in full similarity to the displacement of /H20849Bloch /H20850do- main walls in unbounded ﬁlms. 6The laminar regime corre- sponds to applied magnetic ﬁelds lower than a criticalvalue—named the Walker ﬁeld—up to which domain wallvelocity generally increases with the applied ﬁeld. In thatregime, the domain wall moves with a time-independent ve-locity, high mobility, and a stable structure. The turbulentregime is related to applied magnetic ﬁelds higher than theWalker ﬁeld: the domain wall displacement occurs with pe-riodic changes in the wall structure, the wall velocity oscil-lates, and its average value is drastically low. In the present study, we have extended this work to the case of Fe 20Ni80/Cu /Co spin valves, where a soft FeNi layer containing a transverse head-to-head domain wall interactswith a uniformly magnetized Co layer, via magnetostaticcoupling through a nonmagnetic layer. This FeNi/Cu/Co spinvalve structure has been used in many experiments, 1,7,8as the giant magnetoresistance /H20849GMR /H20850effect gives an easy access to the position of the domain wall /H20849in the ideal situation of a single domain wall in one layer /H20850. However, recent experiments8–10revealed that the mechanism of the displace- ment of the domain wall contained in the FeNi layer of a Co/H208497n m /H20850/Cu /H2084910 nm /H20850/FeNi /H208495n m /H20850spin valve, under the com- bined action of a polarized current and a magnetic ﬁeld, isnot well understood. The purpose of our simulations is toadvance the understanding of the processes that occurin these spin valve samples. We begin by studying theaction of an applied magnetic ﬁeld on a spin valveFe 20Ni80/H208494n m /H20850/Cu/H20849L/H20850/Co/H208494n m /H20850, where a FeNi nanostrip containing one transverse magnetic wall is in magnetostatic interaction /H20849but without exchange coupling /H20850with a cobalt nanostrip, via a layer of a nonmagnetic material. The thick-ness of the separating Cu layer is a parameter that is varied inthe simulations. In the following, we ﬁrst describe the staticequilibrium magnetization distribution and, thereafter, inves-tigate the domain wall dynamics in detail. II. STATIC EQUILIBRIUM MAGNETIZATION DISTRIBUTION A proprietary code11was adapted to the case of an inﬁ- nite nanostrip, with a spin valve architecture. To this end, thecharges on the yzsurfaces of the calculation box were re- moved /H20849more accurately, transferred to inﬁnity /H20850and the mag- netostatic ﬁeld, created by each magnetic layer on the otherlayer, was evaluated. The indirect exchange between layerswas not included as, in experiments, the spacer is thickenough so as to avoid it. Figure 1schematically describes the sample architecture and the magnetization distribution. The nanostrip width was T y=120 nm, and both mag- netic layers had the same thickness, T1=T2=4 nm. The thickness of the Cu layer Lwas a variable parameter of the simulations. Outside the calculation box, the magnetizationwas ﬁxed along the xaxis so as to induce a domain wall in the soft layer only. In the calculation box, the magnetizationpointed initially toward the yaxis, corresponding to a trans- a/H20850Permanent address: Département de Physique, Faculté des Sciences, Uni- versité de Yaoundé I, BP 812 Yaoundé, Cameroun. b/H20850Electronic mail: thiaville@lps.u-psud.fr. T x L F e 2 0 N i 8 0 C u C o T 2 T y T 1 x y z FIG. 1. Schematic description of the spin valve Fe20Ni80/Cu /Co nanostrip /H20849perspective view /H20850. The length of the physical system is inﬁnite. The calcu- lation box used in the simulations is drawn at the center. The value of spacerthickness /H20849L/H20850is varied in simulations.JOURNAL OF APPLIED PHYSICS 105, 023905 /H208492009 /H20850 0021-8979/2009/105 /H208492/H20850/023905/8/$25.00 © 2009 American Institute of Physics 105 , 023905-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Thu, 02 Oct 2014 07:41:27verse wall structure. Indeed, the transverse wall is the stable structure at such nanostrip size,12and moreover it is expected to be stabilized by the spin valve structure /H20849see below /H20850. The parameters for the FeNi layer /H20849index 1 /H20850were typical, namely, saturation magnetization M1=8/H11003105A/m /H20849800 G /H20850, ex- change constant A1=1/H1100310−11J/m/H2084910−6erg /cm/H20850with, in addition, a weak perpendicular anisotropy K1=100 J /m3 /H208491000 ergs /cm3/H20850. For the cobalt layer /H20849index 2 /H20850, the magne- tization was initially uniformly aligned with the positive x axis throughout the sample. Parameters were M2=14 /H11003105A/m /H208491400 G /H20850, exchange constant A2=2 /H1100310−11J/m/H208492.0/H1100310−6erg /cm/H20850, and no anisotropy in a ﬁrst step /H20849mimicking a polycrystalline material with very small grains /H20850. The introduction of a perpendicular anisotropy constant K2=5/H11003105J/m3/H208495/H11003106ergs /cm3/H20850in the cobalt layer was tested but did not result in any signiﬁcant change in the results. An important parameter in the simulations is the length of the calculation box. A too short box may constrain thedomain wall structure, whereas a too long box leads to longcalculation times. For the parameters chosen here, we found that a length of T x=960 nm was sufﬁcient. In the FeNi and Co layers, this calculation box was divided into 241 /H1100331 /H110031 cells so that the mesh size was very close to 4 nm. The Cu layer appears in the calculations only via its thickness, asit affects the decay of the stray ﬁeld in one layer arising fromthe magnetization in the other layer. Figure 2/H20849a/H20850shows the equilibrium magnetization distri- bution observed in the FeNi layer when the Cu layer has athickness of L=10 nm. It corresponds to a symmetric trans- verse wall. From the equilibrium magnetization distribution,thexposition of the wall, the wall width, as well as the magnetization proﬁle were evaluated. The position of thewall qwith respect to the center of the calculation box can be obtained by integration of the longitudinal magnetization, q=1 2Ty/H20885mxdxdy . /H208491/H20850 This assumes that mx=+1 /H20849/H110021/H20850at the left /H20849right /H20850edge of the calculation box. For the above magnetization distribution,the position of the center of the wall is +3.48 nm, so that thewall is practically centered within the calculation box. Thewall width was calculated according to Thiele’s deﬁnition, 13 /H9004T=2Ty/H20885/H20873/H11509m/H6023 /H11509x/H208742 dxdy. /H208492/H20850 We use this deﬁnition because, for wall dynamics, this is the relevant quantity. For the magnetization distribution of Fig.2/H20849a/H20850, one ﬁnds /H9004 T=39.08 nm; this value remains unchanged when the length of the calculation box is increased, a proofof the adequacy of the box length. This value proves largerthan the wall width in a single FeNi layer with the samedimensions /H20849/H9004 T=31.69 nm /H20850. The magnetization proﬁles, obtained by averaging over theycoordinate, are shown in Fig. 3/H20849a/H20850. As now well known,4they turn out to be very close to Bloch wall proﬁles, even though the situation is completely different /H20849charged domain walls, no anisotropy /H20850. These proﬁles read (a) (b) x y z FIG. 2. /H20849Color online /H20850Zero ﬁeld equilibrium magnetization distribution in a Fe20Ni80/Cu /Co spin valve nanostrip, with 120 nm width, 4 nm thickness for FeNi and Co, and 10 nm spacing. Only the central part of the calculationbox /H20849length 360 nm /H20850is shown. The panels refer to /H20849a/H20850the FeNi layer, at the bottom of the stack and /H20849b/H20850the Co layer. The red color in /H20849a/H20850/H20851blue color in /H20849b/H20850/H20852indicates the intensity of the positive /H20849negative /H20850value of magnetization ycomponent, with a saturation at /H110060.2. Notice the marked asymmetry in the position of the quasiwall in the Co layer. 0 200 400 600 800 100 0-0.4-0.200.20.40.60.811.2 x(nm )<mi,i= x, y, z>(x)<mx> <my> <mz>(b) 0 200 400 600 800 1000-1-0.500.51 x(nm )<mi,i = x, y, z>( x)<mx> <my> <mz>(a) FIG. 3. /H20849Color online /H20850Magnetization proﬁles /H20849averaged over the nanostrip width /H20850in the Fe20Ni80/Cu /Co spin valve nanostrip, showing /H20849a/H20850the transverse domain wall in the FeNi layer and /H20849b/H20850the quasiwall in the Co layer. The vertical lines mark the center of the domain wall and highlight the asymmetric position of the quasiwall.023905-2 Ndjaka, Thiaville, and Miltat J. Appl. Phys. 105 , 023905 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Thu, 02 Oct 2014 07:41:27mx=−t h/H20873x /H9004/H20874,my=1 ch/H20873x /H9004/H20874, /H208493/H20850 where /H9004is the wall width parameter. For these proﬁles, Thiele’s width is exactly /H9004, and the average value of the transverse component obeys /H20855my/H20856Tx=/H9266/H9004. /H208494/H20850 By ﬁtting /H20855mx/H20856/H20849x/H20850and /H20855my/H20856/H20849x/H20850to the Bloch wall proﬁles, the values /H9004x=44.76 nm and /H9004y=40.56 nm were obtained, the latter being close to the value /H9004y/H11032=39.72 nm resulting from the average value /H20855my/H20856=0.13. Thus, all values are slightly different, so that the wall is not perfectly described by the Bloch proﬁle. This arises from the magnetization nonunifor-mity in the ydirection due to the large width of the nanostrip in comparison to the exchange length of FeNi /H208495n m /H20850. The Co layer equilibrium magnetization distribution is represented in Fig. 2/H20849b/H20850. We observe that there is no wall in the Co layer but simply a ﬂuctuation of magnetization. Theinﬂuence of the stray ﬁeld of a Néel wall in one layer on the/H20849otherwise uniform /H20850magnetization of an adjacent layer was noticed early 14,15and called a quasiwall.16The Thiele width of this quasiwall is found to be 2687.37 nm. This very largevalue does not at all correspond to a geometrical width butrelates to the weaker magnetization gradients existing in thequasiwall /H20851see the deﬁnition in Eq. /H208492/H20850/H20852. The position of the center of the quasiwall, determined by that of the maximumabsolute value of /H20855m y/H20856/H20849x/H20850, amounts to /H1100217.76 nm. This re- ﬂects the relative positions of both structures in Fig. 2.I n addition, Fig. 2/H20849b/H20850reveals a ydisplacement of the quasiwall from the nanostrip symmetry axis. The magnetization proﬁle of the Co layer equilibrium magnetization distribution is shown in Fig. 3/H20849b/H20850. Again, we see that the center of the quasiwall is shifted /H20849inx/H20850as com- pared to the center of the wall. In the quasiwall, /H20855my/H20856/H20849x/H20850is negative, this being the direct analog of the coupling of the transverse magnetizations in a Néel wall and its associatedquasiwall. 15The quasiwall shows a small perpendicular mag- netization mzthat is much larger than in the wall. All these phenomena are consequences of the magneto- static ﬁeld created in the Co layer by the magnetization ofthe wall located in the FeNi layer. This ﬁeld is represented inFig. 4, where two contributions are distinguished. First, the wall located in the FeNi layer contains “volume” charges−M s/H20849divm/H20850, positive here as the wall is head-to-head, which create a magnetic ﬁeld orienting the magnetization of the Co layer mainly in the direction of the positive zaxis, with in- plane ﬁeld components radiating from the wall center /H20851Fig. 4/H20849a/H20850/H20852. Second, the surface charges located in the xzsurfaces of the FeNi layer create a magnetic ﬁeld that pushes the Coquasiwall magnetization in the direction of the negative y axis /H20851Fig. 4/H20849b/H20850/H20852, giving rise to the negative /H20855m y/H20856/H20849x/H20850in the quasiwall. The shift in the center of the quasiwall compared to the center of the wall is stemming from the xcomponent of the ﬁeld due to the volume charges: the rotation, out of theeasy /H20849x/H20850axis, of the Co magnetization is favored by a ﬁeld opposing its easy axis magnetization. As the Co layer is mag-netized in the same direction as the domain on the left side of the FeNi wall, the largest ﬂuctuation occurs on the left of thedomain wall. The inﬂuence of the thickness of the Cu layer on these parameters was investigated. One observes, as shown in Fig.5, that the width of the wall located in FeNi layer increases when the thickness of the separating Cu layer decreases,from the asymptotic value of /H9004 T=31.69 nm at inﬁnite Cu thickness. Similarly, the components /H20855my/H20856in the FeNi and Co layers increase /H20849in absolute value /H20850when the Cu thickness decreases, with opposite signs. This corresponds to a stabili-zation of the transverse wall structure, through ﬂux closure (a) (b) x y z FIG. 4. /H20849Color online /H20850Magnetostatic ﬁeld created in the cobalt layer by the wall in the FeNi layer, evaluated for the equilibrium conﬁguration. Thearrow coloring in red, black, or blue correspond to a negative, nil, andpositive value of the zﬁeld component, respectively. The arrow length cor- responds to the magnitude of the in-plane /H20849x-y/H20850component of the ﬁeld, with a maximum value of 0.002 times the Co magnetization /H2084935 Oe, 2.8 kA/m /H20850. The maps show /H20849a/H20850the ﬁeld created by volume charges within the FeNi layer and /H20849b/H20850the ﬁeld created by lateral surface charges along the edges of the FeNi layer. 0 1 02 03 04 05 06 07 08 09 0 1 0 0-0.1-0.0500.050.10.150.2 Cu thickness (nm) <my>FeNi Co net 0102030405060∆∆∆∆T_ FeNi(nm) FIG. 5. /H20849Color online /H20850Variation, with Cu thickness, in the Thiele domain wall width in FeNi /H20849ﬁlled symbols, left scale /H20850and of the average myvalues /H20849open symbols, right scale /H20850in the FeNi /H20849circles /H20850and Co /H20849squares /H20850layers. The curve without symbols shows the average net myvalue. The horizontal strokes on the right indicate the values for an isolated FeNi layer.023905-3 Ndjaka, Thiaville, and Miltat J. Appl. Phys. 105 , 023905 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Thu, 02 Oct 2014 07:41:27inside the Co layer /H20849akin the stabilization of a Néel wall in inﬁnite bilayers /H20850. Thus, the width of the transverse wall in- creases /H20849as shown by the /H20855my/H20856for FeNi /H20850and, as the magne- tization gradient in the wall decreases, the Thiele domain wall width increases. For the quasiwall, however, the Thieledomain wall width decreases /H20849but starting from inﬁnity at inﬁnite spacing, not shown /H20850. One thus visualizes the penetra- tion depth of the magnetostatic ﬁeld created by the magneti-zation of each layer on the other. III. MAGNETIZATION DYNAMICS UNDER APPLIED MAGNETIC FIELD In micromagnetics, magnetization dynamics obeys the Landau–Lifshitz–Gilbert equation, namely, /H11509m/H6023 /H11509t=/H92530H/H6023eff/H11003m/H6023+/H9251m/H6023/H11003/H11509m/H6023 /H11509t, /H208495/H20850 wheremis the local unit magnetization vector, /H92530=/H92620/H20841/H9253/H20841is the gyromagnetic ratio, and /H9251is the dimensionless Gilbert damping coefﬁcient.17A fourth order Runge–Kutta integra- tion of Eq. /H208495/H20850simultaneously for FeNi and Co layers was performed, under the inﬂuence of an applied magnetic ﬁeld.The gyromagnetic ratio was /H92530=2.21/H11003105m/H20849As /H20850−1for all simulations. The damping coefﬁcient was set to 0.02 for FeNi and 0.05 for Co. The long-term dynamics could beaccessed by moving the calculation box in order to maintainthe wall always centered. The magnetization distributionused to start the calculations was the zero ﬁeld equilibriummagnetization distribution represented in Fig. 2. The mag- netic ﬁeld was applied in the longitudinal direction of thesample, instantaneously at time 0, so that the transient periodduring which the wall accelerates was also calculated. Inorder to ﬁnd the adequate time step, the energy loss wasmonitored. 18For a time step of /H9004t=25 fs, the computed value/H9251numwas equal to the set /H9251value, with a 10−5absolute precision. For each value of the applied magnetic ﬁeld, thedetails of wall and quasiwall dynamics were calculated,namely, the temporal evolution of the position of the wall,the wall width, the instantaneous wall velocity, the averagevalues /H20855m y/H20856and /H20855mz/H20856of wall magnetization components, as well as the wall magnetization angle. That angle, denoted by /H9278, was deﬁned recently.19Schematically, for a transverse wall,/H9278corresponds to the angle of the wall magnetic mo- ment away from the yaxis in the yzplane; for a vortex wall, it measures the yposition of the vortex. The angle /H9278was determined, in each layer, by fourth order Runge–Kutta inte-gration of its deﬁning equation, d /H9278 dt=1 2Ty/H20885dm/H6023 dt·/H20873m/H6023/H11003/H11509m/H6023 /H11509x/H20874dxdy . /H208496/H20850 The results show that the dynamics of the FeNi/Cu/Co spin valve depends on whether the applied magnetic ﬁeld ishigher or lower than the so-called Walker ﬁeld, whose valueis very close to that obtained for a single FeNi nanostrip,namely, 32 /H11021H w/H1102133 Oe /H208492546–2626 A/m /H20850. Figure 6, ob- tained for an applied magnetic ﬁeld Happ=5 Oe /H20849398 A/m /H20850, illustrates the behavior below the Walker ﬁeld and Fig. 7, obtained for an applied magnetic ﬁeld Happ=50 Oe /H208493979A/m /H20850, characterizes the behavior beyond the Walker ﬁeld. We observe in Fig. 6/H20849a/H20850that the temporal evolution of the position of the wall located in FeNi increases linearlywith time, identical to the position of the quasiwall located inCo. In the transient period, the separation of the two wallsappears to increase slightly. Such a dynamic deformationmay be expected but could also be only an artifact of the method used to determine the position of the quasiwall. Thevalue of /H20855m y/H20856FeNiis opposite to that of /H20855my/H20856Co/H20851Fig.6/H20849b/H20850/H20852,a s in statics, and changes weakly during the transient period. The value of /H20855mz/H20856FeNiincreases markedly during the transient period, a signature of the dynamic distortion of the trans- verse wall, whereas that of /H20855mz/H20856Cohardly changes, as it is determined by the volume charge of the domain wall /H20851Fig. 6/H20849c/H20850/H20852. This behavior is conﬁrmed by Fig. 6/H20849d/H20850, which plots the temporal evolution of the angle of the magnetization inthe wall and quasiwall. The minimum and maximum valuesofm zare low and constant in the wall /H20851Fig.6/H20849e/H20850/H20852and quasi- wall /H20851Fig.6/H20849f/H20850/H20852once the transient is over, indicating that no vortex was injected. Thus, the dynamics below the Walkerﬁeld is essentially the same as that characteristic of a singlelayer, with a nearly unchanged quasiwall attached to thewall. Looking now at the regime beyond the Walker ﬁeld, Fig. 7/H20849a/H20850shows that the wall and the quasiwall always progress in an identical way, but now the temporal evolution exhibitssteps and the instantaneous velocity oscillates. As is wellknown, 5this reﬂects the periodic reversal of the ymoment of the wall through injection and displacement of antivortices.In Fig. 7/H20849b/H20850, the temporal evolution of /H20855m y/H20856shows that quasi- wall moment follows the variation in the wall moment, keep- ing an opposite sign. For /H20855mz/H20856, Fig. 7/H20849c/H20850reveals that the Co moment is very constant, as seen before, whereas the FeNi moment becomes very large during the periods where /H20855my/H20856is maximum, i.e., when a transverse wall structure with no an- tivortex is present. In order to display the antivortex exis-tence, Fig. 7/H20849e/H20850shows the time evolution of the minimum and maximum m zvalues in the FeNi layer. When the mini- mum mzis very close to /H110021 or the maximum to 1, an anti- vortex exists and the magnetization at its center providesthese extrema. In order to check that it is indeed an antivor-tex /H20849cross Bloch line /H20850and not a vortex /H20849circular Bloch lines /H20850, one must look at the conﬁgurations of the magnetization orcalculate the winding number of the magnetization in thevicinity of this structure. The antivortex lifespan is /H110113n s for this applied ﬁeld, and this time decreases with increasingapplied magnetic ﬁeld. 20On the contrary, Fig. 7/H20849f/H20850, repre- senting the corresponding quantities for the Co layer, indi-cates only minor changes. The angle of the magnetization ofthe wall varies quasilinearly as expected /H20851Fig. 7/H20849d/H20850/H20852, while the angle of the magnetization of the quasiwall remains al-most zero. Therefore, here also the behavior is qualitativelythe same as for a single layer, with a quasiwall that justfollows the transformations of the wall structure. From the curves representing the temporal evolution of the position of the wall center, the steady-state wall velocity/H20849for applied magnetic ﬁelds lower than the Walker ﬁeld /H20850,o r its average value /H20849when the applied magnetic ﬁeld is higher than the Walker ﬁeld /H20850was estimated. These values are drawn023905-4 Ndjaka, Thiaville, and Miltat J. Appl. Phys. 105 , 023905 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Thu, 02 Oct 2014 07:41:27in Fig. 8, together with those for a single FeNi layer. The most apparent difference is the increase in domain wall ve-locity in the double layer nanostrip, and the most strikingsimilitude is the absence of shift in the Walker ﬁeld for smalldamping in the Cobalt layer. This last result, surprising atﬁrst sight, means that, even if the magnetostatic interactionwith the Cobalt layer stabilizes the transverse wall structurein the FeNi layer, it little affects the antivortex injection pro-cess. In addition, the velocity-ﬁeld relation just below theWalker ﬁeld is slightly modiﬁed, with a maximum velocityreached at a ﬁeld lower than the Walker ﬁeld. Within theone-dimensional model, 4this can be explained by a variation in the wall width when the effective transverse anisotropyterm is not negligible in comparison to the main /H20849axial /H20850ef- fective anisotropy. To further illustrate the inﬂuence of theCo layer on wall dynamics, Fig. 8also shows the velocity for the same spin valve nanostrip, when the Gilbert coefﬁcientof Co layer is large /H20849 /H9251Co=0.5 /H20850. In this case, the wall veloci- ties prove lower, and the Walker ﬁeld becomes larger. Cal-culations were also performed for negative ﬁelds. Some small differences could be seen. They originate from thebreaking of symmetry due to the interaction with the Colayer, which is apparent on the equilibrium structure in Fig.2/H20849b/H20850that breaks the left-right symmetry. However, these dif- ferences are much smaller than those shown in Fig. 8, so that we do not discuss them in detail. In order to explain the inﬂuence of the Co layer, we focus on the initial regime for low applied ﬁelds. In steadymotion, Thiele’s relation is exact and reads v=/H92530/H9004T /H9251Happ, /H208497/H20850 where the Thiele domain wall width /H9004Thas to be evaluated for the moving domain wall structure. As we study here thezero ﬁeld limit, we are entitled to use the static structure. Forthe more complex spin valve nanostrip studied here, Thiele’s2121.52222.52323.5 q1-q2(nm) -1000100200300400500600700800q(nm) FeNi Co(a) 00.0010.0020.0030.004 <mz> FeNi Co(c) 00.010.020.03ΦΦΦΦ(rad.) FeNi Co(d) x1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.04-0.0200.020.040.060.080.1 time (ns)extrema of mzin FeNi max min(e) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.04-0.0200.020.040.060.080.1 time (ns)extrema of mzin Co max min(f)-0.0500.050.10.15<my> FeNi Co(b) FIG. 6. /H20849Color online /H20850Time evolution, under an applied magnetic ﬁeld Happ=398 A /m/H208495O e /H20850,o f /H20849a/H20850the positions of the wall and the quasiwall, as well as their difference /H20849right scale /H20850,/H20849b/H20850the average transverse magnetizations /H20849in FeNi and Co /H20850,/H20849c/H20850the average perpendicular magnetizations, /H20849d/H20850the magnetization angle of the wall in FeNi and of the quasiwall in Co, /H20849e/H20850the maximum and minimum perpendicular magnetization components in FeNi, and /H20849f/H20850same for Co. The ﬁeld is applied instantaneously at time 0, and only the ﬁrst 5 ns of the evolution are shown.023905-5 Ndjaka, Thiaville, and Miltat J. Appl. Phys. 105 , 023905 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Thu, 02 Oct 2014 07:41:27analysis12can be readily extended as it simply expresses the dynamic equilibrium of forces, F/H6023+D/H6025v/H6023=0/H6023, /H208498/H20850 where F/H6023andD/H6025represent the external force and the dissipa- tion matrix, respectively. In the case of a spin valve nanos-trip, the force arises only from the layer with the domainwall but the dissipation occurs in both layers, so that onewrites F /H6023=2/H92620M1S1Happx/H6023 /H208499/H20850 and01000200030004000q(nm) FeNi Co(a) -50050q1-q2(nm) 0 5 10 15 20 25-101 time (ns)mzin FeNi max min(e)-0.02-0.0100.010.02 -0.02-0.02-0.02-0.02<mz> FeNi Co(c)-0.2-0.100.10.2 <my> FeNi Co(b) 0 5 10 15 20 2 5-0.0300.030.06 time (ns)mzin Coma x min(f)-505101520 ΦΦΦΦ(rad.)FeNi Co(d) x 1000 FIG. 7. /H20849Color online /H20850Time evolution, under an applied magnetic ﬁeld Happ=3979 A /m/H2084950 Oe /H20850,o f /H20849a/H20850the positions of the wall and the quasiwall, and their difference /H20849right scale /H20850,/H20849b/H20850the average transverse magnetizations in FeNi and Co, /H20849c/H20850the perpendicular magnetizations, /H20849d/H20850the magnetization angle of the wall in FeNi and of the quasi-wall in Co, with lines drawn at multiples of /H9266,/H20849e/H20850the maximum and minimum perpendicular magnetization component in FeNi, and /H20849f/H20850same for Co. Note that the time span is 25 ns here. In /H20849e/H20850, the values do not reach exactly /H110061 when an antivortex is present, as the center of this structure may move in between the mesh points. 0 1 02 03 04 05 00100200300400500600700 Applied magnetic field (mT)Wall velocity (m /s) FIG. 8. Wall velocity as a function of the applied magnetic ﬁeld, for a spinvalve nanostrip FeNi /H208494n m /H20850/Cu /H2084910 nm /H20850/Co /H208494n m /H20850/H20849ﬁlled symbols /H20850or just the FeNi /H208494n m /H20850layer of the nanostrip /H20849open symbols /H20850. The damping of Co is 0.05 /H20849ﬁlled circles /H20850or 0.5 /H20849ﬁlled squares /H20850, whereas it is 0.02 for FeNi. The velocity is the stationary value below the Walker ﬁeld, and the average valueabove.023905-6 Ndjaka, Thiaville, and Miltat J. Appl. Phys. 105 , 023905 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Thu, 02 Oct 2014 07:41:27D/H6025v/H6023=−/H92620 /H92530/H20875/H92511M1/H20885 1/H20873/H11509m/H6023 /H11509x/H208742 dxdydz +/H92512M2/H20885 2/H20873/H11509m/H6023 /H11509x/H208742 dxdydz/H20876v/H6023, /H2084910/H20850 where S1=TyT1is the cross section of the FeNi layer. By expressing the wall velocity in the form of relation /H208497/H20850,w e deduce from relations /H208498/H20850to/H2084910/H20850that an effective inverse mobility /H20849Happ /v, that may be called a magnetic viscosity /H20850 exists for the spin valve structure, implying /H20873/H9251 /H9004T/H20874 sv=/H92511 /H90041+M2T2 M1T1/H92512 /H90042. /H2084911/H20850 This relation, which refers to the Thiele domain wall width and damping coefﬁcient in both layers, could also be inter-preted as an effective damping, or as an effective Thieledomain wall width, in a spin valve sample. We prefer how-ever to formulate it as an effective viscosity with additivecontributions from both layers, the weighting factor beingthe magnetic moment of each layer. Note that, from Eqs. /H208499/H20850 and /H2084910/H20850, it is immediate to generalize Eq. /H2084911/H20850to the cases where the gyromagnetic ratio, the thickness, the width T y, etc., are not the same in both layers. The values obtainedfrom Eq. /H2084911/H20850are very close to those deduced from the curves representing the temporal evolution of the wall. Forexample, under an applied ﬁeld of 5 Oe /H20849398 A/m /H20850, the cal- culated velocity is 159.1 m/s, whereas that deduced from thecurves of the temporal evolution of the wall position is 158.5m/s. In addition, relation /H2084911/H20850allows for some predictions. An increase in /H92512raises the value of /H20849/H9251//H9004T/H20850sv; consequently, the velocity of the wall should decrease. On the other hand, when the Cu thickness increases, the value of /H90041decreases, so that the wall velocity should decrease /H20849the effect arising from the variations of /H90042is smaller /H20850. This is precisely what is observed in Fig. 8. The values /H90041and/H90042deduced from the equilibrium magnetization distributions were used to predict the mobilityof the wall for various Cu thickness /H20849Fig.9/H20850. The results ofnumerical calculations, indicated by the symbols in Fig. 9, conﬁrm these predictions fully. In addition, from the ﬁgure,one may get the impression that a special value /H9251Cocfor the Gilbert coefﬁcient of the Co layer exists, below which themobility decreases when the thickness of the Cu layer in-creases, and above which it increases. The numerical value is /H9251Coc/H110150.2 for the parameters chosen. It has, however, no physical meaning as a damping constant as, from Eq. /H2084911/H20850, one understands that the existence of this special value isonly the indication that the Thiele domain wall width in theCo layer is close to a rational function of Thiele’s domainwall width in the FeNi layer, when the spacer thickness var-ies. Therefore, Fig. 9shows that, even if the coupling to another layer does not change qualitatively the wall dynam-ics, the values of the wall velocity /H20849and Walker ﬁeld /H20850is af- fected. The effect gets more pronounced with decreasingspacer thickness. The effect of interaction is twofold: on theone hand, ﬂux closure increases the domain wall width of thetransverse wall, hence its mobility increases. On the otherhand, if the second layer is characterized by a large damping,the wall velocity can be strongly reduced. IV. CONCLUSION The domain wall motion in a narrow Fe 20Ni80/Cu /Co spin valve, under the inﬂuence of an applied magnetic ﬁeld,has been studied by micromagnetic simulations. The FeNilayer contained a transverse domain wall /H20849a narrow and thin nanostrip was considered /H20850, and the Co layer was uniformly magnetized along the nanostrip axis. The equilibrium mag-netization distribution in the spin valve shows that the walllocated in FeNi layer widens when compared to a wallwithin a single layer nanostrip, and that a quasiwall is cre-ated in the Co layer. The transverse magnetizations of thewall and quasiwall are antiparallel, as for the coupling be-tween a Néel wall and a quasiwall in an inﬁnite bilayer. Formoderate applied ﬁelds, the wall and the quasiwall move atthe same velocity. The inﬂuence of the second layer on thewall dynamics was found to result from the competition oftwo effects. First, the ﬂux closure widens the transverse walland therefore increases its velocity. However the quality ofthe Co layer is also important, as the wall velocity decreaseswhen the Gilbert coefﬁcient of this layer increases. Theseresults show that the use of a spin valve for the detection, bythe current in plane /H20849CIP /H20850GMR effect, of the domain wall motion in a nanostrip, requires some care in the interpreta-tion of experimental data. Moreover, on top of the purelymagnetostatic coupling considered in this work, indirect ex-change coupling may occur for thinner spacers, and even amagnetization dynamic coupling by spin diffusion across themetallic spin valve thickness. ACKNOWLEDGMENTS The stay of J.M.N. in Orsay was supported by an invited professor fellowship from the Université Paris-Sud 11 and bya cooperation grant from the French government. 1T. Ono, in Handbook of Magnetism and Advanced Magnetic Materials , Micromagnetism Vol. 2, edited by H. Kronmüller and S. Parkin /H20849Wiley, New York, 2007 /H20850, pp. 933–941.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.7 αCoW all mobility (m²/As)L=i n f i n i t y 40 nm20 10 0 FIG. 9. /H20849Color online /H20850Wall mobility as a function of the damping constant of the cobalt layer in a FeNi /H208494n m /H20850/Cu/H20849L/H20850/Co/H208494n m /H20850spin valve nanostrip. The damping parameter for FeNi is /H9251=0.02. Curves were calculated from the equilibrium structure /H20849the Thiele domain wall width in each layer /H20850and the damping coefﬁcients in each layer. Points are the results of numericalsimulations under a ﬁnite applied ﬁeld of 5 Oe /H20849398 A/m /H20850.023905-7 Ndjaka, Thiaville, and Miltat J. Appl. Phys. 105 , 023905 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Thu, 02 Oct 2014 07:41:272L. Thomas and S. Parkin, in Handbook of Magnetism and Advanced Mag- netic Materials , Micromagnetism Vol. 2, edited by H. Kronmüller and S. Parkin /H20849Wiley, New York, 2007 /H20850, pp. 942–982. 3R. P. Cowburn, in Handbook of Magnetism and Advanced Magnetic Ma- terials , Micromagnetism Vol. 2, edited by H. Kronmüller and S. Parkin /H20849Wiley, New York, 2007 /H20850, pp. 983–1002. 4A. Thiaville and Y. Nakatani, in Spin Dynamics in Conﬁned Magnetic Structures III , edited by B. Hillebrands and A. Thiaville /H20849Springer, Berlin, 2006 /H20850, pp. 161–206. 5Y. Nakatani, A. Thiaville, and J. Miltat, Nature Mater. 2, 521 /H208492003 /H20850. 6A. P. Malozemoff and J. C. Slonczewski, Magnetic Domain Walls in Bubble Materials /H20849Academic, New York, 1979 /H20850. 7T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito, and T. Shinjo, Science 284,4 6 8 /H208491999 /H20850. 8J. Grollier, P. Boulenc, V. Cros, A. Hamzi ć, A. Vaurès, A. Fert, and G. Faini, Appl. Phys. Lett. 83, 509 /H208492003 /H20850. 9C. K. Lim, T. Devolder, C. Chappert, J. Grollier, V. Cros, A. Vaurès, A. Fert, and G. Faini, Appl. Phys. Lett. 84, 2820 /H208492004 /H20850. 10J. Grollier, P. Boulenc, V. Cros, A. Hamzi ć, A. Vaurès, A. Fert, and G.Faini, Appl. Phys. Lett. 95, 6777 /H208492004 /H20850. 11J. Miltat and M. Donahue, in Handbook of Magnetism and Advanced Magnetic Materials , Micromagnetism Vol. 2, edited by H. Kronmüller and S. Parkin /H20849Wiley, New York, 2007 /H20850, pp. 742–764. 12R. D. McMichael and M. J. Donahue, IEEE Trans. Magn. 33,4 1 6 7 /H208491997 /H20850. 13A. Thiele, Phys. Rev. Lett. 30, 230 /H208491973 /H20850. 14H. W. Fuller and D. L. Sullivan, J. Appl. Phys. 33, 1063 /H208491962 /H20850. 15S. Middelhoek, Appl. Phys. Lett. 5,7 0 /H208491964 /H20850. 16J. C. Slonczewski and S. Middelhoek, Appl. Phys. Lett. 6, 139 /H208491965 /H20850. 17J. Miltat, G. Albuquerque, and A. Thiaville, in Spin Dynamics in Conﬁned Magnetic Structures I , edited by B. Hillebrands and K. Ounadjela /H20849Springer, Berlin, 2002 /H20850,p .1 . 18G. Albuquerque, J. Miltat, and A. Thiaville, J. Appl. Phys. 89,6 7 1 9 /H208492001 /H20850. 19A. Thiaville, Y. Nakatani, F. Piéchon, J. Miltat, and T. Ono, Eur. Phys. J. BB60,1 5 /H208492007 /H20850. 20J.-Y. Lee, K.-S. Lee, S. Choi, K. Y. Guslienko, and S.-K. Kim, Phys. Rev. B76, 184408 /H208492007 /H20850.023905-8 Ndjaka, Thiaville, and Miltat J. Appl. Phys. 105 , 023905 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Thu, 02 Oct 2014 07:41:27
1.3609236.pdf	Spin Hall effect-driven spin torque in magnetic textures A. Manchon and K.-J. Lee Citation: Applied Physics Letters 99, 022504 (2011); doi: 10.1063/1.3609236 View online: http://dx.doi.org/10.1063/1.3609236 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/99/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Coupled Dzyaloshinskii walls and their current-induced dynamics by the spin Hall effect J. Appl. Phys. 116, 023909 (2014); 10.1063/1.4889848 Domain wall motion driven by spin Hall effect—Tuning with in-plane magnetic anisotropy Appl. Phys. Lett. 104, 162408 (2014); 10.1063/1.4873583 Exchange magnetic field torques in YIG/Pt bilayers observed by the spin-Hall magnetoresistance Appl. Phys. Lett. 103, 032401 (2013); 10.1063/1.4813760 Erratum: “Spin hall effect-driven spin torque in magnetic texture” [Appl. Phys. Lett. 99, 022504 (2011)] Appl. Phys. Lett. 99, 229905 (2011); 10.1063/1.3665407 Magnetization instability driven by spin torques J. Appl. Phys. 97, 10C703 (2005); 10.1063/1.1849591 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.12.234.99 On: Mon, 15 Dec 2014 03:10:20Spin Hall effect-driven spin torque in magnetic textures A. Manchon1,a)and K.-J. Lee2,b) 1Division of Physical Science and Engineering, KAUST, Thuwal 23955, Saudi Arabia 2Department of Materials Science and Engineering, Korea University, Seoul 136-713, Korea (Received 9 April 2011; accepted 17 June 2011; published online 13 July 2011) Current-induced spin torque and magnetization dynamics in the presence of spin Hall effect in magnetic textures is studied theoretically. The local deviation of the charge current gives rise to acurrent-induced spin torque of the form ð1/C0bMÞ/C2½ ð u 0þaHu0/C2MÞ/C1$/C138M, where u0is the direction of the injected current, aHis the Hall angle and bis the non-adiabaticity parameter due to spin relaxation. Since aHandbcan have a comparable order of magnitude, we show that this torque can signiﬁcantly modify the current-induced dynamics of both transverse and vortex walls. VC2011 American Institute of Physics . [doi: 10.1063/1.3609236 ] The study of the interplay between magnetization dynam- ics and spin-polarized currents through spin transfer torque1–5 (STT) has culminated with the observation of current-induced domain wall motion and vortex oscillations, revealing tremen- dously rich physics and dynamical behaviors.6–8The precise nature of STT in domain walls is currently the object of numerous investigations both experimentally and theoreti- cally. One of the important issues is the actual magnitude ofthe so-called non-adiabatic component of the spin torque b. 6–9 In addition, the nature of spin torque in the presence of spin-orbit coupling (SOC) has been recently uncovered.Although SOC has been long known to generate magnetiza- tion damping and spin relaxation, recent studies have sug- gested that speciﬁc forms of structure-induced spin-orbit coupling could act as a source for the spin torque. 10However, in the case of impurity-induced SOC, incoherent scattering averages out the spin accumulation so that no SOC-inducedspin torque can be generated in homogeneous ferromagnets. 11 Nevertheless, SOC-induced asymmetric spin scattering byimpurities in ferromagnetic materials generates anomalousHall effect (AHE), creating a charge current transverse to both the injected electron direction and the local magnetiza- tion. 12Interestingly, the Hall angle aH, deﬁned as the amount of deviated charge current, can be as large as a few percent in thin ﬁlms,12which is on the same order of magnitude as the non-adiabatic coefﬁcient b.6–8Therefore, it seems reasonable to wonder whether anomalous charge currents could have a sizable effect on domain wall velocities. In this Letter, we study the inﬂuence of such a transverse charge current on current-induced domain wall motion. We show that this AHE-induced charge current generates an additional torque component, proportional to the Hall angle,along the direction perpendicular to both the charge injection direction u 0and to the local magnetization Mð/aH ½ðu0/C2MÞ/C1$/C138MÞ. The current-driven magnetization dynam- ics in transverse and vortex walls is analyzed using Thiele formalism. The mechanisms underlying AHE have been studied experimentally and theoretically for more than 60 years (seeRef. 12for a comprehensive review). For the transport re- gime we are interested in (good metal regime, with a conduc-tivity /C2510 4/C0106X/C01cm/C01), the transport is dominated by scattering-independent mechanisms, i.e., intrinsic/side-jump contributions.12Disregarding the effect of band structure- induced SOC, we will treat the spin transport within the ﬁrst order Born approximation, only accounting for the anoma- lous velocity arising from side-jump scattering.13,14 We adopt the conventional s-dHamiltonian, where the electrons responsible for the magnetization and the ones re- sponsible for the current are treated separately and coupledthrough an exchange constant J. We also take into account an impurity potential V impand its corresponding spin-orbit coupling acting on the itinerant electrons. The one-electronHamiltonian reads ^H¼^p2 2mþJ^r/C1Mþn mð^r/C2$VimpÞ/C1^pþVimp;(1) In Eq. (1), the hat ^denotes an operator while the bold charac- ter indicates a vector. ^ris the vector of Pauli spin matrices, nis the spin-orbit strength (as an estimation, n/C2510/C017/C010/C019 s), and Vimp¼Vimpð^rÞis the impurity potential, which is spin- dependent (2 /C22 matrix) in principle. The magnetization direc- tionM(r,t)¼(sinhcos/,s i n hsin/,c o s h)v a r i e ss l o w l yi n time and space, so that the itineran t electron spins closely fol- low the magnetization direction ( adiabatic approximation). In this picture, the velocity operator is14 ^v¼/C0i/C22h m$þn m^r/C2$Vimp: (2) The expectation value of the velocity in the presence of spin- orbit coupling has been worked out by several authors13,14 and can be written h^vi¼1 i/C22hh½^r;H/C138i /C25vþn^Tv/C2^r; (3) ^T¼Rs^IþDs^r/C1M: (4) Here,Rs¼1/s:þ1/s;,Ds¼1/s:/C01/s;,srbeing the spin- dependent electron momentum relaxation time. The form of the anomalous velocity displayed in Eq. (3)is the extensiona)Electronic mail: aurelien.manchon@kaust.edu.sa. b)Electronic mail: kj_lee@korea.ac.kr. 0003-6951/2011/99(2)/022504/3/$30.00 VC2011 American Institute of Physics 99, 022504-1APPLIED PHYSICS LETTERS 99, 022504 (2011) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.12.234.99 On: Mon, 15 Dec 2014 03:10:20of the anomalous velocity derived in Ref. 13to non-collinear magnetic textures. The description of diffusive non-collinear spin transport in ferromagnets has been intensively addressed over the pastten years 15–17using different approaches. As an example, for slowly varying magnetization in the presence of anomalous velocity, Eq. (2), the relaxation time approximation of the Boltzmann formalism yields a spinor current of the form15 J¼ ^Cð^E/C0$^lðrÞÞ þ ^CHð^E/C0$^lðrÞÞ /C2M; (5) where ^C¼1 2C0ð^IþPM/C1rÞis the normal conductance, ^CH¼1 2CH0ð^IþPHM/C1rÞis the anomalous conductance, and^lðrÞis the local spin-dependent electro-chemical poten- tial.Eis the electric ﬁeld and P(PH) is the polarization of the (anomalous) conductivity. This form is very similar tothe one derived by Zhang, 13extended to non collinear mag- netization situations. The charge and spin currents are calcu- lated using the spinor deﬁnition Je¼Tr½J^I/C138;Js¼/C0lB eTr½J^r/C138: (6) In addition, the spin density continuity equation can be extracted from Eq. (1)using Ehrenfest’s theorem and in the lowest order in SOC @tm¼/C0$/C1J s/C01 sJdm/C2M/C0dm ssf; (7) where m¼n0Mþdm(n0is the equilibrium itinerant spin density) and the spin current is deﬁned as: Js¼h h^r/C10^v þ^v/C10^riik,^vbeing the velocity operator deﬁned in Eq. (2). The inner brackets h…idenote quantum mechanical averag- ing and the outer brackets h…ikrefer to k-state average, /C10 being the direct product. By simply injecting the spin current Eq.(6)into Eq. (7), we obtain the explicit spin continuity equation @tm¼lB e½ðPC0EþPHCH0E/C2MÞ/C1$/C138M /C01 sJdm/C2M/C0dm ssf: (8) To obtain a tractable form of the spin torque, we assume P/C25PHandaH¼CH0/C0. After manipulating Eq. (8)(see Ref. 2), the spin torque T¼1 sJdm/C2Min adiabatic approxi- mation reads T¼ð1/C0bM/C2Þ½/C0 n0@tMþbJ½u/C1$/C138M/C138; (9) where bJ¼lBPC0E/e,u¼u0þaHu0/C2M,u0¼C0E/jC0Ej being the injected current direction. One recognizes the renormalization torque ð/@tMÞ, the usual adiabatic and non-adiabatic spin torque2,3(/bJandbbJ), and the AHE- induced torques ( /aHbJand/baHbJ). Note that recently Shibata and Kohno have derived a similar form for the spin torque in magnetic texture in the case of skew scattering.18 In the case of slowly varying magnetic texture ( @tM!0), the spin torque becomes T¼bJð1/C0bM/C2Þ½ðu0þaHu0/C2MÞ/C1$/C138M:(10)To extract the dynamics induced by these additional terms, we analyze the current-driven domain wall motion using Thiele free energy formalism19for rigid domain wall motion /C16 @tM¼ð /C0 v/C1$ÞM/C17 . Thiele’s dynamic equation yields ð dVFþG/C2ðvþbJuÞþD/C1ðavþbbJuÞhi ¼0 F¼$W;G¼/C0Ms cð$h/C2$/Þsinh; Dij¼/C0Ms cð$i/$j/sin2hþ$ih$jhÞ:(11) Here, Frefers to the external force, Gthe gyrocoupling vec- tor, and the Dthe dissipation dyadic exerted on the domain wall. The magnetic energy is W¼A Msð$MÞ2þK MsðM/C2xÞ2 þKd MsðM/C2zÞ2/C0H/C1M, where Ais the ferromagnetic exchange, Msthe saturation magnetization, K(Kd) is the ani- sotropy (demagnetizing) energy, and His the external ﬁeld. Let us ﬁrst consider a magnetic wire along xcontaining an out-of-plane transverse wall deﬁned by hðxÞ¼2arctan esx D;/¼/ðtÞ(s¼61). The external force reduces to F¼2s HzMs/Dz, while the gyrocoupling force vanishes ð$/¼0Þ. The ﬁnal velocity is then v¼1 a/C18 scHzMs/C0bJ/C18 bux/C0aHp 4uzsin//C19/C19 : (12) Interestingly, the AHE-induced spin torque only acts on the domain wall when injecting the current perpendicular to the magnetic wire ( ux¼0,uz¼1). Still, in this latter case, the velocity depends on sin /which is in principle time depend- ent. This quantity can be determined through the Landau-Lifshitz-Gilbert equation @ t/¼cHzþs Dðb/C0aÞbJp 4aHuzsin//C0caHKcos/sin/: (13) Above Walker breakdown ð@t/6¼0;hsin/it¼0Þ, the ve- locity, Eq. (12), does not show any dependence on the per- pendicular current. On the other hand, below Walkerbreakdown ( @ t/¼0), Eq. (13) provides an implicit expres- sion for the angle /. In the absence of external ﬁeld ( Hz¼0) and in the presence of in-plane anisotropy ( HK=0),//C250 and the velocity is directly proportional to the Hall angle: v/C25aH ap 4bJ. Since aHcan be as large as a few percent,12the expected velocity is similar to the once driven by the non-adiabatic spin torque when the current is injected along the structure ( u x¼1,uz¼0). In the absence of in-plane anisotropy ﬁeld ( HK¼0), the domain wall velocity becomes independent of the current density and reduces to v¼s acDHza/C02b a/C0b. This indicates that the anomalous current only distorts the domain wall struc-ture, without inducing any displacement. In contrast with transverse walls, vortex walls present a 2-dimensional texture that couples longitudinal and trans-verse current-induced velocities. 20,21The vortex wall is located at the center of a magnetic layer and in the vortex region, the angles are hr2<r2 0¼2stan/C01r r0;hR2>r2>r2 0¼p=2; (14)022504-2 A. Manchon and K. Lee Appl. Phys. Lett. 99, 022504 (2011) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.12.234.99 On: Mon, 15 Dec 2014 03:10:20sin/r2<R2¼cx r;cos/r2<R2¼/C0cy r; (15) where r¼xexþyey,c(s) is the chirality (polarity) of the vor- tex. In principle, the vortex extends up to a radius R, beyondwhich the domain wall can be modeled as transverse walls. In the present case, we do not consider the action of these outer transverse walls and concentrate on the vortex wallcore dynamics. From Thiele free energy, Eq. (11), we get the coupled equations for longitudinal and transverse velocities aCv x/C0vy¼/C0 bCþaH 2hi bJux; (16) vxþaCvy¼/C0 1/C0baH 2Dhi bJux; (17) where C¼1þ1 2lnq,D¼ 1þln2q 1þq2, and q¼R/r0. This yields the velocities21 vx¼/C01þabC2þaH 2ðaC/C0bDÞ 1þa2C2bJux; (18) vy¼ðb/C0aÞC þaH 2ð1þabCDÞ 1þa2C2bJux: (19) It clearly appears that AHE signiﬁcantly inﬂuences the motion of a vortex core by enhancing the transverse velocity vy. As an illustration, Fig. 1(a) displays the current-induced polar angle of the core, h¼tan/C01vy=vx, as a function of q, for different values of the Hall angle aH. It indicates that the presence of AHE clearly enhances the polar angle by several degrees.Whereas the polar angle is linear as a function of non- adiabaticity [Fig. 1(b)], the inﬂuence of AHE-induced torque can be quite signiﬁcant, especially in the case of sharp vortex core (see Fig. 1(b), inset). These results show that AHE can contribute to more than half of the transverse velocity in the case of current-driven vortex wall motion. In conclusion, we showed that in the presence of SOC, the spin transfer torque exerted on magnetic textures has the general form ð1/C0bMÞ/C2½ ð u0þaHu0/C2MÞ/C1$/C138M. Whereas the additional AHE-induced torque can induce domain wall motion when injecting the current perpendicular to a trans- verse wall, it can also signiﬁcantly affect the velocity of vor-tex cores by increasing the transverse velocity. K.J.L. acknowledges ﬁnancial support from NRF grant funded by the Korea government (MEST) (Grant No. 2010- 0023798). 1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); L. Berger, Phys. Rev. B 54, 9353 (1996). 2S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). 3A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 (2005). 4S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 (2005). 5G. Tatara, H. Kohno, J. Shibata, Y. Lemaho, and K.-J. Lee, J. Phys. Soc. Jpn.76, 054707 (2007). 6C. Burrowes, A. P. Mihai, D. Ravelosona, J.-V. Kim, C. Chappert, L. Vila, A. Marty, Y. Samson, F. Garcia-Sanchez, L. D. Buda-Prejbeanu, I. Tudosa, E. E. Fullerton, and J.-P. Attan, Nat. Phys. 6, 17 (2010). 7M. Eltschka, M. Wotzel, J. Rhensius, S. Krzyk, U. Nowak, M. Klaui, T. Kasama, R. E. Dunin-Borkowski, L. J. Heyderman, H. J. van Driel, and R. A. Duine, Phys. Rev. Lett. 105, 056601 (2010). 8L. Heyne, J. Rhensius, D. Ilgaz, A. Bisig, U. Rudiger, M. Klaui, L. Joly, F. Nolting, L. J. Heyderman, J. U. Thiele, and F. Kronast, Phys. Rev. Lett. 105, 187203 (2010). 9S.-M. Seo, K.-J. Lee, H. Yang, and T. Ono, Phys. Rev. Lett. 102, 147202 (2009). 10A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008); K. Obata and G. Tatara, Phys. Rev. B 77, 214429 (2008). 11A. Manchon and S. Zhang, Phys. Rev. B 79, 094422 (2009). 12N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010). 13S. Zhang, Phys. Rev. Lett. 85, 393 (2000). 14S. K. Lyo and T. Holstein, Phys. Rev. Lett. 29, 423 (1972). 15S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002). 16J. Barnas, A. Fert, M. Gmitra, I. Weymann, and V. K. Dugaev, Phys. Rev. B72, 024426 (2005). 17A. Brataas, Yu. V. Nazarov, and G.E.W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); ibid.,Eur. Phys. J. B 22, 99 (2001). 18J. Shibata and H. Kohno, J. Phys.: Conf. Ser. 200, 062026 (2010). 19A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973). 20J. He, Z. Li, and S. Zhang, Phys. Rev. B 73, 184408 (2006). 21J.-H. Moon, D.-H. Kim, M. H. Jung, and K.-J. Lee, Phys. Rev. B 79, 134410 (2009). FIG. 1. (Color online) (a) Polar angle as a function of qforaH¼0, 0.01, 0.02, 0.05; (b) Polar angle as a function of non-adiabaticity b/aforaH¼0, 0.01, 0.02, 0.05. Inset: Dh¼h(aH¼5%)/C0h(0) as a function of non-adia- baticity for q¼5, 10, 30, 60.022504-3 A. Manchon and K. Lee Appl. Phys. Lett. 99, 022504 (2011) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.12.234.99 On: Mon, 15 Dec 2014 03:10:20
1.4714772.pdf	Fast switching of a ground state of a reconfigurable array of magnetic nano-dots Roman Verba, Gennadiy Melkov, Vasil Tiberkevich, and Andrei Slavin Citation: Appl. Phys. Lett. 100, 192412 (2012); doi: 10.1063/1.4714772 View online: http://dx.doi.org/10.1063/1.4714772 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v100/i19 Published by the American Institute of Physics. Related Articles Spin-spin relaxation of protons in ferrofluids characterized with a high-Tc superconducting quantum interference device-detected magnetometer in microtesla fields Appl. Phys. Lett. 100, 232405 (2012) Suppression of the precessional motion of magnetization in a nanostructured synthetic ferrimagnet Appl. Phys. Lett. 100, 222411 (2012) Spin-orbit field switching of magnetization in ferromagnetic films with perpendicular anisotropy Appl. Phys. Lett. 100, 212405 (2012) Ferromagnetic properties of single walled carbon nanotubes doped with manganese oxide using an electrochemical method Appl. Phys. Lett. 100, 192409 (2012) Modulation of domain wall dynamics in TbFeCo single layer nanowire J. Appl. Phys. 111, 083921 (2012) Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 05 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsFast switching of a ground state of a reconfigurable array of magnetic nano-dots Roman Verba,1,a)Gennadiy Melkov,1Vasil Tiberkevich,2and Andrei Slavin2 1Faculty of Radiophysics, Taras Shevchenko National University of Kyiv, Kyiv 01601, Ukraine 2Department of Physics, Oakland University, Rochester, Michigan 48309, USA (Received 20 March 2012; accepted 26 April 2012; published online 10 May 2012) We show numerically that a ground state (ferromagnetic or chessboard antiferromagnetic) and microwave absorption frequency of a dipolarly coupled two-dimensional array of axially magnetized magnetic nano-dots can be switched by application of bias magnetic ﬁeld pulses (duration 30–70 ns). Switching to the ferromagnetic state can be achieved by applying a rectangular ﬁeld pulse along thedot axis while switching to the antiferromagnetic state requires the ﬁeld pulse oriented in the dot plane and having a sufﬁciently long trailing edge (tail). Our results prove that arrays of magnetic nano-dots can be used as materials having rapidly reconﬁgurable magnetic and microwaveproperties. VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4714772 ] Magnonic crystals (MC)—artiﬁcial structures with peri- odic variation of magnetic parameters—are promising candi- dates for application in microwave technology since the spinwave (SW) spectra and, therefore, the microwave absorption properties of these structures depend on the structure geome- try and can be tailored to the demand. 1–5It is of a particular practical interest to create MCs capable of operating without permanent bias magnetic ﬁeld but having microwave absorp- tion frequency that can be rapidly switched by application of a bias magnetic ﬁeld pulse. This can be achieved if the MC has more than one stable magnetic conﬁguration (ground state) by switching between these states. Possibility of suchdynamical control of MC’s properties has been demonstrated recently in the case of a one-dimensional MC—an array of alternating long magnetic nano-wires of two differentwidths. 6–8In such a case switching between the different ground states is possible under the application of a longitudi- nal (along the wire long axis) bias ﬁeld since the nano-wiresof different widths have different reversal magnetic ﬁelds. This method, however, may not be suitable for practical application because (i) the quasistatic magnetization reversalprocess in a wire is slow and (ii) the spectrum of microwave absorption (or ferromagnetic resonance (FMR)) of the sys- tem has a doublet structure caused by the presence of non-identical magnetic elements. 9 In this letter, we investigate the possibility of controlla- blefastswitching of the microwave absorption frequency in a two-dimensional array of identical magnetic nano-dots coupled by magnetodipolar interaction. The dots are assumed to be cylindrical particles with radius Rand height hmade from a soft magnetic material (e.g., permalloy having saturation magnetization Ms¼8/C1105A=m, gyromagnetic ratio c¼2p/C128 GHz =T, and Gilbert damping constant aG¼0:01) and ordered in a square lattice with lattice con- stant a(Fig. 1(a)). Below we consider dots which are axially magnetized single domain particles in the absence of anexternal magnetic ﬁeld that is realized if dot radius is compa-rable or smaller than material exchange length (typically, several tens of nanometers) and if dot’s aspect ratio h=R>2 (easy-axis shape anisotropy). 10 The two simplest ground states of such an array— ferromagnetic (FM, see Fig. 1(a)) and chessboard antiferromagnetic (CAFM, see Fig. 1(b))—can be stable simultaneously in a wide range of array’s parameters (Fig. 1(d)). The difference of FMR frequencies in these two states signiﬁcantly exceeds the FMR linewidth of Py and canreach several GHz (see Figs. 1(c)and1(d)). 9,11 It is clear that such an array of dipolarly coupled mag- netic dots can be switched to the FM state by applying amagnetic ﬁeld pulse directed along the dot axis and having a sufﬁciently large amplitude. Switching to the CAFM state is more subtle since it is practically impossible to apply pulsesof different directions to different nano-dots. Nonetheless, it will be shown below that using a spatially homogeneous ﬁeld pulse directed in the dot plane and having proper a h 2RM0 FM CAFM(a) 2468 1 010 8 64 2 01GHz1GHz 2GHz2GHz 3GHz3GHza xyz f21(b) c)((d) FIG. 1. (a), (b) Sketch of the considered array in the FM and CAFM ground states, respectively. (c) FMR power absorption spectra of the array in theFM and CAFM states (array parameters: h/R¼5,a/R¼5). (d) Difference of the FMR frequencies f 21in two different ground states as a function of geo- metric parameters a/Randh/R. The region where one or both ground states are unstable is dashed.a)Author to whom correspondence should be addressed. Electronic mail:verrv@ukr.net. 0003-6951/2012/100(19)/192412/3/$30.00 VC2012 American Institute of Physics 100, 192412-1APPLIED PHYSICS LETTERS 100, 192412 (2012) Downloaded 05 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsparameters, the array can be switched into an almost ideal CAFM state. The switching dynamics in a coupled dot array is inves- tigated by numerical integration of a system of Landau-Lif-shitz-Gilbert equations for magnetization vectors M jof each dot ( jis the dot number) in a macrospin approximation so that the magnetization distribution in each dot is assumed tobe uniform dMj dt¼/C0cMj/C2Beff;j/C0aGc MsMj/C2ðMj/C2Beff;jÞ;(1) with the effective ﬁeld Beff;j¼BeðtÞ/C0l0MsX k^Njk/C1MkþBTðtÞ: (2) Here, BeðtÞis the external magnetic ﬁeld, ^Njkis the mutual demagnetization tensor,9,12the random thermal ﬁeld BTðtÞ represents an isotropic vector Gaussian white noise with var- iance /C232¼2aGkBT=ðcMsVÞ;kBis the Boltzmann constant, T is the absolute temperature, and V¼pR2his the dot volume. Equation (1)is solved using mid-point rule technique.13,14 SW spectra are calculated by the method described in Ref. 9. For all further calculations, the following parameters are used: material parameters of Py, dot aspect ratio h/R¼5, lat- tice constant a¼5R,/C23¼6/C210/C09T/C2s1=2(which corre- sponds to the room temperature T¼300 K for dots with R¼10 nm), and sizes of simulated array—20 /C220 dots with periodic boundary conditions. First, we consider switching of the array into the FM ground state. This can be achieved by applying out-of-plane ﬁeld pulse of a sufﬁciently large magnitude, at which onlythe ideal FM state remains stable. The critical ﬁeld magni- tude B c;zis determined from the condition that an isolated defect (a single dot with an opposite magnetization direction)in an FM matrix loses its stability B c;z¼l0MsðNxx sþFzz 0/C02Nzz sÞ: (3) Here, ^Ns¼^Njjis the self-demagnetization tensor of a dot and^Fkis the array’s demagnetization tensor, deﬁned in Ref. 9. The duration of the ﬁeld pulse required to achieve the switching is mainly determined by the time needed to excitea large-angle precession from the thermal level. It can be estimated as s z’lnðBc;zMsV=kBTÞ=½2aGcðBz/C0Bc;zÞ/C138. For the considered system Bc;z¼0:355 T and sz¼38 ns at Bz¼0:4 T. This estimated value of szis close to the value sz/C2530 ns obtained from the numerical simulations (Fig. 2(a)). The magnitude of the switching ﬁeld can be reduced using the well-known method in one-particle reversal— applying a tilted ﬁeld.15,16One may apply an in-plane ﬁeld pulse with the magnitude Bc;x(see Eq. (4)below), sufﬁcient to magnetize all dots in plane, and a small out-of-plane pulse. The out-of-plane pulse should be longer than the in-plane one to avoid the random reversal caused by strong magnetostatic interaction between the dots. As an example, we simulated the reversal dynamics under the simultaneousapplication of the B x¼Bc;x¼0:248 T pulse with duration 20 ns and the Bz¼0:075 T pulse with duration 30 ns(Fig. 2(f)). This pulse sequence successfully switches the array to the ideal FM state (irrespectively of the initialarray’s conﬁguration) and reduces the ﬁeld magnitude jB ej by more than 25% compared to a single Bz-pulse. Now we will consider the switching of the dot array into a CAFM state. Since this state corresponds to the global energy minimum (the true ground state) in a zero external ﬁeld,17one may expect that after putting the array into an in- plane state (which corresponds to unstable saddle point) the array with high probability will relax into the CAFM state. The critical ﬁeld Bc;xneeded to magnetize all the dots in the in-plane direction can be found as a ﬁeld at which the in-plane FM state becomes unstable. For the considered geometry this instability occurs at the wavevectork¼pe x=aþpey=a, which exactly corresponds to the CAFM periodicity, and the critical ﬁeld is equal to Bc;x¼l0MsðFxx 0/C0Fzz jÞ (4) independently of the angle between the ﬁeld and the basis lattice vectors. To avoid the inﬂuence of the initial state (“memory effects”) the duration of the Bx-pulse should be of the order of sx’ln½ðBx/C0Bc;xÞMsV=kBT/C138=½2aGcðBx/C0Bc;xÞ/C138 /C2420/C030 ns, which will ensure the relaxation of magnetiza- tion down to the thermal level. A typical remanent state of a dot array after the applica- tion of a rectangular in-plane ﬁeld pulse ( Bx¼0:275 T ; sx¼20 ns) is shown in Fig. 2(d). The total magnetic moment in this state vanishes, and the short-range correlations are of the CAFM type (on average, the magnetizations of the near- est neighbors are opposite), but on a large scale this statelooks nothing like the ideal CAFM state (compare Figs. 2(b) and2(d)). The reason for such a behavior is that the CAFM state is double-degenerate—two different, but equivalentconﬁgurations are related by inversion of the magnetization of all dots. When the applied in-plane ﬁeld is abruptly removed, many local independent clusters with CAFM FIG. 2. Shapes of the bias magnetic ﬁeld pulses (a), (f) and remanent states of the array after their application (b)-(e). Duration of the out-of-plane ﬁeld pulses sz¼30 ns, rectangular part of the in-plane pulses sx¼20 ns, duration of the trailing edge (tail) sf¼50 ns. Dashed lines represent critical ﬁelds for transition into the out-of-plane FM (blue) and in-plane FM (red) states. In (e), one cluster of dots with the ideal CAFM periodicity is shown by greenbackground; other dots form a different ideal CAFM cluster.192412-2 Verba et al. Appl. Phys. Lett. 100, 192412 (2012) Downloaded 05 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsperiodicity appear, in which one of the two possible CAFM conﬁgurations is realized. The relaxation of each of the CAFM clusters towards its local energy minimum is very fast due to the large uncompensated magnetodipolar ﬁelds.Therefore, the collection of many small CAFM-clusters in the remanent state shows the short-range order of the CAFM-type, but the long-range CAFM order is absent. To promote formation of the long-range CAFM order, one can use magnetic ﬁeld pulses with a slowly decreasing amplitude, i.e., pulses with relatively long trailing edge (tail) (see Fig. 2(f)). In the unstable region B x<Bc;x, the instanta- neous growth rate of ﬂuctuations near the in-plane FM stateis proportional to ðB c;x/C0BxðtÞÞ. For slowly decreasing applied ﬁeld BxðtÞ, the magnetization ﬂuctuations grow much slower, and the different local CAFM clusters willhave more time to adjust their mutual conﬁgurations. As a result, one may expect that the average size of a CAFM clus- ter in the remanent state will increase. These qualitative arguments are supported by the results of our numerical simulations. Figure 2(e)shows a typical re- manent state of a dot array after the application of an in-planeﬁeld pulse with an exponential tail B xðtÞ/C24expð/C0t=sfÞfor sf¼50 ns. One can clearly see that there are only two CAFM clusters with a large number of dots in each of them. Simula-tions performed for different s fshowed that the average size of a remanent CAFM cluster monotonically increases with the increase of sf, up to the point (in order of 200 ns) when the whole computational region (20 /C220 dots) becomes ﬁlled with a single CAFM cluster. In a real-life situation, with arrays consisting of millions of dots, it would be, perhaps, practically impossible to achieve the ideal CAFM state, and the remanent state will always con- tain many CAFM clusters, which would lead to a certain inho-mogeneous broadening of the FMR peak. To study this effect, we calculated the FMR absorption spectra in the remanent state for different duration s fof the ﬁeld pulse trailing edge (tail) (see Fig. 3). If the in-plane ﬁeld decreases abruptly (sf¼0), the resulting FMR spectrum has an asymmetric shape with a broad maximum shifted about 1 GHz below theFMR frequency f CAFM of the ideal CAFM state (Fig. 3(a)). The FWHM linewidth of the absorption spectrum is about 8 times larger than the linewidth Df0¼2aGfof the ideal structure9(Fig. 3(b)). With increasing sfthe FMR spectrum narrows and becomes more symmetric (Fig. 3(a)). At sf ¼50 ns the absorption spectrum has only minor differences compared to the ideal case, and the FMR linewidth exceeds Df0only by a few percents (see Fig. 3(b)). Thus, the micro- wave properties of such remanent states are practically indis-tinguishable from the properties on the ideal CAFM ground state. In conclusion, we demonstrated that the magnetic ground state and the microwave absorption frequency of an array of dipolarly coupled magnetic nano-dots can be switched by short magnetic ﬁeld pulses with a typical dura-tion of several tens of nanoseconds. While one can easily switch the array into the ideal FM state, switching into theCAFM state is more difﬁcult, and the remanent state of the dot array will always contain clusters with two possibleCAFM conﬁgurations. The size of the clusters increases with the increase of the tail duration s fof the switching pulse, and forsf/C2150 ns, the microwave response of the array becomes practically indistinguishable from the response of an ideal periodic structure. This work was supported in part by the grant DMR- 1015175 from the National Science Foundation of the USA,by the grant from DARPA, by the Contract from the U.S. Army TARDEC, RDECOM, by the Grant No. M/90-2010 from the Ministry of Education and Science of Ukraine, andby the Grant No. UU34/008 from the State Fund for Funda- mental Research of Ukraine. 1G. Gubbiotti, S. Tacchi, G. Carlotti, N. Singh, S. Goolaup, A. O. Adeyeye, and M. Kostylev, Appl. Phys. Lett. 90, 092503 (2007). 2A. V. Chumak, A. A. Serga, B. Hillebrands, and M. P. Kostylev, Appl. Phys. Lett. 93, 022508 (2008). 3A. V. Chumak, A. A. Serga, S. Wollf, B. Hillebrands, and M. P. Kostylev, Appl. Phys. Lett. 94, 172511 (2009). 4S. Neusser and D. Grundler, Adv. Mater. 21, 2927 (2009). 5G. Gubbiotti, S. Tacchi, M. Madami, G. Carlotti, A. O. Adeyeye, and M. Kostylev, J. Phys. D: Appl. Phys. 43, 264003 (2010). 6J. Topp, D. Heitmann, M. P. Kostylev, and D. Grundler, Phys. Rev. Lett. 104, 207205 (2010). 7S. Tacchi, M. Madami, G. Gubbiotti, G. Carlotti, S. Goolaup, A. O. Adeyeye, N. Singh, and M. P. Kostylev, Phys. Rev. B 82, 184408 (2010). 8J. Ding, M. Kostylev, and A. O. Adeyeye, Phys. Rev. Lett. 107, 047205 (2011). 9R. Verba, G. Melkov, V. Tiberkevich, and A. Slavin, Phys. Rev. B 85, 014427 (2012). 10K. L. Metlov and K. Yu. Guslienko, J. Magn. Magn. Matter. 242–245 , 1015 (2002). 11P. V. Bondarenko, A. Yu. Galkin, B. A. Ivanov, and C. E. Zaspel, Phys. Rev. B 81, 224415 (2010). 12M. Beleggia and M. De Graef, J. Magn. Magn. Mater. 278, 270 (2004). 13M. d’Aquino, C. Serpico, and G. Miano, J. Comp. Phys. 209, 730 (2005). 14M. d’Aquino, C. Serpico, G. Coppola, I. D. Mayergoyz, and G. Bertotti, J. Appl. Phys. 99, 08B905 (2006). 15L. He, W. D. Doyle, and H. Fujiwara, IEEE Trans. Magn. 30, 4086 (1994). 16M. Bauer, J. Fassbender, B. Hillebrands, and R. L. Stamps, Phys. Rev. B 61, 3410 (2000). 17J. E. L. Bishop, A. Yu. Galkin, and B. A. Ivanov, Phys. Rev. B 65, 174403 (2002).FIG. 3. (a) Absorption spectra in a remanent state after applying in-planeﬁeld pulses with different tail durations s f(averaged over 5 simulations); (b) dependence of the linewidth Dfin the dot absorption spectra on the tail dura- tion sf. Pulse amplitude Bx¼0:275 T, duration of the rectangular part sx¼20 ns, initial state of the array—FM. Dashed line in (b) is a guide to the eye.192412-3 Verba et al. Appl. Phys. Lett. 100, 192412 (2012) Downloaded 05 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions
1.4968543.pdf	Voltage-controlled magnetization switching in MRAMs in conjunction with spin-transfer torque and applied magnetic field Kamaram Munira , Sumeet C. Pandey , Witold Kula , and Gurtej S. Sandhu Citation: J. Appl. Phys. 120, 203902 (2016); doi: 10.1063/1.4968543 View online: http://dx.doi.org/10.1063/1.4968543 View Table of Contents: http://aip.scitation.org/toc/jap/120/20 Published by the American Institute of Physics Voltage-controlled magnetization switching in MRAMs in conjunction with spin-transfer torque and applied magnetic field Kamaram Munira, Sumeet C. Pandey, Witold Kula, and Gurtej S. Sandhu Emerging Memory Technology Development, Micron Technology, Inc., Boise, Idaho 83707, USA (Received 23 June 2016; accepted 10 November 2016; published online 28 November 2016) Voltage-controlled magnetic anisotropy (VCMA) effect has attracted a signiﬁcant amount of attention in recent years because of its low cell power consumption during the anisotropymodulation of a thin ferromagnetic ﬁlm. However, the applied voltage or electric ﬁeld alone is not enough to completely and reliably reverse the magnetization of the free layer of a magnetic random access memory (MRAM) cell from anti-parallel to parallel conﬁguration or vice versa. An addi-tional symmetry-breaking mechanism needs to be employed to ensure the deterministic writing process. Combinations of voltage-controlled magnetic anisotropy together with spin-transfer torque (STT) and with an applied magnetic ﬁeld ( H app) were evaluated for switching reliability, time taken to switch with low error rate, and energy consumption during the switching process. In order to get a low write error rate in the MRAM cell with VCMA switching mechanism, a spin-transfer torque current or an applied magnetic ﬁeld comparable to the critical current and ﬁeld of the free layer isnecessary. In the hybrid processes, the VCMA effect lowers the duration during which the higher power hungry secondary mechanism is in place. Therefore, the total energy consumed during the hybrid writing processes, VCMA þSTT or VCMA þH app, is less than the energy consumed during pure spin-transfer torque or applied magnetic ﬁeld switching. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4968543 ] I. INTRODUCTION In magnetic random access memory (MRAM), informa- tion is stored in the free layer of a magnetic tunnel junction (MTJ).1An insulating spacer layer separates the ﬁxed and free ferromagnetic layers (Fig. 1). The stored data can be sensed by measuring the tunneling resistance. In order to compete with existing memory technology as a non-volatile candidate, an MRAM bit cell should have high tunneling magnetoresistance, and the free layer must have enough ther- mal stability ( >60) against stochastic thermal switching to retain data for at least ten years.2To write information to the MRAM cell, the spin orientation of the free layer needs to be manipulated. The writing process should be fast (within 10–20 ns) and have low power ( <0.1 pJ) with a very low error rate (10/C09). In the ﬁrst generation of MRAMs, a write operation was performed by an external magnetic ﬁeld induced by ﬁeld lines.3,4This inductive writing has high power consumption and a high probability for cross-interference as the devices are scaled down. The free layer can also be switched by using the spin- transfer torque (STT) where the angular momentum from a spin-polarized current is directly transferred to the free layer.5,6With its ability to write and read faster, smaller cell sizes than ﬁrst generation MRAM, and non-volatility, STT- MRAM technology has attracted a lot of interest in the lasttwo decades. 7,8In particular, magnetic tunnel junction stacks, consisting of a MgO barrier and CoFeB magnetic layers, have attracted considerable attention, thanks to their combination of high perpendicular magnetic anisotropy and high tunneling magnetoresistance.9,10An STT-MRAM cell has a 1-MTJ/1-CMOS transistor memory cell structure, where the CMOS transistor acts as the access device.11As the device dimensions are scaled by j, the voltage and current are also scaled by j.12As the size of the MTJ is scaled to under 20 nm, the switching current density will increase if the thermal stability is to be kept the same. STT-MRAM technology will ultimately be limited by the scaling of the CMOS technology. A CoFeB free layer with a damping of about 0.005 would be able to be scaled down to 40 nm for the 1-MTJ/1-T architecture of a 1 Gb MRAM chip with data retention of 10 years.13Since MTJs are highly resistive devices, the energy consumed by writing can be several tens of fJ due to ohmic heating.14Due to the roadblocks in scaling and highly inefﬁcient energy usage during the writing process, it is quite important to explore other writing schemes in MRAM devices to reduce the write power consumption. Another method to manipulate the magnetization of the free layer is through an electric ﬁeld applied across the MTJ.15–17Pure voltage-controlled magnetic anisotropy (VCMA) switching in precessional regime has also been stud- ied where the electric ﬁeld is turned off at an optimal time when the precessing magnetization is tilted toward the intended destination.22,23The voltage-controlled magnetic anisotropy (VCMA) mechanism does not cause a static rever-sal of magnetization directly, only a modulation of the anisot- ropy. At best, the perpendicular anisotropy of the free layer is canceled, and the magnetization prefers to lay in-plane in the presence of the electric ﬁeld (Fig. 1). Therefore, additional driving forces need to be supplied for bidirectional switching. The estimated energy required for a single domain reversal is around 0.01 fJ, 17which is three orders of magnitude lower 0021-8979/2016/120(20)/203902/8/$30.00 Published by AIP Publishing. 120, 203902-1JOURNAL OF APPLIED PHYSICS 120, 203902 (2016) than that required by spin-transfer torque switching. Spin transfer torque18,19and externally applied ﬁelds20,21,35,36as the driving forces have been extensively studied. However, the reliability of the combined processes has not been properly determined. Ref. 21has shown that with a static magnetic ﬁeld, a free layer with a thermal stability of 33 can beswitched with an error rate of 0.1%. The reference shows thaterror rate can be further reduced by lower damping. In this paper, using a simple macrospin Landau- Lifshitz-Gilbert model, we study four types of combinedswitching mechanisms, where the VCMA effect is used tonegate the perpendicular anisotropy of the free layer and aspin-transfer torque (STT) current or a magnetic ﬁeld ( H app) normal to the free layer is utilized to bias the switching mag-netization towards the correct direction. The four switchingmechanisms were studied in this paper in terms of switching reliability, the time taken to reverse the magnetization, and the energy consumption during the switching process.Parasitic RC components do affect the energy consumptionand delay; in our simple model, we chose to neglect thoseeffects. The four switching mechanisms are: (A) combinedVCMA þSTT; (B) VCMA effect used to modulate the anisotropy so the magnetization of the free layer with per-pendicular anisotropy is switched in plane, and after the elec-tric ﬁeld is turned off, a spin transfer torque is introduced to drive the magnetization of the free layer to its ﬁnal orienta- tion; (C) combined VCMA þH app; and (D) as (B), but the STT is replaced by Happ. In Section II, the electric ﬁeld efﬁciency of the anisot- ropy modulation in the VCMA switching is discussed. The best electric ﬁeld efﬁciencies available in the literature arereviewed, and the lowest efﬁciency number required to can-cel the perpendicular anisotropy of the free layer with a ther-mal stability of 60 is estimated. In Section III, the model used to study the switching mechanisms is described. Theassumptions made in the model for switching mechanism Band the voltage polarities for STT and VCMA are discussed.The performances of the four types of switching proﬁle areevaluated in Section IV, and the summary of switching reli- ability and error rates, and energy consumption are listed inTables IandII. II. ELECTRIC FIELD EFFICIENCY IN VOLTAGE-CONTROLLED MAGNETIC ANISOTROPY The VCMA mechanism can be explained by the spin- dependent charge accumulation/depletion effect at freelayer-insulator interface. 24,25It is important that the VCMA effect is as strong as possible so the perpendicular anisotropyof the free layer can be negated at a voltage below the break-down voltage of the MTJ. The efﬁciency is deﬁned as b¼ dKef ftfre e E; (1) where dKeffis the change in anisotropy for the change in electric ﬁeld E and tfreeis the thickness of the free layer.26 The sign of the VCMA efﬁciency is not as robust as spin- transfer torque where electron ﬂow from the ﬁxed layer to the free layer will favor the parallel conﬁguration while elec- tron ﬂow from the free layer to the ﬁxed later will favor theanti-parallel conﬁguration. The size and sign of the VCMAeffect are found to be dependent on the free layer material,the capping layer, 27and the insulator.28For any sign of the applied electric ﬁeld, anisotropy may increase, decrease, orstay the same. 29In the simple model studied in this work, we assume that a positive voltage applied across the MTJ willswitch decrease the perpendicular anisotropy. Insulators with a higher dielectric constant have a higher VCMA efﬁciency due to an increased dwell time for the elec-trons at the interface. The timescale of the switching is around20 ps. 17A high VCMA efﬁciency has been reported in epitaxi- ally grown V/Fe/MgO layers with bof 1150 fJ/Vm.30As s h o w ni nF i g . 2, for such a high efﬁciency, a voltage of 0.5 V is required for a change of anisotropy, dKeff,o f5 . 7 5 /C2105J/m3. The best number for VCMA efﬁciency available in the litera- ture for a fabricated CoFeB-MgO junction is 90 fJ/V m,13 which is not high enough to switch the magnetization of afree layer with a thermal stability of 60 (45 nm diameter,1 nm thick free layer, and saturation magnetization of1.1/C210 6A/m) within 1 V. A bof at least 312 fJ/V m is needed to switch the free layer within 0.5 V. As we will seein Section III, a much higher bis required ( /C241600 fJ/V m) to switch efﬁciently within 10 ns. The energy consumed duringthe VCMA process is 0.5 CV 2þI2Rtwhere the capacitance of the junction is taken to be 0.1 fF (the measured capaci-tance of a typical MTJ at Micron Technology, Inc.) and V isthe applied voltage, I is the tunneling current, R is the resis-tance, and t is amount of time for which the voltage source isapplied. If the insulator thickness is more than 2.5 nm toblock the tunneling current, I 2Rtis negligible, and the total energy dissipation can be calculated as 0.5 CV2(Fig. 2(b)). III. MAGNETIZATION DYNAMICS OF THE FREE LAYER The thermal stability of the free layer with perpendicular anisotropy can be calculated by FIG. 1. Parallel to anti-parallel switching in an MRAM cell. The magnetiza- tion of the free layer is represented by the polar angle, h, which is measured from the positive Z axis. When the free layer and the ﬁxed layer are in paral- lel conﬁguration, his zero. In the presence of an electric ﬁeld, the voltage- controlled magnetic anisotropy mechanism modulates the perpendicular anisotropy of the free layer and, at best, rotates the magnetization of the free layer to in-plane conﬁguration ( h¼p/2). A second mechanism is needed to guide the magnetization to h¼p. In this study, the switching is considered a success when the angle hcrosses 3 p/4.203902-2 Munira et al. J. Appl. Phys. 120, 203902 (2016) D¼l0Hef f KMSVol 2kBT; (2) where l0is the vacuum permeability, Vol is the volume of the free layer, MSis the saturation magnetization, kBis the Boltzmann constant, and T is the temperature. Hef f K¼ HK/C0l0MSis the effective anisotropy ﬁeld resulting from the crystalline anisotropy, the interfacial anisotropy of thefree layer with the insulator, and the demagnetizationﬁeld. 31Under the macrospin and zero temperature assump- tions, the critical spin-transfer torque current needed to switch is IC¼2qal0Hef f KMSVol /C22hg¼4qaDkBT /C22hg; (3) where ais the Gilbert damping constant, q is the electronic charge, and gis the spin polarization of the current exerting the torque on the free layer. While we desire the device tohave high thermal stability, the critical current and the timeneeded to switch also increase with the thermal stability.Since the cells have to switch within a short time with low write error probability, a current greater than the critical cur- rent, I/C29I C, is needed to switch in the precessional regime.31 The critical applied ﬁeld needed to switch the free layer is Hef f K. The free layer was modeled using a macromagnetic sim- ulator that solves the time-dependent Landau-Lifshitz-Gilbert equation in the single-domain approximation 32 d^m dt¼/C0c^m/C2Hef f KþHapp/C16/C17 /C0ca^m/C2^m/C2Hef f KþHapp/C16/C17 /C0cg/C22hJ ql0MStfre e^m/C2^m/C2^mp; (4) where cis the gyromagnetic ratio and ~Happis the applied ﬁeld. J is the applied spin current density and ^mand ^mpare the magnetization of the free and spin transfer torque current. ^m/C2^m/C2^mPis the Slonczewski spin-transfer torque term that induces the switching. The thermal perturbation is repre-sented by a Langevin random ﬁeld ~H Lthat can be added to the effective magnetic ﬁeld term. The ﬁeld ~HLis related to the system temperature T by~HL¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2akBT l0cMSVols ~G; (5) where ~Gis Gaussian white noise with a mean of 0 and a stan- dard deviation of 1.33T is taken to be 300 K in this study. Each switching mechanism was studi ed by performing 500 000 simu- lations where the starting magnetization of the free layer was chosen from a distribution of ini tial magnetizations that included the effects of thermal perturbation at 300 K. The mean of the ini- tial magnetization distribution is around 0 rad (spin up). The geometric and magnetic parameters of the free layer are chosen to produce a thermal stability Dof 60 with a cir- cular disk of 45 nm in diameter and 1 nm thick. The satura- tion magnetization and damping constant are taken to be that of CoFeB, 1.1 /C2106A/m and 0.01, respectively.34The anisotropy of the disk is to be 9.16 /C2105J/m3, making the effective anisotropy, Keff, to be 1.56 /C2105J/m3. The energy barrier of the circular disk can be expressed as Keffsin2hVol. The VCMA was implemented in the simulator by makingthe magnitude of the anisotropy time dependent. The equa- tions below show the anisotropy shape over one temporal period t VCMA .triserepresents the time the signal takes to increase or decay exponentially KtðÞ¼Kef f/C0dKef f1/C0e/C0t trise/C0/C1 /C0l0 2M2 Sfort<tVCMA;(6) KtðÞ¼Kef f/C0dKef fe/C0t/C0tVCMA trise/C16/C17 /C0l0 2M2 Sfort>tVCMA:(7) Fig. 3shows the time-dependent anisotropy K(t) for tVCMA of 8 ns and triseof 200 ps. In our simple model, the STT voltage polarity is ﬁxed in such a way to ensure spin up to down switching. For the same voltage polarity as the STT, an electric ﬁeld across the MTJ will switch the anisotropy from perpendicular to in- plane. As discussed in Section II, anisotropy modulations with electric ﬁeld in MTJs are much more complicated. Switching is considered a success when the magnetiza- tion passes 3 p/4 rad (Fig. 1), with the ﬁnal state being around prad. Switching is considered a success when magnetization crosses p/2 rad during the write process in STT- MRAMs.31,40The criteria are made stricter in this study to FIG. 2. (a) Change in the perpendicular anisotropy, dKeff, with respect to the applied voltage for various electric ﬁeld efﬁciency values. An electric ﬁeld sensi- tivity of 312 fJ/V m is needed to cancel the perpendicular anisotropy of a 45 nm diameter and 1 nm thick free layer with a thermal stability of 60 and saturationmagnetization of 1.1 /C210 6A/m. (b) The energy (0.5 CV2) consumed during the VCMA process.203902-3 Munira et al. J. Appl. Phys. 120, 203902 (2016) ensure that the magnetization is being properly guided by the second switching force and is headed in the right direction.In the case of low spin-transfer torque current or appliedmagnetic ﬁeld, even after crossing the plane, thermal pertur-bation can kick back the magnetization to the starting point.Therefore, it is safer to have stricter criteria for successfulswitching. Pure spin-transfer torque switching is also held tothe stricter standard for fair comparison in Section IV B. We are aware that switching mechanism B is unrealistic since is it not possible to separate the VCMA and the STTwhen a voltage source is applied across the MTJ for thininsulators (1 nm) in a simple free layer-insulator-ﬁxed layerMTJ. In Refs. 35and36, the thickness of the insulator MgO was set to be greater than 2.5 nm to suppress the STT effectin order to study VCMA þHapp contribution to the switch- ing. We decided to separate the two effects intentionally inswitching mechanism B to study the error rates in this paper,for reasons explained in Section IV B . IV. RESULTS AND DISCUSSION A. Switching with voltage-controlled magnetic anisotropy effect While the low energy switching of the VCMA mecha- nism is highly desirable, it is important that the magnetiza-tion of the free layer switches in-plane within a reasonabletime. Fig. 4(a) shows the maximum, minimum, and average times for dK eff, ranging from 2 /C2105J/m3to 8/C2105J/m3.When dKeffis 3/C2105J/m3(enough to cancel the perpendicu- lar anisotropy of a free layer with Keffof 1.56 /C2105J/m3, thermal stability of 60, and saturation magnetization of1.1/C210 6A/m), the average time to switch in-plane is 13 ns, with the maximum being near 34 ns and the minimum near6 ns. This delay is quite high when compared to the 10 nsswitching time for DRAM or 70 ps for SRAM. 2Also, there is a big variation in the switching time (6 ns to 34 ns), as seen in Fig. 4(b). In order to switch within an average of 5 ns, a dKeffvalue greater than 6 /C2105J/m3is needed. When dKeffis increased to 8 /C2105J/m3, the standard deviation of the switching time spread reduces quite considerably. FordK eff¼8/C2105J/m3, the maximum time for is 11 ns, the average time is 3 ns, and the minimum time is 2 ns. As MRAM devices are further scaled down, Keffof the free layer will need to be increased to maintain thermal sta-bility. Materials will be needed with ever higher electric ﬁeldefﬁciency. B. Switching mechanisms A and B: VCMA1spin-transfer torque In switching mechanism A, a voltage is applied across the MTJ with a 1 nm insulator (spin-transfer torque effectwill not be blocked). Both VCMA and STT effects will tryto rotate the magnetization. For VCMA, the desired endpoint is at h¼p/2 rad (Fig. 1), and for STT, it is at h¼prad. In Fig. 5(a), for combined VCMA þSTT switching mecha- nism, where bis 1600 fJ/V m at 0.5 V ( dK eff¼8/C2105J/m3) and I/IC¼0.72, the magnetization distribution does not switch completely. Because of the high electric ﬁeld efﬁ-ciency and high dK eff, the VCMA effect is much stronger than STT. Therefore, the energy proﬁle of the free layer hasits minimum at the angle p/2 rad. The mean of the distribu- tion at 50 ns is at this angle. If the voltage source is removed, half of the distribution will switch to the intended destinationwhile the other half will return to the origin. In Fig. 5(b), an electric ﬁeld efﬁciency of 312 fJ/V m is used. The VCMA effect is not overpowering the STT in thiscase and is just strong enough to cancel the perpendicularanisotropy. dK effis 1.56 /C2105J/m3,2 . 1 8 /C2105J/m3,a n d 3.12/C2105J/m3at 0.5, 0.7, and 1 V, respectively ( I/IC¼0.72, FIG. 3. Time-dependent anisotropy Keff(t) for a tVCMA of 8 ns and a triseof 200 ps. FIG. 4. (a) The maximum, minimum,and average times taken for the magne- tization distribution of the free layer to rotate to the in-plane orientation for various dK effvalues including the effect of thermal perturbations at 300 K. In order to switch to the in-plane conﬁgu-ration within an average of 5 ns, a dK eff value greater than 6 /C2105J/m3is needed. (b) Time taken to switch to the in-plane orientation when dKeffis 3/C2105J/m3and 8/C2105J/m3.203902-4 Munira et al. J. Appl. Phys. 120, 203902 (2016) 1, and 1.5). Ideally, the slope of the error probability vs. time curve determines the speed of the switching. A steeper slopewill indicate faster switching. The reported slope for pure STT as shown in Fig. 5(b) is less steep than that observed experimentally. 37This is because our simple model is assum- ing the free layer to be a single ferromagnetic domain.Therefore, mechanisms that aid switching, e.g., domain nucle-ation, and in-coherent switching, are not taken into account. Amicromagnetic model, which is outside the scope of thispaper, is needed to study such complicated mechanisms. 38 When compared to pure STT switching, the combined mechanism cuts down the STT current by half. For pure STTswitching with I/I C¼3, error rate is 10/C09at 11 ns. For VCMA þSTT hybrid switching mechanism, an I/ICof 1.5 anddKeffof 3.12 /C2105J/m3will deliver the same low write error rate at 20 ns. As the STT current required from the combined switch- ing effects of the VCMA and STT in (A) is still high, wedecided to separate the two processes in (B) (Figs. 6(a) and 6(b)). Separating the two processes will enable them to per- form their intended functions without interference from theother. When the two processes are separated, VCMA will beused to rotate the magnetization from the starting point to in-plane orientation, and then, STT will be used to nudge themagnetization of the free layer toward the ﬁnal destination.Also, in this switching mechanism, the more power-hungrySTT will not be used until required. An effective ﬁeld is applied across the tunnel junction for a t VCMA of 3 and 8 ns andtriseof 200 ps with dKeffof 8/C2105J/m3(b¼1600 fJ/V m at 0.5 V). The spin-transfer torque current, I, is applied FIG. 5. (a) Combined VCMA and STT switching mechanism A. A voltage of 0.5 V is applied across the tunnel junc- tion for b¼1600 fJ/V m and I/ IC¼0.7. Since the VCMA effect overpowers theSTT, the magnetization prefers to be in in-plane orientation with the mean of distribution at h¼p/2. (b) Switching mechanism A for b¼312 fJ/V m at 0.5 V, 0.7 V, and 1 V and I/I C¼0.7, 1, and 1.5, respectively, and pure STT switching for I/IC¼1.5, 2, and 3. An electric ﬁeld sensitivity of 312 fJ/V m reduces the STT current needed by half for comparable performance. FIG. 6. (a) Switching mechanism B for b¼1600 fJ/V m at 0.5 V. The electric ﬁeld is turned on for tVCMA of 3 ns. Spin transfer torque current, I, is applied immediately afterward. The error rate curves saturate after a certain point. (b) tVCMA increased to 8 ns. The dotted lines in the plot for tVCMA¼8n s are approximations.TABLE I. The write error rates and energy consumption for pure STT and switching mechanisms A and B. tVCMA indicates the amount of time the elec- tric ﬁeld is applied across the MTJ. It is not applicable in the case of pure STT switching. ‘ON’ indicates that the electric ﬁeld is on the entire time (switching proﬁle A). Delay accounts for the time required to reach the errorrate speciﬁed. The * indicates that this is an approximation. Switching b t VCMA I/IC Delay Error Energy Mechanism (fj/Vm) (ns) (ns) rate (pJ) STT N/A N/A 1.5 25 10/C020.8579 STT N/A N/A 2 25 10/C041.602 STT N/A N/A 3 22 10/C092.9446 A 312 ON 0.72 25 10/C020.1923 A 312 ON 1 25 10/C060.3926 A 312 ON 1.5 20 10/C090.6864 A 1600 ON 0.72 25 1000.1923 B 1600 8 0.2 22 <10/C020.0088 B 1600 8 0.3 21.8 10/C030.0196 B 1600 8 0.5 14.9 10/C050.0258 B 1600 8 0.7 13.5 10/C060.0423 B 1600 8 1 11 10/C090.0471203902-5 Munira et al. J. Appl. Phys. 120, 203902 (2016) immediately afterward. When the electric ﬁeld is on, the error probability is 100% because the magnetization switches in-plane at most. As soon as the electric ﬁeld is turned off and the spin-transfer torque current is turned on,the error rate drops quickly as most of the magnetization dis- tribution hovering in-plane switches to the ﬁnal state quickly. For a t VCMA of 3 ns, the error rates for I/IC¼0.2, 0.3, and 0.5 remain constant after a certain point in time. This is because 3 ns of VCMA is not enough to rotate the magneti- zation of the free layer to the in-plane position, as shown in Fig. 7(a). The mean of the distribution is still not centered at an angle of p/2 rad. The error rates decrease, but when tVCMA is increased to 8 ns, which is enough time for the magnetization distribution to rotate to in-plane orientation,a ss h o w ni nF i g . 7(a).F o r I/I C¼0.3, the error rate at 25 ns drops from 10/C01to 10/C03when tVCMA is increased from 3 ns to 8 ns. The reason why the error rate becomes constant after a certain time can be understood with the aid of Fig. 7(b). When the electric ﬁeld is turned off and the spin-transfer tor-que with I/I C<1 is applied, the effective barrier between the two bi-stable states, Eb¼Keff(1/C0I/IC)2), prevents the mag- netization distribution closer to the starting point (near 0 rad)from reaching the ﬁnal state ( prad). When I/I C¼1i s applied, the barrier disappears. The dotted lines in the plot fortVCMA¼8 ns are approximations since 500 000 data points were not enough to complete the error curves. We assume that for tVCMA¼8 ns and I/IC¼1, there is no inﬂec- tion point in the error curve, and the error rate drops to 10/C09 within 11 ns. A more accurate error rate can be calculated by solving the Fokker-Planck equation for the two-step switch- ing process.39,40 Table Ilists the energy consumed and write error rates during the STT and VCMA þSTT writing processes in Figs. 5and6. The energy consumed during the spin-transfer tor- que switching was calculated by 0.5 CV2þIV t where the voltage V is estimated from the experimental results in Ref. 41and t is the amount of time taken to reach the best error rate. For the switching mechanism B, VCMA is applied and then STT; we assumed spin-transfer torque current, I, isnegligible when the time is shorter than tVCMA . Switching mechanism A for b¼312fJ/Vm at 1 V and I/IC¼1.5 reaches an error rate of 10/C04at 22 ns. The error rate is comparable to pure STT switching with I/IC¼3. However, the energy con- sumed for pure STT switching is roughly 4.3 times higher than the combined mechanism. To get a lower error rate(down to 10 /C09) for switching mechanism B, a high bvalue of 1600 fJ/Vm and an I/ICof 1 are needed. C. Switching mechanism C and D: VCMA 1applied field A similar study was done for VCMA þapplied magnetic ﬁeld switching (Figs. 8and9). Figs. 8(a) and8(b) show switching mechanism C for b¼1600 fJ/V m and b¼312 fJ/ V m, respectively. Similar to the combined VCMA þSTT switching, bin Fig. 8(a) is too high and the magnetization distribution centers at p/2 rad. b¼312fJ/Vm improves the error rate, but a Happ=Hef f Kof 1 is needed to reach an error rate of 10/C04within 25 ns. In Figs. 9(a)and9(b), the error rates for switching mecha- nism D follow the same trend as B. The high power appliedﬁeld switching is separated from the VCMA mechanism inorder to reduce energy consumption. Error rates improve ast VCMA is increased from 5 ns to 8 ns. For Happ=Hef f K¼0:5, error rates drop from 10/C03to 10/C06when tVCMA is increased. The error rates drop to acceptable numbers when tVCMA is 8 ns, and an effective ﬁeld equivalent to Hef f Kis applied. FIG. 7. (a) The magnetization distribu- tions of the free layer after an electric ﬁeld is applied for 3 ns and 8 ns. dKeff is set to 8 /C2105J/m3. At 3 ns, the mean of the distribution is still not centered at 0.5 prad. (b) Energy potential of the free layer for I/IC¼0, 0.3, and 1. For I/IC>0, the effective barrier is reduced. The barrier disappears when the overdrive current is equal to or greater than 1. TABLE II. The write error rates and energy consumption for switching mechanisms C and D. tVCMA indicates the amount of time the electric ﬁeld is applied across the MTJ. ‘ON’ indicates that the electric ﬁeld is on the entire time (switching proﬁle C). Delay accounts for the time required to reach theerror rate speciﬁed. The indicates that this is an approximation. Switching b t VCMA Happ Delay Error Energy Mechanism (fj/Vm) (ns) =Hef f K (ns) rate (pJ) C 312 ON 0.5 25 <10/C0272.25 C 1600 ON 0.5 25 10072.25 C 312 ON 1 25 10/C04289.2 D 1600 8 0.5 14.9 10/C0619.0162 D 1600 8 1 11 *10/C0934.7045203902-6 Munira et al. J. Appl. Phys. 120, 203902 (2016) Table IIlists the energy consumed during the VCMA þHappwriting process. The current required for the magnetic ﬁeld induction can be estimated using Biot-Savart’s law,42 B¼l0I=2pd. d is the distance between the ﬁeld line and the free layer, and for our calculation, it is taken to be 22 nm.4 The energy consumed while the ﬁeld is applied is I2Rt,w h e r e I is the current required to induce the magnetic ﬁeld, R is theresistance of the active area, 2and t is the amount of time taken to reach the best error rate. The best error rate for C is 10/C04 when b¼312 fJ/V m and an Happ=Hef f Kof 1 is used. However, energy consumption is very high at 289.2 pJ. Fort VCMA of 8 ns and Happ=Hef f Kof 0.5, switching mechanism D reduces the error rate to 10/C04while energy consumption reduces to 19 pJ. V. CONCLUSION While using the VCMA during the writing process con- sumes a small amount of energy, the mechanism is not sufﬁ-cient to completely reverse the magnetization of the free layerof the MRAM from anti-parallel to parallel conﬁguration orvice versa. An additional switching force (spin-transfer torqueor external magnetic ﬁeld) is required to complete the switch-ing of the free layer. In this paper, we investigated the perfor-mance of VCMA þspin torque, and VCMA þappliedmagnetic ﬁeld writing mechanisms for their switching reliabil- ity, the time taken to switch, and the energy consumed duringthe writing process. The free layer has a diameter of 45 nm and is 1 nm thick. The thermal stability of the free layer is 60, the perpendicular anisotropy is 1.56 /C210 5J/m3,s a t u r a t i o n magnetization is 1.1 /C2106A/m, and Gilbert damping is 0.01. Error rate is calculated at 300 K. The four types of combinedswitching mechanisms studied are: (A) combinedVCMA þSTT; (B) the VCMA effect is used to modulate the anisotropy to switch the magnetization of the free layer to in-plane orientation, and after the VCMA is turned off, an STTcurrent is introduced to drive the magnetization of the freelayer to its ﬁnal destination; (C) combining the VCMAþexternally applied ﬁeld; and (D) same as B, except that the STT is replaced by H app. For switching mechanisms A and C, high electric ﬁeld anisotropy modulation ( b¼1600 fJ/V m) overpowers the secondary forces giving the direction to the desired destina-tion, resulting in the magnetization rotating to the in-planeorientation and then to either of the bi-stable positions ( h¼0 orp) after the voltage source is removed. For a lower electric ﬁeld efﬁciency of 312 fJ/V m, the error rates are comparableto cases with just spin-transfer torque switching with doublethe overdrive current with signiﬁcant reduction in writeenergy consumption. For switching mechanism C, where FIG. 8. (a) Combined VCMA and Happ switching mechanism C. A voltage of 0.5 V is applied across the tunnel junc-tion for b¼1600fJ/Vm and H app=Hef f K ¼0:5. Since the VCMA effect over- powers the applied ﬁeld, the magneti- zation prefers to be in in-plane orientation with the mean of distribu- tion at h¼p/2. (b) Switching mecha- nism C for b¼312fJ/Vm at 0.5 V and Happ=Hef f K¼0:5 and 1. FIG. 9. (a) Switching mechanism D for b¼1600 fJ/V m at 0.5 V. The electric ﬁeld is turned on for tVCMA of 5 ns. A magnetic ﬁeld, Happ, is applied immedi- ately afterward. The error rate curves remain constant after a certain point. (b)tVCMA increased to 8 ns. The dotted lines in the plot for tVCMA¼8n s a r e approximations.203902-7 Munira et al. J. Appl. Phys. 120, 203902 (2016) VCMA and magnetic ﬁeld are applied simultaneously, the error rate is still high (10/C04) for a high write energy of 289.2 pJ for Happ=Hef f Kof 1 after 25 ns. In order to lower the write energy, the VCMA and sec- ondary force are separated in switching mechanisms C and D. VCMA was used to rotate the magnetization to in-planeorientation using energy in the fJ range with bof 1600 fJ/V m at 0.5 V and t VCMA of 8 ns. A spin-transfer torque current or an applied ﬁeld comparable to the critical current and ﬁeldof the free layer is applied afterwards to direct the magneti- zation to the right destination. It is necessary to use a current or ﬁeld comparable to the critical ones to produce a very lowerror rate (10 /C09). The energy consumption for the hybrid switching is much lower than just STT, magnetic ﬁeld, and combined VCMA þSTT or VCMA þapplied ﬁeld as the high-energy secondary force is used for a shorter time. ACKNOWLEDGMENTS We gratefully acknowledge J. 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1.3238314.pdf	Tunable steady-state domain wall oscillator with perpendicular magnetic anisotropy A. Bisig, L. Heyne, O. Boulle, and M. Kläui Citation: Applied Physics Letters 95, 162504 (2009); doi: 10.1063/1.3238314 View online: http://dx.doi.org/10.1063/1.3238314 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/95/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Controlled domain wall pinning in nanowires with perpendicular magnetic anisotropy by localized fringing fields J. Appl. Phys. 115, 17D506 (2014); 10.1063/1.4864737 Effects of notch shape on the magnetic domain wall motion in nanowires with in-plane or perpendicular magnetic anisotropy J. Appl. Phys. 111, 07D123 (2012); 10.1063/1.3677340 Enhanced current-induced domain wall motion by tuning perpendicular magnetic anisotropy Appl. Phys. Lett. 98, 132508 (2011); 10.1063/1.3570652 Tunable magnetic domain wall oscillator at an anisotropy boundary Appl. Phys. Lett. 98, 102512 (2011); 10.1063/1.3562299 Current-induced domain wall motion in a nanowire with perpendicular magnetic anisotropy Appl. Phys. Lett. 92, 202508 (2008); 10.1063/1.2926664 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.156.157.31 On: Wed, 25 Mar 2015 08:55:15Tunable steady-state domain wall oscillator with perpendicular magnetic anisotropy A. Bisig, L. Heyne, O. Boulle, and M. Kläuia/H20850 Fachbereich Physik, Universität Konstanz, Universitätsstraße 10, 78457 Konstanz, Germany /H20849Received 10 August 2009; accepted 4 September 2009; published online 22 October 2009 /H20850 We theoretically study domain wall oscillations upon the injection of a dc current through a geometrically constrained wire with perpendicular magnetic anisotropy. The frequency spectrum ofthe oscillation can be tuned by the injected current density and additionally by the application of anexternal magnetic ﬁeld. Our analytical calculations are supported by micromagnetic simulationsbased on the Landau–Lifshitz–Gilbert equation. The simple concept of our localized steady-stateoscillator might prove useful as a nanoscale microwave generator with possible applications intelecommunications or for rf-assisted writing in magnetic hard drives. © 2009 American Institute of Physics ./H20851doi:10.1063/1.3238314 /H20852 The recent discovery that a spin-polarized current can exert a torque on a magnetization through transfer of spinangular momentum has opened a new way to manipulatemagnetization. 1Spin-polarized currents can be used to re- verse a magnetization in multilayer pillar elements2and in- teract with domain walls, leading to current induced domainwall motion in the direction of the electron ﬂow. 3,4In the nanopillar geometry, the spin transfer torque can compensatefor the magnetic damping and this leads to sustained magne-tization precession dynamics that are converted into micro-wave emission by magnetoresistive effects. 5Recently, new schemes based on the oscillations of magnetic domain walls due to spin-polarized currents have been studied.6–9These schemes led to the concept of spin torque oscillator /H20849STO /H20850 based on domain walls, which may prove useful as for ap-plications in telecommunications or for rf-assisted writing inmagnetic hard drives. Microwave generation due to the small angle precession of a magnetic free layer in nanopillar structures 5leads to small output power, so here the challenge is to increase thepower of the STO. In complicated nanopillar multilayerstructures, where an out-of-plane magnetic injector layer isused to polarize the current, full angle precessions of a mag-netic free layer in combination with a magnetic tunnel junc-tion gives rise to higher output power. 10However, a simpler and easier to fabricate system of magnetization exhibitingfull angle precessions upon current injection is an oscillatingmagnetic domain wall. Here, the question is whether sus-tained precession of a pinned domain wall induced by spintransfer can be obtained. The spin transfer torque acts in thecase of domain walls differently from antidamping. This wasrecognized early on by Berger in a seminal paper 11and re- cently it was shown that domain wall oscillation can be ob-tained through the Walker precession phenomena, where thewhole domain wall structure oscillates periodically at micro-wave frequencies. 6However, the large current density nec- essary to attain this Walker regime has so far always led todomain wall motion in the standard wire geometry in softmagnetic materials, making it impossible to pin the domainwall during the oscillation.Several schemes have been proposed to solve this prob- lem, including the use of wires with an artiﬁcial local gradi-ent of the magnetic damping 6or extremely narrow wires with nanoscale lateral dimensions7resulting in lower critical current for Walker precession, but these approaches are notvery realistic for the design of future devices. For three di-mensional geometrically conﬁned domain walls which arecomplicated to fabricate, inside a magnetic bridge betweentwo electrodes, a similar behavior is presented. 9While the concept of a domain wall oscillator seems appealing, asimple and more realistic system is missing. Furthermore, inthe approaches put forward so far, there is no possibility totune the frequency independent of the output power, which isa key prerequisite for a device. In this letter, we show that in perpendicularly magne- tized materials, Walker precession of a pinned domain wallcan be easily obtained by pinning the domain wall in asimple notch geometry. By properly choosing the constric-tion geometry dimensions, a small domain wall demagnetiz-ing ﬁeld can be attained, while keeping the domain wallstrongly pinned. The steady-state oscillations are describedby analytical equations based on the Landau–Lifshitz–Gilbert /H20849LLG /H20850equation with spin torque terms. Micromag- netic simulations are performed and show that precessionoccurs at zero applied ﬁeld and is associated with large angleoscillation of the domain wall magnetization. The oscillationfrequency depends linearly on the injected current and can betuned over a wide range. In addition, higher order modes arerevealed which can be controlled by a small transverse mag-netic ﬁeld. These results open a new route for a novel kind ofa spin transfer oscillator operating at zero ﬁeld with a simplegeometry and with potential for large output power in moresophisticated implementations. First, we consider a one dimensional system with per- pendicular anisotropy where the magnetization Mis turning from M z//H20841M/H20841=1 to Mz//H20841M/H20841=−1 /H20849see Fig. 1/H20850. We describe this system with an analytical model, based on the LLGequation with spin torque terms,12,13 /H11509m /H11509t=−/H92530m/H11003Heff+/H9251m/H11003/H11509m /H11509t−/H20849u·/H11612/H20850m+/H9252m /H11003/H20851/H20849uS·/H11612/H20850m/H20852, /H208491/H20850 where m=M /MSis the unit vector along the local magneti-a/H20850Also at: Zukunftskolleg, Universität Konstanz, 78457 Konstanz, Germany. Electronic mail: mathias@klaeui.de.APPLIED PHYSICS LETTERS 95, 162504 /H208492009 /H20850 0003-6951/2009/95 /H2084916/H20850/162504/3/$25.00 © 2009 American Institute of Physics 95, 162504-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.156.157.31 On: Wed, 25 Mar 2015 08:55:15zation direction and Heffis the effective magnetic ﬁeld in- cluding the external ﬁeld, the anisotropy ﬁeld, the magneto-static ﬁeld, and the exchange ﬁeld. The spin current driftvelocity u=j eP/H9262B/eM Sdescribes the spin current associated with the electric current in a ferromagnet, where Pis the spin polarization of the current, /H9262Bis the Bohr magneton, eis the /H20849positive /H20850electron charge, and jeis the current density.14The dimensionless constant /H9252describes the degree of nonadiaba- city between the spin of the nonequilibrium conduction elec-trons and the local magnetization. For the magnetization dynamics of this system, the pinned domain wall can be described by two collective co-ordinates; the domain wall center position q/H20849t/H20850and the do- main wall tilting angle /H9274/H20849t/H20850. Following the approach of Jung et al. ,15we obtain two equations, describing our system, re- ferred to as the one dimensional collective coordinatesmodel, /H9004 0/H9274˙−/H9251q˙=/H9252u+/H92530/H90040 2MS/H20873/H11509/H9280 /H11509q/H20874, /H208492/H20850 q˙+/H9251/H90040/H9274˙=−u−/H92530/H90040 MSKdsin 2/H9274, /H208493/H20850 where /H90040is the constant domain wall width, Kdis the effec- tive domain wall anisotropy, and /H9280/H20849q/H20850is the domain wall potential energy per unit cross-sectional area, representingthe pinning potential of the wire constriction geometry.16 In the case where the effective wall anisotropy Kd=Ky−Kxvanishes, corresponding to a geometry where the Bloch and Néel domain wall have the same magnetostaticenergy and no energy barrier in between, Eqs. /H208492/H20850and /H208493/H20850 lead to a steady-state oscillation of the polar angle /H9274/H20849t/H20850, /H9274˙=−u /H9251/H90040. /H208494/H20850 This in-plane rotation corresponds to a continuous and re- peated transition from Bloch to Néel wall and vice versa/H20849Fig.1/H20850. Note that in the non-adiabatic case steady-state os- cillations are only possible in the presence of a pinning po-tential even when the magnetostatic energy difference K d vanishes since otherwise the domain wall will start to move. However, in realistic wire geometries, inhomogeneous demagnetization ﬁelds lead to energy barriers between theBloch and the Néel walls and the effective wall anisotropynever vanishes completely, even by proper tuning of the con-striction width wand wire thickness t. Hence, for /H20841K d/H20841/H110220 the domain wall starts to rotate when the current density jeis larger than a certain threshold value Jcgiven by17 Jc=e/H92530/H90040 P/H9262B/H20841Kd/H20841. /H208495/H20850 This critical current density represents a minimal spin torque, which has to be applied in order to turn the magnetization from a Néel to a Bloch wall. In this case /H9274˙is no more constant but oscillating and Eq. /H208494/H20850still holds with /H20855/H9274˙/H20856=u//H9251/H90040for current densities large compared to the criti- cal current density. Note that both, the linear dependence of the oscillation frequency of the injected current density equa-tion /H20851Eq. /H208494/H20850/H20852, as well as the critical current density J care independent of the nonadiabatic spin torque constant /H9252. To conﬁrm this one dimensional picture, micromagnetic simulations are performed. We consider a system consistingof a ferromagnetic structure with perpendicular magnetic an-isotropy with a geometrical conﬁnement, as shown in Fig. 1. The outer dimensions of the wire are 500 /H1100360/H110037n m 3, and for this material Kdis minimal at a constriction width wof 16 nm. Note that the constriction with wdepends strongly on the material parameters and can be easily made larger forother out-of-plane materials. We consider only the effect ofthe adiabatic spin torque /H20849 /H9252=0 /H20850, assuming typical material parameters for Co/Pt multilayer with perpendicularmagnetic anisotropy:18,19MS=1.4/H11003106A/m, A=1.6 /H1100310−11J/m, the effective perpendicular magnetic aniso- tropy Keff=2.7/H11003105J/m3, and Gilbert damping parameter /H9251/H110050.15. The results of our micromagnetic simulations employing the LLG micromagnetic simulator20are shown in Fig. 2, il- lustrating the linear dependence of the domain wall oscilla-tion frequency on the injected current density j efor two val- ues of /H9251. For the simulation, the system is divided into a rectangular mesh with ﬁnite elements of 2 /H110032/H110037n m3, smaller than the exchange length for Co/Pt which is lex =3.6 nm. The linear behavior is in agreement with the simple one dimensional model and can be observed over abroad frequency range between 500 MHz and 3.5 GHz. Thecritical current density J cis marked by red points /H20849see Fig. 2/H20850, and it is not dependent on /H9251and calculated as Jcsim FIG. 2. /H20849Color /H20850Domain wall oscillation frequency fas a function of the injected spin current drift velocity ufor constant /H9251/H11005 /H208490.15, 0.25 /H20850.T h e oscillation frequency shows a linear dependence on the current density overa broad frequency range between 500 MHz and 3.5 GHz. /H20849A/H20850The domain wall proﬁle is symmetric under rotation for low current density j e=1.79 /H110031011A/m2./H20849B/H20850At high current density je=1.34/H110031012A/m2, the domain wall shows asymmetric oscillations, this leads to a nonharmonic behavior. FIG. 1. /H20849Color online /H20850Schematic illustration of the geometrically conﬁned structure. The arrows represent the magnetization conﬁguration inside thestructure, which can be either a Bloch or a Néel domain wall.162504-2 Bisig et al. Appl. Phys. Lett. 95, 162504 /H208492009 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.156.157.31 On: Wed, 25 Mar 2015 08:55:15=1.34/H110031011A/m2/H20849corresponding to spin drift velocity u =1.11 m /s/H20850. The theoretical value Jc=1.2/H110031010A/m2 given by Eq. /H208495/H20850, assuming a constant domain wall width /H90040=/H20881A/Keff=3.2 nm, is lower than the value extracted from our simulations Jcsim/H11022Jc. This is due to the fact that the po- tential landscape for the polar angle /H9274is modulated by the wire constriction geometry in a way that additional spintorque has to be applied in order to turn the magnetizationinside the domain wall and this is not taken into account inthe one dimensional calculations. The nonlinearity at higher current densities can be ex- plained by the deformations exhibited by the domain walljust before depinning /H20851see Fig. 2, insets /H20849A/H20850and /H20849B/H20850/H20852. For a low current density j e=1.79/H110031011A/m2, the domain wall center position qis not shifted q=/H208490/H110060.5 /H20850nm /H20851inset /H20849A/H20850/H20852, whereas for high current density je=1.34/H110031012A/m2, the center position is pushed to the left hand side due to the spintorque /H20851inset /H20849B/H20850/H20852. So far the frequency of the domain wall oscillations can only be tuned by the injected current density so that fre-quency and output power cannot be varied independently.By the application of an external magnetic ﬁeld H ext=Hyeˆy in the plane of the wire, the potential landscape of the polar angle is modulated, leading to fundamental changes inthe power spectrum of the oscillations. The power spectraof the magnetization dynamics for various ﬁeld strengths /H92620Hy=0–3 mT and constant current density je=1.79 /H110031011A/m2/H20849corresponding to spin drift velocity u =1.48 m /s/H20850are plotted in Fig. 3. For/H92620Hy=0 mT it shows a sharp peak at the oscillation frequency f=747 MHz, whereas for higher ﬁeld strength /H92620Hy/H110222.0 mT, the oscillation frequency strongly decreases continuously down to f =525 MHz, indicated by the red dashed line. For higher ﬁeldstrengths, additional peaks at higher frequencies appear. Notethat in the graph, the ﬁrst peak corresponding to the funda-mental oscillation frequency is scaled down by a factor of 10for comparison. In conclusion, we have shown that Walker precession of a pinned domain wall can be easily obtained in perpendicu-larly magnetized materials, where the domain wall is pinnedby the geometrical constriction of the wire. By properly choosing the wire constriction dimensions, a small K dcan be attained, while keeping the domain wall strongly pinned.This combined with the high damping /H9251results in Walker precession at low current density. This STO can be tunedover a broad frequency range by the injected current density.When an external ﬁeld is applied, the power spectrum ismodiﬁed leading to strongly nonharmonic oscillations open-ing a novel way for tuning the frequency of the domain wallSTO. Finally, we would like to mention that for a realistic device, a high output power is a key criterion. This can beachieved by the coupling of multiple STO’s and by signalenhancement through a magnetic tunnel junction, fabricatedin top of the wire constriction yielding a three terminal de-vice. The latter also opens an additional way of manipulatingthe oscillation frequency: the current in the plane of the wire,which excites the oscillation and sets the frequency, can beindependently tuned from the current that ﬂows verticallyacross the tunnel barrier and determines the output power,which leads to a versatile microwave source. The authors would like to acknowledge the ﬁnancial support by the DFG /H20849SFB 767, KL1811 /H20850, the Landesstiftung Baden Württemberg, the European Research Council via itsStarting Independent Researcher Grant /H20849Grant No. ERC- 2007-Stg 208162 /H20850scheme, EU RTN SPINSWITCH /H20849MRTN- CT-2006–035327 /H20850, and the Samsung Advanced Institute of Technology. 1M. D. Stiles and J. Miltat, in Spin Dynamics in Conﬁned Magnetic Struc- tures , edited by B. Hillebrands and A. Thiaville /H20849Springer, Berlin, 2006 /H20850, Vol. 3, pp. 225–308. 2J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph,Phys. Rev. Lett. 84, 3149 /H208492000 /H20850. 3A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 /H208492004 /H20850. 4M. Kläui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, L. J. Heyderman, F. Nolting, and U. Rüdiger, Phys. Rev. Lett. 94, 106601 /H208492005 /H20850. 5S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoe- lkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850425,3 8 0 /H208492003 /H20850. 6J. He and S. Zhang, Appl. Phys. Lett. 90, 142508 /H208492007 /H20850. 7T. Ono and Y. Nakatani, Appl. Phys. Express 1, 061301 /H208492008 /H20850. 8M. Franchin, T. Fischbacher, G. Bordignon, P. de Groot, and H. Fangohr, Phys. Rev. B 78, 054447 /H208492009 /H20850. 9K. Matsushita, J. Sato, and H. Imamura, J. Appl. Phys. 105, 07D525 /H208492009 /H20850. 10D. Houssameddine, U. Ebels, B. Delaët, B. Rodmacq, I. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda, M.-C.Cyrille, O. Redon, and B. Dieny, Nature Mater. 6, 447 /H208492007 /H20850. 11L. Berger, Phys. Rev. B 33, 1572 /H208491986 /H20850. 12Z. Li and S. Zhang, Phys. Rev. B 70, 024417 /H208492004 /H20850. 13S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 /H208492004 /H20850. 14A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 /H208492005 /H20850. 15S. W. Jung, W. Kim, T. D. Lee, K. J. Lee, and H. W. Lee, Appl. Phys. Lett. 92, 202508 /H208492008 /H20850. 16L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, and S. S. P. Parkin, Nature /H20849London /H20850443, 197 /H208492006 /H20850. 17G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 /H208492004 /H20850. 18O. Boulle, J. Kimling, P. Warnicke, M. Kläui, U. Rüdiger, G. Malinowski, H. J. M. Swagten, B. Koopmans, C. Ulysse, and G. Faini, Phys. Rev. Lett. 101, 216601 /H208492008 /H20850. 19P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier, J. Ferré, V. Baltz, B. Rodmacq, B. Dieny, and R. L. Stamps, Phys. Rev. Lett. 99, 217208 /H208492007 /H20850. 20M. R. Scheinfein, LLG Micromagnetics Simulatior /H20849http://llgmicro.home.mindspring.com /H20850. FIG. 3. /H20849Color online /H20850Fourier spectrum of the magnetization component Mx as a function of frequency fforje=1.79/H110031011A/m2and various external ﬁeld strength /H92620Hy=/H208490 , 1 0 , 2 0 , a n d3 0m T /H20850. At zero ﬁeld, a sharp peak in- dicates an oscillation frequency of f=747 MHz, whereas for higher ﬁeld strenghts the oscillation frequency is shifted to lower frequencies down to f=525 MHz /H20849indicated by the red dashed line /H20850. Furthermore, higher order oscillations are visible by the peaks forming at higher frequencies.162504-3 Bisig et al. Appl. Phys. Lett. 95, 162504 /H208492009 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.156.157.31 On: Wed, 25 Mar 2015 08:55:15
5.0018801.pdf	Appl. Phys. Lett. 117, 102402 (2020); https://doi.org/10.1063/5.0018801 117, 102402 © 2020 Author(s).Hybrid magnetoacoustic metamaterials for ultrasound control Cite as: Appl. Phys. Lett. 117, 102402 (2020); https://doi.org/10.1063/5.0018801 Submitted: 18 June 2020 . Accepted: 27 August 2020 . Published Online: 10 September 2020 O. S. Latcham , Y. I. Gusieva , A. V. Shytov , O. Y. Gorobets , and V. V. Kruglyak ARTICLES YOU MAY BE INTERESTED IN A reconfigurable magnetorheological elastomer acoustic metamaterial Applied Physics Letters 117, 104102 (2020); https://doi.org/10.1063/5.0015645 Spin current generation and detection in uniaxial antiferromagnetic insulators Applied Physics Letters 117, 100501 (2020); https://doi.org/10.1063/5.0022391 Controlled giant magnetoresistance and spin–valley transport in an asymmetrical MoS 2 tunnel junction Applied Physics Letters 117, 102401 (2020); https://doi.org/10.1063/5.0018869Hybrid magnetoacoustic metamaterials for ultrasound control Cite as: Appl. Phys. Lett. 117, 102402 (2020); doi: 10.1063/5.0018801 Submitted: 18 June 2020 .Accepted: 27 August 2020 . Published Online: 10 September 2020 O. S. Latcham,1 Y. I.Gusieva,2 A. V. Shytov,1 O. Y. Gorobets,2 and V. V. Kruglyak1,a) AFFILIATIONS 1University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom 2Igor Sikorsky Kyiv Polytechnic Institute, 37 Prosp. Peremohy, Kyiv 03056, Ukraine a)Author to whom correspondence should be addressed: V.V.Kruglyak@exeter.ac.uk ABSTRACT We propose a class of metamaterials in which the propagation of acoustic waves is controlled magnetically through magnetoelastic coupling. The metamaterials are formed by a periodic array of thin magnetic layers (“resonators”) embedded in a nonmagnetic matrix. Acoustic waves carrying energy through the structure hybridize with the magnetic modes of the resonators (“Fano resonance”). This leads to a rich set ofeffects, enhanced by Bragg scattering and being most pronounced when the magnetic resonance frequency is close to or lies within acousticbandgaps. The acoustic reﬂection from the structure exhibits magnetically induced transparency and Borrmann effect. Our analysis showsthat the combined effect of the Bragg scattering and Fano resonance may overcome the magnetic damping, ubiquitous in realistic systems. This paves a route toward the application of such structures in wave computing and signal processing. Published under license by AIP Publishing. https://doi.org/10.1063/5.0018801 The minimization of energy loss in modern computing devices calls for unorthodox approaches to signal processing. 1,2For instance, proposals to employ spin waves3as a data carrier to save energy in nonvolatile memory devices have promoted growth in the research area of magnonics.4However, these hopes are hampered by the short propagation distance of spin waves, caused by the magneticdamping. 5,6Magnetostrictive materials offer a route to circumvent this. Indeed, acoustic waves have longer attenuation lengths as compared to spin waves at the same frequencies. In magnetostrictive materials, acoustic waves can still couple to spin waves, forming hybrid magneto- acoustic waves.7–13Thus, one regains the option of magnetic control and programmability, catering to the design of systems that evoke ben- eﬁts of both acoustics and magnonics in terms of the energy efﬁciency. The recently studied magnetoacoustic devices11and metamaterials13 were typically formed using alternating magnetostrictive materials, so that the full acoustic and magnonic spectra were hybridized. To reduce the inﬂuence of the magnetic damping, we explored systems in which the magnetic loss was restricted to an isolated, thin-ﬁlmmagnetostrictive inclusion (“resonator”), hosting a single spin-wave mode, that of the ferromagnetic resonance (FMR). 14The FMR mode hybridized with acoustic waves only near the Kittel frequency,3 which led to their resonant scattering in a magnetoacoustic versionof the Fano resonance. 15The FMR mode’s frequency and linewidth (and therefore the strength of the Fano resonance) were determinedby the bias magnetic ﬁeld and by the magnetic damping, respectively. Our analysis highlighted the need to enhance the (generally, weak) magnetoelastic interaction and to suppress the (generally, strong) magnetic damping, which was partly achieved by adopting an obli-que incidence geometry. A question arises as to whether the effects of the magnetoelastic coupling could be enhanced even further due to Bragg scattering in magnetoacoustic metamaterials 13formed by peri- odic arrays of the magnetoacoustic resonators introduced in Ref. 14. In this Letter, we demonstrate that, by combining individual magnetoacoustic resonators into one-dimensional (1D) arrays (similar to locally resonant phononic crystals),16one can indeed signiﬁcantly enhance their effect on incident acoustic waves. The acoustic reﬂectiv-ity of such a metamaterial exhibits a peak due to the magnetoacoustic Fano resonance. The peak’s height and shape can be tuned at frequen- cies in the proximity of phononic bandgaps. In particular, its behaviornear the two edges of a bandgap exhibits a strong asymmetry, which is linked to the Borrmann effect. 17Inside the bandgaps, we identify behavior reminiscent of the magnetically induced transparency.15 These features of our prototypical metamaterial could be employed to process acoustic signals and engineer reconﬁgurable magnetoacoustic devices. The building blocks of our metamaterials are thin ferromagnetic slabs (resonators) of thickness d, inﬁnite in the Y–Zplane, separated by nonmagnetic spacer layers of thickness dsðds/C29dÞ,a ss h o w ni n Appl. Phys. Lett. 117, 102402 (2020); doi: 10.1063/5.0018801 117, 102402-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplFig. 1(a) . The slabs are magnetized by a bias magnetic ﬁeld HB¼HB^z and have saturation magnetization Ms. The elastic properties of the magnetic and spacer materials may differ. The shear stress producedby propagating transverse acoustic waves perturbs the magnetization,as described by the standard magnetoelasticity theory. 7,9,18–20The hybridization between the acoustic waves and the magnetization pre-cession manifests itself as a Fano-like peak in the frequency depen-dence of the acoustic reﬂectivity [ Fig. 1(b) ]. 14This peak occurs near the Kittel frequency of the slab, fFMR, and is therefore controlled by the bias magnetic ﬁeld. The strength of the coupling between the propa-gating acoustic and localized magnetic modes is noticeably enhancedfor an oblique incidence [ Fig. 1(b) ]. However, for realistic values of the magnetoelastic coupling, B, a noticeable effect requires rather small values of the Gilbert damping, e.g., a’10 /C03. We aim to increase the interaction time of the acoustic waves with the magnetic slabs by slowing the waves down in the vicinity ofphononic bandgaps. Hence, an enhancement of the magnetoacousticresponse of such a structure can be expected when this anticrossing istuned to the proximity of the phononic bandgap. So, we arrange theslabs into arrays, either containing Nmagnetic elements or semi- inﬁnite. Let the nth resonator be situated at x n¼nL,w h e r e L ¼dþdsis the period of the array. Acoustic waves are obliquely inci- dent on the array from the left. The magnetoacoustic response of ﬁnitearrays is characterized by the reﬂection, R N, transmission, TN,a n d absorption, AN, coefﬁcients. Using the transfer matrix method,21these coefﬁcients can be expressed via the reﬂection, r, and transmission, t, coefﬁcients in the forward direction together with the respective coefﬁ-cients ~r,a n d ~t, in the backward direction. For normal incidence, reci- procity between forward and backward reﬂection is maintained(r¼~r). However, at oblique incidence, rand ~racquire different phases. The transmission and reﬂection coefﬁcients for an individualslab are derived by considering the modes inside the slab and match-ing interfacial displacements and stresses with those of the incomingand outgoing elastic waves. The magnetoelastic interaction inside theslab can be included in the matching procedure adding relevant con-tributions to the stress 14,22,23or the acoustic impedance.14For a thin slab, one can neglect exchange contribution to the effective magneticﬁeld and treat magnetodipole interaction by introducing relevantdemagnetizing coefﬁcients. 14As illustrated in Fig. 1(b) , the resulting coefﬁcients t,~t,r,a n d ~rexhibit a strong frequency dependence, indica- tive of the resonant hybridization between the acoustic and magneticmodes. The spectral function (derived in the supplementary material ) of a phononic crystal with embedded magnetic slabs, as shown inFig. 1(c) , exhibits a magnetically tunable anticrossing with the usual phononic bandgap dispersion. The transverse acoustic displacement U¼Uðx;y;tÞ^zdue to an obliquely incident acoustic wave inside the nth nonmagnetic spacer layer,ðn/C01ÞL<x<nL/C0d,i sg i v e nb y Uðx;y;tÞ¼e /C0ixtþiqyyAnei/xþBne/C0i/x/C2/C3 ; (1) where qrepresents the wave number in the nonmagnetic layer and /x¼qx½x/C0ðn/C01ÞL/C138.I nw h a tf o l l o w s ,w er e t a i no n l yt h e x-depen- dence of the wave function. The amplitudes AnandBnare acoustic, traveling to the right and to the left in the nth nonmagnetic layer, respectively. Then, for a wave of unit amplitude incident from the leftonto a ﬁnite array, we have A 0¼1;B0¼RN,AN¼TN,BN¼0. To form the transfer matrix Mfor a single period of the array, amplitudesatx¼nLandx¼ðnþ1ÞLcan be related via forward ( t,r) and back- ward ( ~t;~r) transmission and reﬂection coefﬁcients. Waves in neigh- boring segments can be matched by treating them as “black boxes” with the given transmission and reﬂection coefﬁcients. Hence, wew r i t ef o rt h ei n t e r f a c eb e t w e e nt h e nth and ðnþ1Þth segment: A nþ1expð/C0ivhÞ¼tAnþ~rBnþ1expðivhÞ; Bn¼~tBnþ1expðivhÞþrAn;(2) where vh¼xdscoshﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ q=Cp i st h ea c o u s t i cp h a s ed e l a yw i t h i nt h e spacer layer. The transfer matrix Mlinks the vector ðAnþ1;Bnþ1Þto (An,Bn) and is constructed by inverting Eq. (2)as M¼t/C0~rr~t/C01½/C138 expivhðÞ ~r~t/C01expivhðÞ /C0r~t/C01exp/C0ivhðÞ ~t/C01exp/C0ivhðÞ8 < :9 = ;: (3) The action of Mcan be represented by its eigenvalues l6and the respective eigenvectors. The eigenvalues that solve the characteristic equation l2/C02Tlþd¼0 are given by l6¼T7ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ T2/C0Dp , whereD/C17detM¼lþl/C0and 2T/C17TrM¼lþþl/C0.F r o mE q . (3),w eﬁ n dt h a t D¼t=~t, which has an absolute value of one. As usual, we ﬁnd that the two eigenvalues of Meither both lie on the unit circlejlj¼1 or one is inside and the other is outside. In our system, FIG. 1. (a) The problem geometry is schematically shown. The metamaterial is formed by a 1D array of thin-ﬁlm magnetoacoustic resonators embedded in a non- magnetic matrix. Individual resonators scatter acoustic waves incident from bothsides. A bias magnetic ﬁeld H Bis applied in the resonator’s plane. (b) The fre- quency dependence of the reﬂection coefﬁcient, r, for incidence angles ranging from 0/C14to 20/C14is shown for an isolated Co resonator in a silicon nitride matrix. The vertical line indicates the Kittel frequency for a ﬁeld of l0HB¼50 mT and a¼10/C03. The inset shows the corresponding transmission, t, and absorption, a, coefﬁcients. (c) The spectral function, S(f,k), of acoustic waves in the metamaterial is shown. The frequency of the anticrossing is controlled by the bias magnetic ﬁeld, which has a value of l0HB¼50 mT here.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 102402 (2020); doi: 10.1063/5.0018801 117, 102402-2 Published under license by AIP Publishingthe energy is dissipated due to the Gilbert damping. Hence, we can deﬁne l6so that jlþj<1, representing the wave propagating to the right. For a ﬁnite array of Nresonators, the full transfer matrix MN¼MNretains the eigenvector basis with eigenvalues lN 6.T h ei n i - tial and ﬁnal state amplitudes are then projected onto a reciprocal of this basis, multiplied by the eigenvalues, and resolved to obtain for theﬁnite array’s reﬂection coefﬁcient, R N¼R11/C0l2N/C0/C1 1/C0nl2N ðÞ; (4) where R1is the reﬂection from a semi-inﬁnite array, R1¼rexpð/C0ivhÞ~tl/C0/C0t~t/C0r~r ðÞ expivhðÞhi/C01 ; (5) andnis deﬁned as n¼t~t/C0r~r ðÞ expivhðÞ/C0~tlþ t~t/C0r~r ðÞ expivhðÞ/C0~tl/C0: (6) The transmission coefﬁcient of a ﬁnite length array has the form TN¼1/C0nðÞ lN þ 1/C0nl2N: (7) Detailed derivation of Eqs. (4)and(7)i sg i v e ni nS e c .Io ft h e supple- mentary material . The absorbance is found as A2 N¼1/C0jRNj2 /C0jTNj2. In what follows, we omit the explicit dependence of the quan- tities nandlupon the frequency, x, and the phase delay, vh. To illustrate how RNdepends on the number of elements in a ﬁnite array, we have performed detailed calculations for an array of resonatorswith parameters equal to those from Ref. 24:m a s sd e n s i t y q¼8900 kg m /C03, magnetoelastic coupling coefﬁcient B¼8:8M Jm/C03, shear modulus C¼76 GPa, gyromagnetic ratio c¼31:7G H zT/C01, and saturation magnetization Ms¼203 kA m/C01;d¼30 nm. The matrix is silicon nitride [ q0¼3192 kg m/C03;C0¼127 GPa ;ds ¼500 nm (Refs. 25and26)].Figure 2 presents the results of the calcula- tions for a generic case, without ﬁne-tuning of the magnetoelastic reso-nance. For N>1, the absolute value of the reﬂection coefﬁcient reaches unity in frequency regions corresponding to the acoustic stopbands (phononic bandgaps). These are caused by the mismatch of the acoustic impedance Z¼ﬃﬃﬃﬃﬃﬃqCpat the surface of the slabs, which occur even in the absence of the magnetoelastic coupling ( B¼0). 27–29Each passband contains N/C01p e a k s ,w h i c ha r ed u et ot h ep h a s ed e l a yo ft h ea c o u s t i c waves increasing by pacross each Brillouin zone.21The magnetoelastic coupling ( B6¼0) manifests itself via an asymmetric peak due to the Fano resonance, positioned at the Kittel frequency fFMR ¼cl0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HBðHBþMsÞp ’8:8G H z a t l0HB¼180 mT.7The fre- quency dependences of TNandANare given in the supplementary material for a complete picture. The rapid oscillation in passbands in Fig. 2 is formed due to the multiple reﬂections within arrays of ﬁnite size. For sufﬁciently largearrays (i.e., when the decay length is smaller than the array size), these oscillations are suppressed. Indeed, the oscillations are suppressed for R 1[calculated using Eq. (5)and shown by the solid line in Fig. 2 ], as expected for N!1 . So, our subsequent analysis is focused on the semi-inﬁnite array. Figure 3 displays the reﬂectivity R1, of a semi-inﬁnite array as a function of the frequency for different values of the bias magnetic ﬁeld.We identify two regimes based on the position of the Kittel frequency, fFMR, relative to phononic bandgaps. Regime I occurs when fFMR is tuned inside a passband, away from band edges. This is shown in Fig. 3(a) , with insets comparing R1andr.T h ep e a ki n R1is lower than that in rboth when fFMRis located in the passbands above and below the stop band, away from band edges. This suppression iscaused by the destructive interference of waves reﬂected backwardfrom different resonators. Regime II occurs when the Kittel frequency, f FMR, either falls within the bandgap [ Fig. 3(b) ] or approaches it from a passband [Fig. 3(c) ]. Here, the resonant scattering becomes highly sensitive to the detuning of fFMRfrom the band edge, differently affecting the scat- tering of acoustic waves with frequencies within the bandgap and adja-cent passbands. In the passbands in close proximity to the bandgap,where the Bragg condition holds, the scattering is enhanced by theconstructive interference of waves reﬂected backward from differentresonators. In the bandgaps, the reﬂectivity is reduced from unity, asseen best in Fig. 3(b) .T h i sm a yb ei n t e r p r e t e da sam a g n e t i c a l l y induced transparency, which is further supported by our analysis ofthe acoustic scattering from ﬁnite arrays, which is described in thesupplementary material . This reduction of reﬂectivity is not symmetric as the bias ﬁeld sweeps the Kittel resonance frequency across the bandgap. The behav-ior at the upper and the lower bandgap edges is distinctly different: thereduction of reﬂectivity is stronger as f FMR approaches the upper bandgap edge. This can be attributed to the Borrmann effect.30,31In a pure phononic crystal ( B¼0), the modes at the band edges are two standing waves, phase shifted by 90/C14.32For one of the modes, the max- ima of the stress occurs within the magnetic slabs, while for the other,this pattern is reversed: the slabs become the nodes. With the Gilbertdamping being the primary mechanism of energy dissipation, absorp-tion is suppressed for the mode that has nodes at the magnetic slabs,similar to Refs. 17and 33. This condition is realized at the lower bandgap edge if the acoustic impedance of the magnetic (M) materialis greater than that of the nonmagnetic matrix (NM), i.e., Z M>ZNM (Fig. 3 ). The situation is reversed when ZM<ZNM.A tt h ei n ﬂ u e n c e dFIG. 2. The frequency dependence of the reﬂection coefﬁcient, RN, calculated using Eq.(4)forN¼1 (i.e., r),N¼3, and N¼9, is compared to that for a semi-inﬁnite array, R1, calculated using Eq. (5). We assume a¼10/C02andl0HB¼180 mT. The solid vertical line indicates the Kittel frequency, fFMR, and the inset is focused around the region of the magnetoacoustic Fano resonance.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 102402 (2020); doi: 10.1063/5.0018801 117, 102402-3 Published under license by AIP Publishingedge, a shift in the edge frequency is induced by proximity to the Kittel frequency fFMR.34This band shift is separate from the induced trans- parency; this becomes apparent when fFMRsweeps a bandgap with a width signiﬁcantly exceeding the Fano resonance linewidth, as shownin the supplementary material . We emphasize that the magnetoelastic effects shown in Fig. 3 remain signiﬁcant even for a realistic damping value of a¼10 /C02.T h i si sac o n s i d e r a b l ei m p r o v e m e n tc o m p a r e dt oa single resonator where this damping value would completely suppress the Fano resonance.14 To characterize the tunability of the acoustic reﬂection coefﬁcient by the bias magnetic ﬁeld, we introduce the ﬁeld modulation coefﬁ- cient f¼@jR1j=@HB, the frequency and ﬁeld dependence of which around the ﬁrst three phononic bandgaps is shown in Fig. 4 .I np r a c - tice, the higher frequency phononic bandgaps could be more difﬁcultt oa c c e s s ,a st h i sw o u l dr e q u i r eal a r g eb i a sm a g n e t i cﬁ e l d( >0.25 T). Hence, we limit our analysis to frequencies around the ﬁrst bandgap [Fig. 4(d) ]. We see that fis signiﬁcantly enhanced when f FMR(solid, black) is tuned to the proximity of the bandgap edges (vertical, dashed,black), as expected for a Fano resonance induced modulation of scat-tering coefﬁcients. 15The strength of the Fano resonance is determined by the interplay between the damping and the strength of the magne- toelastic coupling.14The damping in our metamaterial is modulated by the Borrmann effect. This leads to an asymmetry of the ﬁeld modu-lation coefﬁcient with regard to the lower and higher frequency edgesof the phononic bandgap [ Fig. 4(d) ]. In summary, we have shown that the metamaterial approach is indeed helpful for magnetoacoustics. Our hybrid metamaterials,formed by 1D arrays of resonators, magnify the effect of magnetoelas- tic coupling upon the acoustic scattering, thereby mitigating theGilbert damping to tolerable levels. The scattering is tunable by a biasmagnetic ﬁeld and exhibits a rich and complex behavior, such as the induced transparency and Borrmann effect. The next step toward real- istic structures and devices would be to extend the model into the sec-ond and third dimensions and to consider surface acoustic waves.However, the design strategies presented here will remain useful. Ourresults may help in engineering magnetoacoustic sensors, actuators, radio frequency modulators, and other devices that could beneﬁt from the enhanced magnetic ﬁeld modulation of the amplitude or the phaseof acoustic waves, as demonstrated here. See the supplementary material for (i) the derivation of the reﬂec- tion and transmission coefﬁcients for ﬁnite arrays of scatterers, (ii) the derivation of the phonon spectral function, (iii) additional ﬁgures for the transmission and absorbance for ﬁnite arrays, and (iv) signaturesof magnetically induced transparency. The research leading to these results has received funding from the Engineering and Physical Sciences Research Council of theUnited Kingdom (Grant Nos. EP/T016574/1 and EP/L015331/1)and from the European Union’s Horizon 2020 research and innovation program under Marie Sklodowska-Curie Grant Agreement No. 644348 (MagIC). DATA AVAILABILITY The data that support the ﬁndings of this study are available within the article and its supplementary material .FIG. 3. The frequency dependence of the acoustic reﬂection coefﬁcient, R1, from a semi-inﬁnite array with a¼10/C02is shown for three different values of the Kittel frequency, fFMR, tuned by the bias magnetic ﬁeld, HB. The solid vertical lines indi- cate the position of fFMR. The dashed black curve represents RB¼0, i.e., R1for B¼0. (a) Regime I: fFMR is in the passband, far from the phononic bandgap. The insets compare R1(solid) with r(dotted) at l0HB¼50 mT (left, red) and 150 mT (right, black). Regime II: (b) fFMR atl0HB¼92 mT is inside the bandgap, and (c) fFMRatl0HB¼98 mT is close to the bandgap. FIG. 4. The frequency and ﬁeld dependence of the absolute value of the modulation coefﬁcient, jfj¼j@jR1j=@HBj, is shown around the (a) ﬁrst, (b) second, and (c) third phononic bandgaps. The solid white lines represent fME. (d) The frequency and ﬁeld dependence of the modulation coefﬁcient, f¼@jR1j=@HB, is shown around the ﬁrst phononic bandgap. The position of the bandgap edges at B¼0i sm a r k e dw i t hd a s h e d vertical lines, and the solid black line represents fME. In all panels, a¼10/C02.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 102402 (2020); doi: 10.1063/5.0018801 117, 102402-4 Published under license by AIP PublishingREFERENCES 1G. P. Perrucci, F. H. P. Fitzek, and J. Widmer, “Survey on energy consumption entities on the smartphone platform,” in IEEE Vehicular Technology Conference (2011), Vol. 73, p. 1. 2S. Manipatruni, D. E. 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1.4860946.pdf	Current induced domain wall dynamics in the presence of spin orbit torques O. Boulle, L. D. Buda-Prejbeanu, E. Jué, I. M. Miron, and G. Gaudin Citation: Journal of Applied Physics 115, 17D502 (2014); doi: 10.1063/1.4860946 View online: http://dx.doi.org/10.1063/1.4860946 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The influence of the spin-orbit torques on the current-driven domain wall motion AIP Advances 3, 072109 (2013); 10.1063/1.4813845 Spin-wave excitations induced by spin current through a magnetic point contact with a confined domain wall Appl. Phys. Lett. 101, 092405 (2012); 10.1063/1.4745777 Current-induced motion of a transverse magnetic domain wall in the presence of spin Hall effect Appl. Phys. Lett. 101, 022405 (2012); 10.1063/1.4733674 Micromagnetic analysis of the Rashba field on current-induced domain wall propagation J. Appl. Phys. 111, 033901 (2012); 10.1063/1.3679146 Effect of ac on current-induced domain wall motion J. Appl. Phys. 101, 09A504 (2007); 10.1063/1.2713211 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.118.88.48 On: Sun, 01 Jun 2014 13:15:34Current induced domain wall dynamics in the presence of spin orbit torques O. Boulle,a)L. D. Buda-Prejbeanu, E. Ju /C19e, I. M. Miron, and G. Gaudin SPINTEC, CEA/CNRS/UJF/INPG, INAC, 38054 Grenoble Cedex 9, France (Presented 6 November 2013; received 23 September 2013; accepted 14 October 2013; published online 17 January 2014) Current induced domain wall (DW) motion in perpendicularly magnetized nanostripes in the presence of spin orbit torques is studied. We show using micromagnetic simulations that the direction of the current induced DW motion and the associated DW velocity depend on the relativevalues of the ﬁeld like torque (FLT) and the Slonczewski like torques (SLT). The results are well explained by a collective coordinate model which is used to draw a phase diagram of the DW dynamics as a function of the FLT and the SLT. We show that a large increase in the DW velocitycan be reached by a proper tuning of both torques. VC2014 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4860946 ] The dynamics of magnetic domain walls (DWs) induced by a spin polarized current has attracted a large effort of research during the last ten years motivated not only by promising devices in the ﬁeld of magnetic storage and log-ics 1but also by the wealth of the physics involved. The prin- ciple behind current induced DW motion is the spin transfer effect where the spin current going through the DW is trans-ferred to the DW magnetization leading to spin transfer tor- que and DW motion in the carrier direction. 2Whereas ﬁrst experiments in soft in-plane magnetized stripes were rela-tively well described by this scheme, it was shown later that the spin transfer effect was much more efﬁcient in ultrathin out-of-plane magnetized multilayers, such as Pt/Co/Pt 3or Pt/Co/AlOx multilayers4meaning that other effects were at stack. This was ﬁrst interpreted as the result of very large additional non-adiabatic effects due to the incompleteabsorption of the spin current, 5but the physical mechanisms behind were elusive.3In addition, puzzling experiments were reported, for example in asymmetricPt(4 nm)/Co(0.6 nm)/Pt(2 nm) multilayers, where the DW is moved in the direction or opposite to the current direction when reversing the position of both Pt layers in the stack, incontradiction with the standard spin transfer mechanism. 6,7 Recently, it was shown that the large spin orbit coupling due to the presence of heavy metal (such as Pt or Ta) as well asthe inversion asymmetry due to the interfaces can lead to additional current induced spin-orbit torques. 8–10Two types of torques have been identiﬁed: (1) a ﬁeld like torque (FLT)m/C2JH FLuysimilar to the one exerted by an in-plane mag- netic ﬁeld JHFLuyoriented perpendicularly to the current direction (see Fig. 2); (2) a Slonczewski like torque (SLT) /C0c0JHSLm/C2ðm/C2uyÞ. Recent experimental7,11,12and theo- retical works19–22have underlined that these torques may actually explain the apparent very large efﬁciency of the spintransfer effect reported in these systems. In this paper, we study how the FLT and the SLT affect the current induced DW dynamics. Using micromagneticsimulations, we show that depending on their relative values,the DW can move in one direction or the other with respect to the current direction. The results of micromagnetic simu- lations are well reproduced by a collective coordinate model (CCM) taking into account both torques. This model is usedto predict a phase diagram of the DW direction and velocity as a function of the FLT and SLT and allows the identiﬁca- tion of the torque conditions for maximum velocity. We consider a perpendicularly magnetized stripe with a width of 100 nm. Micromagnetic simulations are based on the Landau-Lifschitz-Gilbert equation to which the currentinduced torques have been added @m @t¼/C0c0 l0MsdE dm/C2mþam/C2@m @t/C0u@m @x þbum/C2@m @x/C0c0m/C2m/C2HSLJuyþc0m/C2JHFLuy; (1) where c0¼l0cwith cthe gyromagnetic ratio, Ethe energy density, and Msthe saturation magnetization. The third term is the adiabatic spin-transfer torque where u¼JPglB=ð2eMsÞ,lBthe Bohr magneton, Jthe current density, Msthe saturation magnetization, Pthe current spin polarisation. The fourth term is the non-adiabatic torque described by the dimensionless parameter b.5For the micro- magnetic simulations, the fo llowing parameters have been used: the exchange constant A¼10/C011A/m, the anisotropy constant K0 an¼1:25/C2106J=m3,Ms¼1.1/C2106A/m, the damping parameter a¼0.5,b¼0, and P¼1. The thickness of the magnetic layer is 0.6 nm. 3 D micromagnetic simulations were carried out using a homemade code.13Note that we do not consider the Dzyaloshins kii-Moryia interaction11,12,14,15so that a Bloch DW equilibrium conﬁguration is observed. Fig.1(a)shows the results of micromagnetic simulations (dots) of the DW velocity as a function of the current density forHSL¼0.025 T/(1012A/m2), and different values of the FLT. The FLT strongly affects the current induced DWmotion: depending on the value of the FLT, the DW velocity is positive or negative, meaning the DW moves in the direc- tion or opposite to the current direction, and the DW velocityamplitude depends non-monotonically on the FLT. a)Author to whom correspondence should be addressed. Electronic mail: Olivier.boulle@cea.fr. 0021-8979/2014/115(17)/17D502/3/$30.00 VC2014 AIP Publishing LLC 115, 17D502-1JOURNAL OF APPLIED PHYSICS 115, 17D502 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.118.88.48 On: Sun, 01 Jun 2014 13:15:34To better understand these results, we consider a CCM which assumes that the DW keeps its static structure during its motion and the DW is described by its position qand the DW angle w(see Fig. 2).5,16 We ﬁrst assume small current densities so that the DW structure and the magnetization in the domains are littleaffected by the spin orbit torques and one can assume a standard Bloch proﬁle. The polar and azimuthal angles hand uare then assumed as h¼2arctan ðexpf½x/C0q/C138=DðwÞgÞand u¼w/C0p=2 with wconstant. Here DðwÞ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃ A=jp is the DW width with j¼ðK anþKsin2wÞ1=2where Kan ¼K0 an/C0l0M2 s=2;Han¼2Kan=ðl0MsÞ;K¼l0MsHk=2 with Hkthe DW demagnetizing ﬁeld. The integration of Eq. (1) over this DW proﬁle leads to the following equations: _wþa_q D¼bu Dþc0JHSLp 2sinw; (2) _q D/C0a_w¼c0Hk 2sin2wþu Dþc0HFLJp 2sinw: (3) The SLT enters the equations as an additional force on the DW which is proportional to sin w, i.e., to the component of the DW magnetization along the current direction. It is thus zero for a perfect Bloch conﬁguration whereas it is maximal for a N /C19eel conﬁguration. The effect of the SLT on the DW can be seen as an effective non-adiabatic parameter bSLsinw with bSL¼c0HSLDpeMs=ðglBPÞ. For the value of HSL/C240.07 T/(1012A/m2) measured in Pt,10one obtains a large value bSL/C243. The SLT can thus account for the large bvalues reported in the literature in perpendicularly magnetized nano- stripes.3A simple expression of the DW velocity can be deduced from Eqs. (2)and (3)in the steady state regime (_w¼0) v¼bu aþc0HSLJD ap 2sinw: (4)The DW angle wwill thus determine the direction and the amplitude of the DW velocity. The effect of the SLT is in fact similar to an easy axis magnetic ﬁeldH SL¼m/C2JHSLuy¼sinwJHSLuz. For the conﬁguration of Fig.2andJ>0, a positive (resp. negative) sin wleads to HSL aligned upward (resp. downward) and thus moves the DW in the electron (resp. current) direction. The steady state value of wis the result of a balance between the different in-plane components of the currentinduced torques (see Fig. 2). In the steady state regime (_w¼0), there are two stable positions for the angle wclose to 0 and pand at low current density, wcan be switched hys- teretically between these two positions when sweeping cur- rent (see Fig. 1(b)). For small angles wclose to 0 and w w¼uðb/C0aÞ=D ac0Hkþ/C15p 2c0JðaHFL/C0HSLÞðþpÞ; (5) v¼bu aþp 2c0HSLJ auðb/C0aÞ p 2c0JðaHFL/C0HSLÞþ/C15ac0Hk;(6) where /C15¼1( r e s p . /C15¼/C01) for w/C240 (resp. w/C24p). The direc- tion of the DW motion is thus set by the relative values of b andaon the one hand, and by the relative value of HSLand aHFLon the other hand. We show in Fig. 1(a) (continuous line), the DW velocity predicted by this CCM. A goodagreement is obtained with the micromagnetic simulations forH FL¼0o r HFL¼0.1 T/(1012A/m2) but the agreement is less satisfactory at larger current densities forH FL¼0.2 T/(1012A/m2). In this case, the large transverse magnetic ﬁeld Ht¼JHFLinduced by the FLT affects the do- main and DW structure and the simple assumption of a BlochDW structure does not hold. To account for the effect of large H t, we consider a CCM which assumes a more complicated DW structurewhere the domain and DW deformation induced by H tis taken into account.17To describe the current induced DW dynamics for such a DW proﬁle, a Lagrangian approach isconsidered. 15,17,18The Lagrange-Rayleigh equations then lead to: ar 4A_q/C0b au/C18/C19 /C0c0HSLJu0sinh0cosðh0/C0wÞþ@fk @w_w¼0; (7) @fk @wð_q/C0uÞ/C0ar 4A1 k2 0þ1 3@1=k0 @w/C18/C192" /C2p2/C0u2 0/C02u2 0 1/C0u0cotan u0 ! # _w¼c0 2Ms@r @w:(8) Here ris the DW energy density, h0¼arccos ð/C0hsinwÞ; u0¼arctan ½ﬃﬃﬃﬃﬃﬃﬃﬃ 1/C0h2p hcosw/C138 with h¼JHFL=Han;k0¼sinu0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðKanþKsin2wÞ=Aq ;fkðw;hÞis a dimensionless parameter deﬁned in Refs. 17and18. From Eq. (7), one can easily derive the DW velocity in the steady state regime ( _w¼0) _q¼bu=aþ4A rac0HSLJu0sinh0cosðh0/C0wÞ: (9)FIG. 1. (a) DW velocity as a function of the current density for HSL¼0.025 T/(1012A/m2) and different values of the FLT deduced from micromagnetic simulations (dots), the standard CCM (continuous line), and the extended CCM taking into account the DW deformation induced by the FLT (dashed line). The DW demagnetizing ﬁeld Hk¼33 mT is used for the CCM simulations. (b) DW velocity and DW angle was a function of J (HFL¼0 and HSL¼0.1 T/(1012A/m2)) calculated from the CCM. FIG. 2. Schematic representation of the different current induced torques acting on the DW magnetization.17D502-2 Boulle et al. J. Appl. Phys. 115, 17D502 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.118.88.48 On: Sun, 01 Jun 2014 13:15:34If one compares Eq. (9)to Eq. (4), one can see that Htonly affects the DW velocity induced by the SLT and this in twoways. First, the DW width in Eq. (4)is replaced by an effec- tive DW width 4 A/r, which is increased by H t. Second, Ht changes the steady state angle wand thus the DW velocity. These two elements lead to an increase of the DW velocity as compared to the standard CCM model. We plot in Fig. 1(a)(dotted lines), the prediction of the model in the case of a high FLT HFL¼0.2T/(1012A/m2). The model allows a bet- ter agreement with the micromagnetic simulations compared to the standard CCM model, in particular for high currentdensities where H tis large. However, one limitation of our model is that it does not take into account the deformation of the domain induced by the SLT, which may be relevant athigh current density and large values of the SLT such that JH SL/C24Han. This extended model can be used to draw a phase dia- gram of the DW dynamics as a function of the FLT and SLT. Fig. 3shows (a) the DW velocity and (b) the DW angle win color scale as a function of the SLT and the FLT for J¼1012A/m2, obtained by solving numerically Eqs. (7)and (8). One can note that the direction of the DW motion depends on the relative values of the SLT andFLT: on the top left of the diagram, for large SLT or low FLT, the DW velocity is positive whereas on the lowerright corner, for large FLT and low SLT, the DW velocity is negative. This change in the direction of the DW motion goes with a switching of the DW chirality (Fig. 3(b)) with p=2<w<pðsinw>0Þfor the positive velocity region and/C0p=2<w<0ðsinw<0Þfor the negative velocity region. As expected from Eq. (4), the direction of the DW motion is determined by the sign of sin w. In these two regions, the DW dynamics stays in the steady state regime whereas in between, a small region with precessional re-gime is observed. Interestingly, the largest DW velocity is predicted at the border of the precessional regime, and corresponds to wclose to 6p/2 where the SLT is maxi- mum. For this particular condition, the SLT and FLT nearly compensate so that large angles wand thus large SLT can be reached. One can show that these borderswith maximal velocity corresponds to the condition H SL/C25aHFL6ðaP/C22hÞðDeMsÞ. A large DW velocity can thus be reached by a proper tuning of the SLT andFLT. Experimentally, whereas the SLT and the FLT due to the spin orbit coupling are more related to the intrin- sic properties of the material, the Oersted ﬁeld generatedby the current ﬂowing in the metallic layers surrounding the magnetic layer also leads to a FLT. For a given cur- rent density, its amplitude can be modiﬁed by playingon the width and thickness of the metallic layers. To conclude, the direction of the current induced DW motion and the associated DW velocity depend on the rela-tive values of the ﬁeld like torque (FLT) and the Slonczewski like torques (SLT). A large increase in the DW velocity can be reached by a proper tuning of both torques. This work was supported by project Agence Nationale de la Recherche, Project No. ANR 11 BS10 008 ESPERADO. 1S. S. P. Parkin et al.,Science 320, 190 (2008). 2L. Berger, J. Appl. Phys. 49, 2156 (1978). 3O. Boulle et al.,Mater. Sci. Eng.: R: Reports 72, 159 (2011). 4I. M. Miron et al.,Nature Mater. 10, 419 (2011). 5A. Thiaville et al.,Europhys. Lett. 69, 990 (2005). 6R. Lavrijsen et al.,Appl. Phys. Lett. 98, 132502 (2011). 7P. P. J. Haazen et al.,Nature Mater. 12, 299 (2013). 8I. M. Miron et al.,Nature 476, 189 (2011). 9L. Liu et al.,Science 336, 555 (2012). 10K. Garello et al.,Nature Nanotechnol. 8, 587 (2013). 11K.-S. Ryu et al.,Nature Nanotechnol. 8, 527 (2013). 12S. Emori et al.,Nature Mater. 12, 611 (2013). 13L. Buda et al.,Compt. Mater. Sci. 24, 181 (2002). 14A. Thiaville et al.,Europhys. Lett. 100, 57002 (2012). 15O. Boulle et al.,Phys. Rev. Lett. 111, 217203 (2013). 16J. Slonczewski and A. P. Malozemoff, Magnetic Domain Walls in Bubble Materials (Academic Press, New York, 1979). 17A. Hubert, Theorie der Dom €anenw €ande in geordneten Medien (Springer, Berlin, 1974). 18O. Boulle et al.,J. Appl. Phys. 112, 053901 (2012). 19S. Seo et al.,Appl. Phys. Lett. 101, 022405 (2012). 20E. Martinez et al.,AIP Advances 3, 072109 (2013). 21K. M. D. Hals and A. Brataas, Phys. Rev. B 88, 085423 (2013). 22J. Linder and M. Alidoust, Phys. Rev. B 88, 064420 (2013). FIG. 3. (a) DW velocity and (b) steady state DW angle win color scale as a function of the SLT and the FLT, obtained by solving numerically Eqs. (7) and(8)forJ¼1012A/m2. The black lines mark out the region associated with precessional motion.17D502-3 Boulle et al. J. Appl. Phys. 115, 17D502 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.118.88.48 On: Sun, 01 Jun 2014 13:15:34
1.369878.pdf	Coupling mechanisms in exchange biased films (invited) T. C. Schulthess and W. H. Butler Citation: J. Appl. Phys. 85, 5510 (1999); doi: 10.1063/1.369878 View online: http://dx.doi.org/10.1063/1.369878 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v85/i8 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 09 May 2013 to 131.211.208.19. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsExchange Bias II Ivan Schuller, Chairman Coupling mechanisms in exchange biased ﬁlms invited T. C. Schulthessa)and W. H. Butler Metals and Ceramics Division, Oak Ridge National Laboratory Oak Ridge, Tennessee 37831-6114 We use an atomistic Heisenberg model in conjunction with the classical Landau Lifshitz equation for the spin motion to study coupling mechanisms between ferromagnetic ~FM!and antiferromagnetic ~AFM !ﬁlms. Calculations for CoO/FM illustrate that there are two coupling mechanisms at work, the spin–ﬂop coupling and an AFM–FM coupling through uncompensateddefects. While the latter accounts for exchange bias and related phenomena, the former gives rise toa large coercivity and perpendicular alignment between FM spins and AFM easy axis. Acombination of the two mechanisms explains apparent discrepancies between reversible andirreversible measurements of the AFM–FM coupling. © 1999 American Institute of Physics. @S0021-8979 ~99!31508-5 # I. THE PROBLEM OF AFM–FM COUPLING ‘‘Exchange bias,’’1which refers to a shift ( Heb) in the magnetization curve away from the zero ﬁeld axis, is prob-ably the most intriguing of several phenomena 2observed when a ferromagnet ~FM!is in contact with an antiferromag- net~AFM !. Consequently most theoretical work1,3–9on the subject has been primarily concerned with the description ofthis asymmetry in the magnetization curve. Despite four de-cades of research since its discovery, the understanding ofthis effect is still not established. The key issue in a theory of exchange bias is understand- ing how the coupling between the AFM and FM leads to aunidirectional anisotropy. The experimentally observed shift in the magnetization curve implies that the two conﬁgura-tions at the respective endpoints of the curve have differentenergies. In the case of an AFM/FM system with largeenough anisotropy in the AFM, only the spins in the FM willinvert upon reversal of the applied ﬁeld. Since the two con-ﬁgurations are not equivalent by inversion symmetry theycan have different energies, depending on the nature of thecoupling between the AFM and FM spins. When the interfacial layer of the AFM is uncompensated and perfectly ﬂat, the existence of a coupling between theFM and AFM is straightforward to understand. 1It is clear that in a simple model in which the interfacial AFM spinsmaintain ~approximately !their initial relative orientations, the initial and ﬁnal conﬁgurations, before and after reversalof the applied ﬁeld will have different energies. For this case,Neel 3and later Mauri et al.4have shown, that realistic values forHebcan be obtained when a domain wall forms in the AFM during the reversal of the FM magnetization. When the interface plane is compensated however, the nature of theAFM–FM coupling is not obvious. Experiments indicate that the loop shift is of similar magnitude for compensated and uncompensated inter-faces. 10,11Furthermore, an interface which is, in principle,uncompensated at the atomic scale may be compensated on average over longer length scales when it is rough. Onlywhen interfacial terraces are much larger than the AFM do-main wall width, would one have the situation where theAFM can break up into domains which have uncompensatedinterfaces with the FM. Otherwise the AFM domains willspan several terraces and their interface to the FM will becompensated. Thus the case of magnetically compensatedAFM interface planes seems to be more relevant for the ex-change bias problem. Using localized atomic spins, Hinchey and Mills 12and recently Koon5demonstrated that, due to frustration of inter- facial spins, the FM magnetization will align perpendicularto the AFM easy axis when the AFM interface plane is com-pensated. This establishes the coupling between the FM andthe AFM when the interface is compensated and is referredto asspin–ﬂop coupling . However, contrary to Koon’s ex- pectation, spin–ﬂop coupling does not lead to the formationof a domain wall in the AFM during FM magnetization re-versal and therefore in itself does not lead to exchangebias. 13 A different route was taken by Malozemoff,6who ex- plained the coupling as due to a random ﬁeld which he at- tributed to interface roughness. This theory is particularlyappealing because it accounts for many of the observationsthat are related to the loop shift. 7However, some of Maloz- emoff’s conclusions with regard to dependence of the ex-change bias on the AFM layer thickness 8and the observed increase of Hebwith decreasing interface roughness in some systems14have led to arguments against his theory.15,5Suhl and Shuller9have recently proposed yet another mechanism which explains the loop shift. They use a quantum mechani-cal description of the spins and show that the emission andreabsorption of virtual spin waves leads to exchange bias. Clearly there are several possible mechanisms that lead to the result that the energies of conﬁgurations with reversedFM magnetization are different which, in turn, implies a uni-directional shift of the magnetization curve. However, atheory has to explain other important effects that are known a!Electronic mail: schulthesstc@ornl.govJOURNAL OF APPLIED PHYSICS VOLUME 85, NUMBER 8 15 APRIL 1999 5510 0021-8979/99/85(8)/5510/6/$15.00 © 1999 American Institute of Physics Downloaded 09 May 2013 to 131.211.208.19. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsto be related to the AFM–FM coupling. Discussing all of these effects would require ant entire book chapter. Thus werestrict ourselves to the following effects which have beenobserved in many different AFM–FM systems: ~1!Exchange bias, positive and negative, as well as related phenomenadiscussed by Malozemoff 7and the dependence of Hebon the interface roughness. ~2!The large coercivity16–18that is known to be related to the coupling because it disappearswhen the AFM is disordered. ~3!The perpendicular align- ment between the FM magnetization and the AFM easyaxis. 16,17,19~4!The observation that reversible measurements of the coupling can yield values several times larger thanthose determined from the loop shift, H eb.20,21 While all of these effects are observed in some AFM/FM systems, some are missing in other systems. For example, acoercivity related to the coupling is observed when Permal-loy~Py!is in contact with FeRh but the system shows no exchange bias. 18The opposite is the case when FeMn or IrMn are used as AFMs: in these systems, the coercivity issmall but the exchange bias can be considerable. The theorythus also has to be able to account for the possible absence ofsome of the effects. In the present work, we start from a microscopic descrip- tion within an extended Heisenberg model and use theLandau–Lifshitz equation to investigate the rather complexmagnetic conﬁgurations that can occur at AFM–FM inter-faces. We use CoO with ~111!interface planes as a model AFM-layer, since it is the system for which many experi-ments are published and, more importantly, in which all ofthe above mentioned effects are observed. Our microscopiccalculations will be limited to idealized situations in whichthe AFM and FM ﬁlms are single crystals and in a singledomain state ~i.e., we exclude the formation of domain walls perpendicular to the interface !. But nevertheless, we are able to show, that even in this simpliﬁed context most relevanteffects can be accounted for. We will discuss implications ofour results to situations that currently cannot be handled witha microscopic approach. II. METHOD AND MODEL In our model, a spin conﬁguration is a set of three- dimensional vectors, M[$mWi%, which are located on atomic sites,iwhere we assume the bilayer to be periodic in the two dimensions parallel to the interface. The energy of the spinconﬁguration consists of four terms, E @M#5EZ1EJ1EA1ED, of which the ﬁrst three are, respectively, the Zeeman energy, EZ5(imWiHWext, the exchange energy, EJ52(iÞjJijsWisWj, withsW5mW/umWu, and the anisotropy energy, EA 5(iKisin2ui. The magnetic moments, mi, the exchange pa- rameters, Jij, and the anisotropy constants, Ki, for CoO/ Py~111!and CoO/Co ~111!bilayers have been speciﬁed elsewhere.22The mangnetostatic contribution to the energy is ED5(iÞj$mWimWj23(mWinˆij)(mWjnˆij)%/uRWi2RWju3, wherenˆijis the unit vector that points into the direction that connect the sites atRWiandRWj. The magnetic moments are subject to the Landau–Lifshitz equation of motion ~EOM !with the Gilbert–Kelley form for the damping term,] ]tmWi52g~mWi∧Hi!1sSmWi∧] ]tmWiD1 umWiu, whereHi@M#52(]/]mWi)E@M#is the local magnetic ﬁeld, gis the gyromagnetic ratio, and sis an arbitrary damping parameter. The results do not depend on the actual values ofthe damping constant because the EOM is only used to ﬁndequilibrium solutions and to determine the stability of thesesolutions. With present computational resources, the treat-ment of domain walls perpendicular to the interface is notfeasible with our method. We thus treat only states in whichthere is a single domain parallel to the interface in order thatthe magnetic conﬁgurations are two-dimensional periodic.This implies that during ﬁeld reversal the magnetization ro-tates coherently. We start with an initial solution of the EOMfor a certain applied ﬁeld. Then we change the values of theapplied ﬁeld in steps and determine the new solution of theEOM. When the applied ﬁeld is reversed and there is anenergy barrier that prevents the FM from switching, the ini-tial solution becomes metastable. By further increasing themagnitude of the ﬁeld one approaches a bifurcation pointwhere the metastable solutions becomes unstable and the FMmagnetization switches to align with the applied ﬁeld. Themagnitude of the applied ﬁeld at the bifurcation point thencorresponds to the coercive ﬁeld. III. SPIN–FLOP COUPLING We begin with a qualitative discussion of the possible spin conﬁgurations in our CoO/FM system. Since we assumethe CoO interface plane to be compensated with a ﬁxeduniaxial anisotropy, we are left with two equivalent AFM FIG. 1. Top view of CoO–FM ~111!interface with compensated AFM interface plane. Filled and open arrows indicate, respectively, the unrelaxedand relaxed moment directions in the AFM layer. FM moment directions aregiven by triangles. The two possible spin-ﬂop states for a given FM mag- netization direction are labeled with AandB, respectively. A WandBWare the corresponding states with reversed FM magnetization and a domain wallin the AFM. The fan of arrows on one of the atoms indicates schematicallyhow the moment directions change going into the AFM from the interfaciallayer ~open arrow !to the interior ~ﬁlled arrow !. The spin-ﬂop states with reversed FM magnetization are denoted by A 8andB8.5511 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 T. C. Schulthess and W. H. Butler Downloaded 09 May 2013 to 131.211.208.19. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsspin-conﬁgurations ~denoted by AandBin Fig. 1 !. If both the AFM and the FM spins are rigidly aligned among them-selves, the energy ~withH ext50!is independent of the rela- tive alignment between FM magnetization and AFM easyaxis. However, when the spins are allowed to cant in order tominimize the energy ~eg., by solving the EOM !, the FM moments align perpendicular to the AFM easy axis whichgives rise to the spin–ﬂop coupling. In the following, we callthe axis parallel to the interface and perpendicular to theAFM easy axis the spin–ﬂop coupling axis . For each AFM spin-conﬁguration there are two ways to align the FM perpendicularly and we are left with four states~A,A 8,B, andB8in Fig. 1 !that have the same energy when no external ﬁeld is applied. The Zeeman term lifts the four-fold degeneracy when a ﬁeld is applied and selects the twostates which have an FM spin-component parallel to the ﬁeldas the new energy minima. We choose the initial ﬁeld direc-tions such, that states AandBhave lowest energy. Since both conﬁgurations are equivalent, we will choose Afor the remainder of the discussion. When the magnetic ﬁeld is re-versed there are, in principle, two possible ﬁnal conﬁgura-tions. The ﬁrst possibility is the state in which a domain wallhas formed in the AFM ~A Win Fig. 1 !. The second possibil- ity is the spin-ﬂop state with reversed FM spins ~A8in Fig. 1!. Since, upon ﬁeld reversal, conﬁguration A8is energeti- cally equivalent to Athe magnetization loop that corre- sponds to the path A!A8!Awill not be shifted. On the other hand, AWhas to accommodate a domain wall and thus has higher energy than A8andA. Therefore the loop of A !AW!Awill show exchange bias. This is as far as we can go with a qualitative discussion. The decision as to which ofthe two paths the spin-system follows depends on the relativeenergy barriers between the states and has to be determinednumerically. When the spin motion is restricted to the plane parallel to the interface, as in the calculation of Koon, 5the pathA !A8is impossible since it requires the spins to come out of plane when they rearrange. However, when the EOM issolved without any restriction, 13the energy barrier of A !A8is lower than the domain wall energy in AW, and the system switches between AandA8when the ﬁeld is cycled, giving rise to a symmetric magnetization curve. This is thecase in both models, 13Koon’s and the present CoO/FM bi-layer. A typical magnetization curve is shown in Fig. 2. The effect of spin-ﬂop is that it hinders the FM magnetizationreversal giving rise to hysteresis with large coercivity. Val-ues forH care given in Table I. IV. A MECHANISM FOR THE LOOP SHIFT The conclusion of the last section is made under the idealized assumption that the interface is perfectly ﬂat. Re-alistically, however, the interface will be rough and containdefects such as dislocations. A simpliﬁed way to incorporatedefects related to interfacial roughness such as steps, islands,or point defects into our calculations, is to replace an FM sitewith a corresponding arrangement of AFM sites on the FMside of the interface. The case of a point defect is illustratedin Fig. 3. The defect moment is coupled to only one of thetwo AFM sublattices. The net interaction of the FM with thetwo AFM sublattices is no longer balanced. This causes theFM to cant away from the spin–ﬂop coupling axis as indi-cated by conﬁguration Din Fig. 3. Conﬁguration Dis only one of four possible states with lowest energy, and we as-sume that it was selected by an external ﬁeld that points intothe second quadrant of the xyplane. When the ﬁeld and the FM magnetization are reversed we arrive at conﬁguration D 8 in Fig. 3. As a consequence of the unbalanced exchange coupling between the FM, the defect and the AFM, the en-ergy ofD 8will be higher and the magnetization curve that corresponds to D!D8!Dwill be shifted. The magnitude of the shift depends on the density of these uncompensated defects ~larger shift for more uncom- pensated defects !. For CoO/Py, however, Takano et al.23 have measured the amount of uncompensated AFM magne- tization along the direction of the applied ﬁeld and haveshown that it correlates with the values of H eb. In our cal- culation, we use a 4 34 unit cell with one point defect per cell and apply a ﬁeld parallel to the interface plane at an FIG. 2. Typical set of magnetization curves for perfect ~diamonds !and rough ~squares !interface. This particular example is for a 200 Å Py ﬁlm withJF–F516meV and AFM anisotropy axis along the @1¯1¯7#.TABLE I. Hcfor ﬂat CoO/Py ~200 Å !interface as well as HcandHebfor interface with uncompensated AFM-defects ~in Oe !. AFM easy axisJF–F ~meV!Flat interface HcInterface with defects Hc Heb @1¯1¯7#16 885 575 75 @1¯01# 16 1625 1250 74 @1¯01# 9.4 1336 1039 38 FIG. 3. Magnetic conﬁguration of an CoO–FM ~111!interface, where one FM site has been replaced by an AFM site. The nomenclature is similar tothat of Fig. 1, where the open triangles in conﬁguration Dindicate that the FM moments are slightly canted away from the ideal spin–ﬂop coupling axis. In conﬁguration D 8the FM spins are simply inverted. Dashed lines highlight the next nearest neighbor interactions between the defect site andone of the two AFM sublattices.5512 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 T. C. Schulthess and W. H. Butler Downloaded 09 May 2013 to 131.211.208.19. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsangle fH510° from the yaxis. For this setup, we ﬁnd that the amount of uncompensated AFM magnetization projected onto the ﬁeld axis is about 1% of the moments in a CoOmonolayer. For this amount and a Py ﬁlm thickness of d FM 5300Å Takano et al.measured a loop shift of 50 Oe ~this corresponds to about 75 Oe when dFM5200Å !. A typical magnetization curve is shown in Fig. 2 and the calculatedloop shifts and coercivities are given in Table I. The calcu-lated values for H ebagree reasonably well with experiment. We conclude this section with a remark on positive ex- change bias. In conﬁguration D, the net uncompensated AFM magnetization points away from the FM spins and theapplied ﬁeld. This is because the negative exchange couplingto the FM overwhelms the Zeeman coupling between the netmagnetization of the defect and the applied ﬁeld. In prin-ciple, however, one could think of a different system inwhich the Zeeman coupling of the defect is more importantfor very large cooling ﬁelds. In this case the spin-conﬁguration is preset into the state with higher energy~which corresponds to D 8!and would switch to the lower energy state upon ﬁeld reversal, resulting in a positive shiftof the magnetization curve. Note that cooling in a very largeﬁeld is the actual requirement to observe positive exchangebias. 24 V. REVERSIBLE MEASUREMENTS In order to determine the spin–ﬂop coupling strength, we apply a small ﬁeld, H', parallel to the interface plane but perpendicular to the spin–ﬂop coupling axis ( fH590°). We then solve the EOM and determine the energy differenceDE5E(H ')2E(0) and the angle, f, between the total mag- netization and the coupling axis. For small enough ﬁelds25 the results satisfy the relations DE5Keff~dFM!sin2f'Keff~dFM!f2, ~1! where we have introduced effective coupling strength, Keff. Since the spin–ﬂop coupling only applies to interfacial spinsand the FM spins are not rigidly coupled to each other, theFM magnetization will twist when the ﬁeld H 'is applied. The energy of this twist is contained in DEand depends on the thickness, dFM, of the FM ﬁlm. Thus Keffshows a thick- ness dependence as well. Coupling the FM spins rigidly toeach other would remedy this problem but would alsochange the spin–ﬂop coupling strength, since the spin-relaxation in the FM also contributes to the effective cou-pling. Our method for calculating K effis equivalent to the experiment of Miller and Dahlberg,20who have applied a small ﬁeld perpendicular to the Co magnetization in aCoO/Co bilayer and determined the angle fwith the aniso- tropic magneto-resistance technique. The theoretical and ex-perimental coupling constants are compared in Fig. 4 wherethe difference in the deﬁnition of the coupling constants be-tween our and the experimental work 20requires the inclusion of the factor 2. The agreement between the calculated andthe experimental results is remarkable, particularly since wehave not adjusted any parameters. The quantitative agree-ment should, however, not be overstated since the model ofthe bilayer and the Heisenberg Hamiltonian used for the cal- culation both contain signiﬁcant approximations. VI. DISCUSSION In the previous three sections we have shown that the perpendicular coupling, the coercivity, the loop shift, and thestrong coupling seen in reversible measurements, can be un-derstood within an atomistic Heisenberg model. The modelused in the calculations, however, is in many respects toosimple to accurately describe the situation in thin ﬁlms. Itenvisions a periodic arrangement of one type of defect andgives the proper loop shift, but it does not explain othereffects related to the loop shift. In a realistic interface, thedefects will be of different types and will be randomly ar-ranged. On average, the interactions between the AFM andthe FM may be fully compensated when the AFM is in asingle domain state. However, when the system is cooled inthe presence of coupling to an ordered FM ﬁlm, the AFMmay break up into domains with walls perpendicular to theinterface such that the exchange coupling due to the defectsno longer cancels. This is precisely the mechanism that leadsto the Imry–Ma type of random ﬁeld. 26Therefore, extending the current model of rough interfaces to realistic lengthscales leads directly to the Malozemoff theory of exchangebias which accounts for many of the effects related to theloop shift. 7The explanation of positive exchange bias which we have given at the end of Sec. IV is still applicable withinthe random ﬁeld model. To complete the list of effects re-lated to the loop shift, we have to discuss the dependence ofH ebon the roughness. Experiments show that interface roughness can both increase27or decrease14the loop shift. Clearly, when the random ﬁeld is solely due to roughnessinduced defects, such as the point defects treated in Sec. IV,one would expect the random ﬁeld and with it H ebto in- crease with increasing roughness. However, when the ran-dom ﬁeld is induced by uncompensated regions that are re-lated to lattice strain ~such as dislocations !one would observe the opposite trend: since roughness decreases latticestrains it would lower the random ﬁeld and H eb. In thin ﬁlms, the magnetization reversal is usually not a coherent rotation as in our calculations. Reversal is ratherinitiated in some region of the ﬁlm and is completed through FIG. 4. Effective AFM–FM coupling strength between CoO and Co vs thickness of the Co ﬁlm. Experimental values ~crosses !are taken from Ref. 20. Theoretical values represent 2 Keffcalculated for JF–F516meV ~squares ! andJF–F512.4meV ~diamonds !.5513 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 T. C. Schulthess and W. H. Butler Downloaded 09 May 2013 to 131.211.208.19. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsgrowth of reversed domains, a process which implies the propagation of domain walls through the ﬁlm. Spin–ﬂopcoupling can now lead to an increase in the coercivity in twodifferent ways. First, some of the AFM domains, to whichthe FM couples, are small enough that their magnetic con-ﬁguration reverses during the inversion of the FM. This leadsto irreversible effects in the AFM that contribute to the co-ercivity. This idea was used by Lin et al. 28and recently by Stiles and McMichael,29as well as by Hou et al.30to explain irreversible effects in polycrystalline AFM/FM systems. In a second mechanism, spin–ﬂop coupling acts like an induced uniaxial anisotropy in the FM layer which reducesthe size of domain walls when the FM is coupled to the AFMand thus increases the coercivity through pinning of thesedomain walls. In free Py the anisotropy is very small ( K Py .203103erg/cm3) which implies large domain walls for which pinning is very unlikely and consequently the materialis magnetically ideally soft. When the AFM–FM coupling isaveraged over the entire ﬁlm thickness, our results in theprevious section would yield a uniaxial anisotropy of theorder of 0.5 310 6erg/cm3for a 200 Å thick-ﬁlm, which is comparable to anisotropies in Fe. Spin–ﬂop coupling, how-ever, is due to relaxation of spins in the interface region andthe anisotropy it induces is therefore concentrated in thatregion. Assuming this region to be about 10 Å and a corre-sponding estimate for the coupling of K eff'2erg/cm2, one is left with an average anisotropy constant of about 2310 7erg/cm3which is about three order of magnitude larger than the bulk value for Py. The domain wall size in the ﬁlmcan thus be expected to reduce from ;10 3Å to below 100 Å and pinning of domain walls in Py would then be realisticwhen the ﬁlm is coupled to an AFM such as CoO. These two mechanisms for the coercivity have quite dis- tinct consequences. In the ﬁrst mechanism, the coercivitydepends on the domain structure in the AFM and thereforecan be expected to depend on the magnetic history of thesample. In the second mechanism, the coercivity only de-pends on the spin–ﬂop coupling and the morphology of the ﬁlm. It is not affected by the AFM domain structure and thusshould be independent of the magnetic history. Since exchange bias and the coupling induced coercivity have their origin at the interface, one expects them to beinversely proportional to the thickness of the FM layer. Forthe loop shift this is well accepted. In the case of the coer-civity, however, this is only a ﬁrst guess which may applywhen the coercivity is due to losses in the AFM. If it is dueto the coupling induced reduction of domain walls discussedabove, the functional dependence of domain wall size anddefect densities at different length scales have to be consid-ered as well. The size of a domain wall depends not only onthe anisotropy, which in the present case is restricted to theinterface region and has no dependence on d FM, but also on the exchange interactions. The net FM exchange energy of adomain wall in the ﬁlm increases linearly with the ﬁlm thick-ness, which implies that the domain wall size will be propor- tional to AdFM. The density of defects which pin the wall is more difﬁcult to discuss and we will restrict ourselves to twomodel situations. ~1!When the defect density is constant at all length scales below a certain threshold, the pinning willbe independent of the domain wall size as long as it is small enough. In this case one expects H c;1/dFM.~2!We assume that the interface is of fractal nature and when the domainwall size, d w, is decreased, the density of defects that can pin the wall increases as 1/ dw. Withdw;AdFM, we thus expectHc;1/dFM3/2. This last result is similar to that recently obtained by Zhang.31 In contrast to the irreversible process of magnetization reversal, the reversible measurements of Miller andDahlberg 20are much simpler to model. Actually, we think that the calculations presented in the last section fully corre-spond to this measurement and thus it does not need furtherexplanation. The combination of spin–ﬂop coupling and de-fect induced random ﬁeld, however, clariﬁes the apparentdiscrepancy between reversible and irreversible measure-ment. While the former senses a ~not necessarily linear !su- perposition of random ﬁeld and spin–ﬂop coupling, the latteronly measures the coupling due to the random ﬁeld. VII. CONCLUSIONS We can therefore conclude that two coupling mecha- nisms, spin–ﬂop coupling and defect induced random ﬁelds,must be present, in order to explain all four classes of phe-nomena which are directly related to AFM–FM coupling.The consequences of the possible absence of one of themechanisms are relatively straightforward. The absence ofuncompensated defects and the corresponding random ﬁeldwould eliminate the loop shift, but if spin–ﬂop couplingwere still present, one should still be able to measure strongcoupling in reversible experiments, observe coupling in-duced coercivities, and the perpendicular coupling. On theother hand, if spin–ﬂop coupling were absent while the de-fect induced random ﬁeld is present, the system would showexchange bias but no coupling induced coercivity. In thiscase, perpendicular coupling would not be observable butmore importantly, the results of reversible and irreversiblecoupling measurement should be similar. 32 When we apply these conclusions to Py interfaced with either NiMn, FeRh, and FeMn or IrMn, we argue that bothcoupling mechanisms are present in NiMn/Py whereas inFeRh/Py, the random ﬁeld should be absent but spin–ﬂopcoupling should be considerable. Thus in both systems oneshould observe strong coupling in reversible measurementsand perpendicular alignment between the FM magnetizationand the AFM easy axis. In FeMn/Py and IrMn/Py, we specu-late that the random ﬁeld is present but the spin–ﬂop cou-pling is much smaller since the coupling induced coercivityis much smaller. In fact, the discrepancy between irreversibleand reversible coupling measurements in IrMn/Py is found tobe at least an order of magnitude smaller than in NiMn/Py. 21 The spin–ﬂop coupling strength seems to be related to theAFM spin-structure of these alloys. NiMn and FeRh withlarge spin–ﬂop coupling are chemically ordered systemswith collinear spin structures 33,34and thus the situation is similar to that of CoO. FeMn and IrMn, for which the currentdiscussion implies much smaller spin–ﬂop coupling arechemically disordered and probably have noncollinear spinstructures 35which may cause the reduction in spin–ﬂop cou-5514 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 T. C. Schulthess and W. H. Butler Downloaded 09 May 2013 to 131.211.208.19. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionspling strength. The absence of exchange bias in FeRh/Py is not only intriguing but, in our opinion, implies a great op-portunity, since comparing interfacial spin structures andmorphologies between NiMn/Py and FeRh/Py could give im-portant information on defect induced coupling and randomﬁelds. ACKNOWLEDGMENTS Research sponsored by the Division of Materials Sci- ences, U.S. Department of Energy under Contract No. DE-AC05-96OR22464 with Lockheed Martin Energy ResearchCorporation. 1W. H. Meiklejohn and C. P. Bean, Phys. Rev. 105, 904 ~1957!. 2J. Nogue´s and Ivan K. Schuller, J. Magn. Magn. Mater. ~in press !. 3L. Nee´l, Ann. Phys. ~Paris!2,6 1~1967!. 4D. Mauri, H. C. Siegmann, P. S. Bagus, and E. Kay, J. Appl. Phys. 62, 3047 ~1987!. 5N. C. Koon, Phys. Rev. Lett. 78, 4865 ~1997!. 6A. P. Malozemoff, Phys. Rev. B 35, 3679 ~1987!. 7A. P. Malozemoff, J. Appl. Phys. 63, 3874 ~1988!. 8A. P. Malozemoff, Phys. Rev. B 37, 7673 ~1988!. 9H. Suhl and I. K. Schuller, Phys. Rev. B 58, 258 ~1998!. 10R. Junblut et al., J. Magn. Magn. Mater. 148, 300 ~1995!. 11J. Nogue´s, D. Lederman, T. J. Moran, Ivan K. Schuller, and K. V. Rao, Appl. Phys. Lett. 68, 3186 ~1996!. 12L. L. Hinchey and D. L. Mills, Phys. Rev. B 34, 1689 ~1986!. 13T. C. Schulthess and W. H. Butler, Phys. Rev. Lett. 81, 4516 ~1998!. 14J. Nogue´s, D. Lederman, T. J. Moran, I. K. Schuller, and K. V. Rao, Appl. Phys. Lett. 68, 3186 ~1996!. 15P. J. van der Zaag, A. R. Ball, L. F. Feiner, R. M. Wolf, and P. A. A. van der Heijden, J. Appl. Phys. 79, 5103 ~1996!. 16T. J. Moran and I. K. Schuller, J. Appl. Phys. 79, 5109 ~1996!.17Y. Ijiri, J. A. Borchers, R. W. Erwin, S.-H. Lee, P. J. van der Zaag, and R. M. Wolf, Phys. Rev. Lett. 80, 608 ~1998!. 18S. Yuasa, M. Ny ´vlt, T. Katayama, and Y. Suzuki, J. Appl. Phys. 83, 6813 ~1998!. 19T. J. Moran, J. Nogue ´s, D. Lederman, and I. K. Schuller, Appl. Phys. Lett. 72, 617 ~1998!. 20B. H. Miller and E. Dan Dahlberg, Appl. Phys. Lett. 69, 3932 ~1996!. 21T. G. Pokhil and S. Mao ~unpublished !. 22See Ref. 13 for details about the model and parameters. In the present work, we do not vary the AFM–FM exchange parameter but rather ﬁx itto the values of the exchange in the AFM. Whenever the value for theFM–FM exchange parameter, J F–F, differs from Ref. 13 as is mentioned in the text. 23K. Takano, R. H. Kodama, A. E. Berkowitz, W. Cao, and G. Thomas, Phys. Rev. Lett. 79, 1130 ~1997!. 24J. Nogue´s, D. Lederman, T. J. Moran, and I. K. Schuller, Phys. Rev. Lett. 76, 4624 ~1996!. 25For ﬁeld larger than a few hundred Oersted, the induced changes in the AFM become notable and change the form of the ﬁeld dependence of thetotal energy. 26Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35, 1399 ~1975!. 27T. J. Moran, J. M. Gallego, and I. K. Schuller, J. Appl. Phys. 78, 1887 ~1995!. 28T. Lin, C. Tsang, R. E. Fontana, and J. Kent Howard, IEEE Trans. Magn. 31, 2585 ~1995!. 29M. Stiles and R. D. McMichael, Phys. Rev. B ~in press !. 30C. Hou, H. Fujiwara, F. Ueda, and H. S. Cho, Mater. Res. Soc. Symp. Proc. ~to be published !. 31S. Zhang, J. Magn. Magn. Matter ~in press !. 32We assume that all effects are related solemnly to the AFM–FM coupling. FeF2/Fe is a system in which, as Moran et al.19pointed out, perpendicular alignment could be related to lattice strain. 33T. C. Schulthess and W. H. Butler, J. Appl. Phys. 93, 7225 ~1998!. 34V. L. Moruzzi and P. M. Marcus, Phys. Rev. B 48, 16106 ~1993!. 35T. C. Schulthess, W. H. Butler, G. M. Stocks, S. Maat, and G. J. Mankey ~unpublished !.5515 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 T. C. Schulthess and W. H. Butler Downloaded 09 May 2013 to 131.211.208.19. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
1.5116748.pdf	J. Appl. Phys. 126, 163903 (2019); https://doi.org/10.1063/1.5116748 126, 163903 © 2019 Author(s).RF voltage-controlled magnetization switching in a nano-disk Cite as: J. Appl. Phys. 126, 163903 (2019); https://doi.org/10.1063/1.5116748 Submitted: 27 June 2019 . Accepted: 30 September 2019 . Published Online: 24 October 2019 Joseph D. Schneider , Qianchang Wang , Yiheng Li , Andres C. Chavez , Jin-Zhao Hu , and Greg Carman ARTICLES YOU MAY BE INTERESTED IN Heat exchange with interband tunneling Journal of Applied Physics 126, 165107 (2019); https://doi.org/10.1063/1.5113870 Recombination of Shockley partial dislocations by electron beam irradiation in wurtzite GaN Journal of Applied Physics 126, 165702 (2019); https://doi.org/10.1063/1.5121416 Thermionic enhanced heat transfer in electronic devices based on 3D Dirac materials Journal of Applied Physics 126, 165105 (2019); https://doi.org/10.1063/1.5123398RF voltage-controlled magnetization switching in a nano-disk Cite as: J. Appl. Phys. 126, 163903 (2019); doi: 10.1063/1.5116748 View Online Export Citation CrossMar k Submitted: 27 June 2019 · Accepted: 30 September 2019 · Published Online: 24 October 2019 Joseph D. Schneider,1,a) Qianchang Wang,1,a) Yiheng Li,2,a) Andres C. Chavez,1 Jin-Zhao Hu,1 and Greg Carman1,b) AFFILIATIONS 1Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, California 90095, USA 2Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China a)Contributions: J. D. Schneider, Q. Wang, and Y. Li contributed equally to this work. b)Author to whom correspondence should be addressed: carman@seas.ucla.edu ABSTRACT Nanomagnetic oscillators are key components for radio-frequency (RF) signal generation in nanoscale devices. However, these oscillators are primarily electric current-based, which is energy ine ﬃcient at the nanoscale due to ohmic losses. In this study, we present an actuation mechanism for magnetization switching using a multiferroic structure that relies on an RF voltage input instead of electricalcurrent. An AC voltage with a DC bias is applied to the piezoelectric substrate and the magnetic nanodisk with perpendicular magneticanisotropy that is attached onto the substrate, which can achieve steady magnetic oscillation when the driven voltage is at ferromagnetic resonance (FMR) of the nanodisk. Changing the DC bias changes t he magnetic anisotropy of the magnetoelastic nanodisk, hence changes the FMR and oscillation frequency. The frequency modulation is quanti ﬁed using the Kittel equation. Parametric studies are con- ducted to investigate the in ﬂuence of voltage amplitude, frequency, waveform, and the thickness of the magnetoelastic nanodisk. This multiferroic approach opens possibilities for designing energy e ﬃcient nanomagnetic oscillators that have both large amplitude and broad frequency range. Published under license by AIP Publishing. https://doi.org/10.1063/1.5116748 INTRODUCTION Nanomagnetic oscillators have been used for many applica- tions such as nanoscale RF signal generators, 1–3microwave-assisted recording, nanoscale magnetic ﬁeld sensors,4and neuromorphic computing hardware.5In a conventional nanomagnetic oscillator, steady magnetic oscillation is achieved when the current-induced spin torque cancels the Gilbert damping.4–10However, current- driven magnetic oscillations are power-consuming at the nanoscaledue to ohmic losses. In contrast, voltage-driven magnetic oscillationis a more energy e ﬃcient control scheme. One of the more promis- ing methods to achieve voltage-based magnetization control is strain-mediated multiferroics, which are based on the magnetoe-lastic/piezoelectric heterostructure. Static magnetization controlusing multiferroics has been demonstrated both numerically andexperimentally. 11–18Additionally, multiferroics have been used for dynamic magnetization control, such as spin wave genera- tion19,20and ferromagnetic resonance driven by surface acousticwaves.21,22Another application where control over dynamic mag- netization oscillations is critical is magnetic dipole antennas,23 which are important for communication in conductive mediums (i.e., seawater). A recent study numerically demonstrated magnetic oscillation in magnetoelastic nanoellipses with in-plane magnetic anisotropy patterned on a piezoelectric substrate.24The ellipse ’s magnetization is controlled with a pair of electrodes located slightly o ﬀ-axis of the ellipse ’s minor length. Applying a voltage to the electrodes causes the magnetization to rotate toward the minor axis of the ellipsebecause of strain-induced magnetic anisotropy. When the voltage is removed, the shape anisotropy of the ellipse causes the magnetiza- tion to return to ellipse major axis. Consequently, by applying an AC voltage to the electrodes, a magnetic oscillation can be achieved but the oscillation amplitude is limited to 90°. Moreover,no feasible frequency modulation mechanism is demonstrated in this study. Therefore, a new design of multiferroic nanomagneticJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 163903 (2019); doi: 10.1063/1.5116748 126, 163903-1 Published under license by AIP Publishing.oscillators with large oscillation amplitude and wide tunable fre- quency range is needed. THEORY In this study, a nanoscale Ni disk with perpendicular magnetic anisotropy (PMA) is simulated as a nanomagnetic oscillator andboth large oscillation amplitude and wide frequency tunabilityare numerically demonstrated. The nanodisk is placed on top ofa piezoelectric substrate with patterned electrodes to actuate the system. The multiferroic oscillator system is evaluated with a 3D ﬁnite element model that couples micromagnetics, electrostatics, and elastodynamics. 12,14,25The model assumes small deformations, linear elasticity, and linear piezoelectricity. Thermal ﬂuctuations would require a stochastic approach, which is beyond the scope of this work and is not considered. The magnetocrystalline anisotropy is assumed negligible. The precessional magnetic dynamics aregoverned by the Landau-Lifshitz-Gilbert (LLG) equation @m @t¼/C0μ0γ(m/C2Heff)þαm/C2@m @t/C18/C19 , (1) where mis the normalized magnetization, μ0is the vacuum permittivity, γis the gyromagnetic ratio, and αis the Gilbert damping parameter. Heffis the e ﬀective magnetic ﬁeld de ﬁned by Heff¼HexþHDemagþHPMAþHME, where Hexis the exchange ﬁeld,HDemag is the demagnetization ﬁeld,HPMA is the e ﬀective PMA ﬁeld, and HMEis the magnetoelastic ﬁeld generated by strain. The PMA ﬁeld is given by HPMA¼/C02KPMAmz^z=(μ0MS).14,26 Assuming the PMA originates from interfacial e ﬀects, the PMA coeﬃcient is KPMA=K i/t, where tis the thickness of the magnetic thinﬁlm, and the interfacial anisotropy is Ki¼2:6/C210/C04J=m2 for Ni. The HMEﬁeld is calculated as27 HME(m,ε)¼/C01 μ0MS@ @m/C26 B1εxxm2 x/C01 3/C18/C19 þεyym2 y/C01 3/C18/C19 þεzzm2 z/C01 3/C18/C19 /C20/C21 þ2B2(εxymxmyþεyzmymzþεzxmzmx)/C27 , (2) where mx,my, and mzare components of normalized magnetiza- tion along the x,y, and zaxes and B1and B2are the ﬁrst and second order magnetoelastic coupling coef ﬁcients. B1and B2are deﬁned by B1¼B2¼3EλS 2(1þν), where Eis Young ’s modulus and λSis the saturation magnetostriction coef ﬁcient of the magnetic material. In Eq. (2), the total strain ( ε) consists of two parts: ε¼εpþεm, where εpis the piezostrain and εm ijis the strain contribution due to isotropic magnetostriction. The strain due to magnetostriction isgiven by ε m ij¼1:5λs(mimj/C0δij=3), where δijis the Kronecker delta.27The piezostrain εpis determined using the linear piezoelec- tric constitutive equations: εp¼sE:σþdt/C1E, (3) D¼d:σþeσ/C1E, (4) where σis the stress, Dis the electric displacement, Eis the electric ﬁeld,sEis the piezoelectric compliance matrix under constant elec- tricﬁeld, dand dtare the piezoelectric coupling matrix and its transpose, and eσis the electric permittivity matrix measured under constant stress. A complete description of the coupled model can be found in the publications of Liang et al.12,25 The precession characteristics of the multiferroic nanomag- netic oscillator are studied by applying several di ﬀerent voltage amplitudes, frequencies, and waveforms for various disk thick- nesses. First, static voltages of 0 V and 1.8 V are applied to the elec- trodes to establish a baseline for the magnetization precession dueto PMA and under the in ﬂuence of in-plane strain for a 2 nm-thick disk. Second, square waves with a minimum value of 0 V and a maximum value of V 0are applied to the electrodes at four diﬀerent frequencies for a 2 nm-thick disk. The choice of squarewaves is motivated by the results of the static cases where a DC voltage is applied to the electrodes. In particular, the four chosenwaveforms are (1) V 0= 1.8 V at 0.8 GHz, (2) V 0= 1.8 V at 1.1 GHz, (3) V 0= 1.8 V at 1.6 GHz, and (4) V 0= 2.0 V at 1.4 GHz. Third, the disk thickness tis varied and three di ﬀerent cases are studied for applied square waves at 3.2 GHz. The three cases are (1)V 0= 4.0 V with t= 2.0 nm, (2) V 0= 4.0 V with t= 1.8 nm, and (3) V0= 4.3 V with t= 1.8 nm. Fourth, 0.55 GHz symmetric and asym- metric square waves with V 0= 1.8 V are applied to the electrodes for a 2 nm-thick disk. The asymmetric waveform is used to produce an ultralow oscillation frequency. Fifth, the e ﬀects of wave- form type on the 2 nm-thick oscillator dynamics are studied.Speciﬁcally, three di ﬀerent applied voltages are used: (1) square wave with V 0= 1.8 V at 1.1 GHz, (2) sinusoidal wave with V¼0:9þ0:9 sin(2 πf0t), and (3) sinusoidal wave with V¼0:9þ4 π/C20:9 sin(2 πf0t), where f0¼1:1 GHz. The initial volt- ages of all applied waveforms start from V 0/2 and ramp toward V 0. Figure 1(a) illustrates the simulated multiferroic nanomagnetic oscillator structure. A PbZr 0.53Ti0.47O3(Ref. 28) PZT-5H (abbrevi- ated as PZT) substrate is used as the piezoelectric material with lateral dimensions of 1500 × 1500 nm2and 800 nm thickness. The PZT ’s top surface is mechanically free and its bottom surface is ﬁxed (i.e., mechanically clamped on a thick substrate), and low- reﬂecting boundary conditions are applied to the four lateral sides. A Nickel magnetic disk with a diameter of 50 nm and a height of 2nm is perfectly adhered in the center of the PZT top surface. Two50 × 50 nm 2electrodes are placed symmetrically adjacent to the Ni disk along the y axis. The edge-to-edge distance between the elec- trode and the magnetic disk is 20 nm. The voltage pulses are always applied to the two electrodes simultaneously, while the bottomsurface of PZT is electrically grounded. The Ni material parametersused in the analysis are α¼0:038, M s¼4:8/C2105A=m, exchangeJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 163903 (2019); doi: 10.1063/1.5116748 126, 163903-2 Published under license by AIP Publishing.stiﬀness Aex¼1:05/C210/C011J=m( u s e di n Hex), and λs¼/C034 ppm, Young ’sm o d u l u s E= 180 GPa, density ρ¼8900 kg =m3,a n d Poisson ’sr a t i o ν¼0:31.29–32 RESULTS Figures 1(b) and 1(c) show the volume average magnetiza- tion (i.e., integrated over the entire volume of the magnetic nano-disk) precession of the Ni disk for applied static voltages of0 V and 1.8 V, respectively. In both ﬁgures, the magnetization is released from a canted out-of-plane direction chosen as m¼(0, 1, 1) =ﬃﬃﬃ 2p .Figure 1(b) shows the 3D trajectory for a 5-ns magnetic precession without applied voltage. For this case, thedominant magnetic anisotropy is the PMA of the disk resulting ina volume average (i.e., integrated over the entire volume of the magnetic nanodisk) e ﬀective magnetic ﬁeldH eﬀdirected along the z axis causing the magnetization to precess around this direc-tion. In contrast, Fig. 1(c) shows the Ni disk magnetization precess around the y axis when 1.8 V is applied to the electrodes. This occurs because the voltage induces a compressive strain along the y axis and a tensile strain along the x axis. Speci ﬁcally, the compressive direction corresponds to a strain-induced mag-netic easy axis and the tensile direction corresponds to a magnetichard axis since Ni is negative magnetostrictive. The two results of Figs. 1(b) and1(c) illustrate a mechanism to generate large ampli- tude oscillations of the disk magnetization. Without appliedvoltage, there are two stable magnetic states for the disk: m z=+ 1 and m z=−1. When a voltage is applied to the electrodes, the magnetization is brought to an intermediate state 90° away from the z axis (i.e., in-plane). By leveraging these two e ﬀects and accu- rately timing the voltage application, it is possible to oscillate themagnetization perpendicularly between m z=+ 1a n dm z=−1.14.26 This demonstrates the potential for 180° oscillations, thus overcoming the de ﬁciency (90° max oscillation) of previously dis- cussed multiferroic designs. Figures 2(a) –2(d) show the e ﬀects of applying a square wave voltage amplitude and frequency on the precession dynamics ofthe multiferroic nanomagnetic oscillator. The applied voltages have maximum values V 0= {1.8, 1.8, 1.8, 2.0} V at frequencies of {0.8, 1.1, 1.6, 1.4} GHz, respectively. In the ﬁgures, the blue dashedlines represent the applied voltage while the solid black lines repre- sent the 2 nm Ni disk ’s volume-averaged perpendicular magnetiza- tion m zas a function of time. Figures 2(a) and 2(c) show a disordered m ztemporal response for V 0= 1.8 V at 0.8 GHz and FIG. 1. (a) 3D illustration of the simulated structure (unit: nanometer). (b) Trajectory of magnetic precession without voltage applied. (c) Trajectory of magnetic precession when +1.8 V is applied to the top electrodes. FIG. 2. (a)–(d) T emporal evolution of perpendicular magnetization m zunder alternate applied voltage with different amplitudes and frequencies: (a) 1.8 V at 0.8 GHz, (b) 1.8 V at 1.1 GHz, (c) 1.8 V at 1.6 GHz, and (d) 2 V at 1.4 GHz. (e) Summary of simulation results of steady oscillation cases and theoretical ﬁtting line derived from the Kittel equation.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 163903 (2019); doi: 10.1063/1.5116748 126, 163903-3 Published under license by AIP Publishing.1.6 GHz, respectively. In contrast, Figs. 2(b) and2(d) show steady mzoscillations for V 0= 1.8 V at 1.1 GHz and 2.0 V at 1.4 GHz frequencies, respectively. These two later results clearly show thatthe multiferroic oscillator achieves uniform large amplitude (i.e.,180°) perpendicular magnetic oscillations at di ﬀerent voltages and diﬀerent frequencies. The oscillation dynamics presented in Figs. 2(a) –2(d) can be explained by comparing the frequency of the applied voltages tothe ferromagnetic resonance (FMR) of the nanomagnetic oscilla-tors. Speci ﬁcally, the uniform oscillation observed for V 0=1 . 8 V at 1.1 GHz [ Fig. 2(b) ] occurs because the applied voltage fre- quency matches the oscillator FMR frequency. This leads to the strain-induced magnetization oscillations matching the intrinsicprecessional magnetization oscillations. Therefore, the magnetiza-tion reoriented by the voltage-induced strain is in phase with theprecessional motion. This is clearly seen in Fig. 2(b) as the mag- netization reaches its extremum values (m z=± 1 ) a t n e a r l y t h e same time the max (1.8 V) or min (0 V) is reached. In contrast,the disordered oscillations for applied voltages of V 0=1 . 8 V a t 0 . 8G H za n d1 . 6G H z[ Figs. 2(a) and2(c)] are caused by the mis- match between the intrinsic precessional motion of the magnetic moments (i.e., FMR frequency) and applied voltage frequency. Speciﬁcally, for the 2 nm disk with an applied voltage amplitude of 1.8 V, the ferromagnetic resonance frequency is 1.1 GHz sooperating at 0.8 GHz or 1.6 GHz produces a disordered oscilla- tion, i.e., signi ﬁcant overshoot or undershoot as the magnetization is reoriented by the voltage-induced strain. Consequently, thisovershooting or undershooting produces disordered nonperiodicoscillations of m z. However, as shown in Fig. 2(d) , increasing the voltage (V 0= 2.0 V) produces uniform periodic oscillations but at a higher frequency, i.e., 1.4 GHz. The reason that the higher applied voltage frequency is now in phase with the precessionalmotion is because the applied voltage shifts the oscillator FMRfrequency to a higher level, i.e., modi ﬁes the magnetic anisotropy with an applied voltage. Further interpretation of the results can be explained as follows. Observing Fig. (2d) ,i n i t i a l l ym=m z= 1, a voltage applied to the electrodes results in a compressive stress along the y axis ofthe disk forcing the magnetization in plane. The voltage pulse isthen released and m = m z=−1, where the 180° switch is due to the timing of the voltage pulse. Next, a voltage pulse is applied resulting in compression on the Ni disk forcing the magnetizationin plane again. Upon release of the voltage, m = m z=1 . I n summary, the magnetization switched from +1 to −1( o n ec o m - plete cycle) in two voltage cycles of 0 –2 V. Furthermore, the current required is calculated using i¼V0ωC, where C is the static capacitance of the structure calculated using Finite ElementMethods and ωis the 2 πmultiplied by the driving frequency. For the case presented in Fig. 2(d) , the current required is 55 μA, whereas spin-torque oscillators require a DC in the milliampere range. 33,34 To better quantify the frequency tunability suggested in Fig. 2 , i.e., an applied oscillating voltage shifts the FMR, an analytical solu-tion for the frequency modulation by voltage is derived from the Kittel equation. For a thin disk, as simulated in this work, the demagnetization factors are approximately N x=N y= 1 and Nz= 0,35. Consequently, the FMR frequency ( f) can be determinedfrom the simpli ﬁed Kittel equation:35 f¼γμ0 2πﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ Heff(HeffþMS)q , (5) where Heff¼HexþHPMAþHME. This derivation assumes that exchange is negligible because we are looking at the k = 0 mode(i.e., FMR); hence, we ignore H exin the e ﬀective ﬁeld. Furthermore, since Ni is negatively magnetostrictive, we can assume that the main contributing strain term to the e ﬀective magnetoelastic ﬁeld (Hme) is the compression strain along the y axis ( εyy). Hence, the magnetoelastic ﬁeld is reduced to Hme¼/C02 μ0MSB1myεyy: (6) The PMA e ﬀect can be approximated as a constant 2000 ppm in-plane uniaxial strain in the magnetoelastic ﬁeld since the minimum uniaxial strain required to overcome PMA is −2000 ppm. Consequently, the total e ﬀective ﬁeld can be written as Heff¼/C02 μ0MSB1my(εyyþ2/C210/C03): (7) In Eq. (7),myis set to be a constant value of 0.2 rather than a tem- porally varying value to better explain the underlying physics. By plugging Eqs. (7)into (5)weﬁnd that the resonance fre- quency ( f) is directly proportional to the square root of the applied strain εyy. Furthermore, a ﬁnite element simulation shows that εyyand voltage amplitude are related by the linear equation: εyy¼/C01463/C2V0(ppm) for this speci ﬁc geometry. The resulting equation of frequency in terms of voltage is drawn in Fig. 2(e) as a black dashed line, and several simulation results are shown as blue dots for comparison. The simulation results are represented by steady oscillation cases for a 2 nm disk with applied voltageamplitudes and frequencies of {1.5, 1.8, 2.0, 3.0, 4.0} V and{0.7, 1.1, 1.4, 2.3, 3.2} GHz, respectively. For comparison, the fourcases discussed in Figs. 2(a) –2(d) are also marked in Fig. 2(e) with black open circles. As seen in Fig. 2(e) , the oscillation frequency increases with applied voltage oscillation amplitudes,and the trend agrees with the analytical solution. This clearlyshows that the multiferroic oscillator produces frequency tuningin the presence of a dynamically oscillating voltage. The magni- tude of the tuning is directly related to the amplitude of the oscillating voltage. Figures 3(a) –3(c) show dynamic magnetization data for applied voltages V 0= {4.0, 4.0, 4.3} V with a frequency of 3.2 GHz and are applied to oscillators with disk thicknesses of {2.0, 1.8, 1.8} nm, respectively. Here, Fig. 3(a) shows that 4.0 V at 3.2 GHz applied voltage can excite uniform magnetic oscillation; however,the oscillation becomes unstable when the thickness decreases to1.8 nm for this voltage, as shown in Fig. 3(b) . In contrast, Fig. 3(c) shows that the uniform oscillation occurs again for the 1.8 nm-thick disk when the applied voltage increases to 4.3 V.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 163903 (2019); doi: 10.1063/1.5116748 126, 163903-4 Published under license by AIP Publishing.The change in uniformity and oscillation frequency for chang- ing disk thickness can be explained by understanding that PMA isa function of Ni disk ’s thickness. Speci ﬁcally, since the PMA of the Ni disk is inversely proportional to the magnet ’s thickness, the 1.8 nm oscillator has stronger PMA. Hence, the thinner disk requires a higher voltage (i.e., larger strain) to overcome the PMA. This means the FMR curve for the 1.8 nm oscillator shifts to theright when compared to the 2 nm oscillator [shown in Fig. 2(e) ], as the intersection point of the FMR curve on the x axis correspondsto the minimum voltage required to overcome PMA. Therefore, the 1.8 nm oscillator requires higher voltage to achieve a steady mag- netic oscillation at the same frequency. Note that 4.3 V is thehighest voltage used here, which corresponds to an electric ﬁeld of ≈5.4 MV/m. Although this electric ﬁeld exceeds PZT ’s 0.7 MV/m coercive ﬁeld, 36in the simulation the electric ﬁeld does not reverse direction, indicating that the high electric ﬁeld will not reverse the polarization. All the cases discussed above have the voltage-on and voltage-o ﬀsquare pulses with the same temporal length within each period, i.e., referred to as symmetric. As discussed previ- ously, the voltage-on requires accurate timing and should matchthe FMR. However, the duration of voltage-o ﬀstage can be adjusted in a more ﬂexible way than constrained to be equal to voltage on. Figures 4(a) and4(b) compare a 1.8 V applied voltage at 0.55 GHz with symmetric and asymmetric pro ﬁles for a 2 nm Ni disk, respectively. In Fig. 4(a) , a steady magnetic oscillation is absent (disordered) because the voltage frequency does not match the FMR of the magnetic disk, which is 1.1 GHz at 1.8 V. In con- trast, the voltage-on portion in Fig. 4(b) matches the FMR of 1 . 1G H z ,b u tt h ev o l t a g e - o ﬀportion is purposely extended. Consequently, a steady magnetic oscillation with an overall muchlower frequency (i.e., 0.55 GHz) is achievable using an asymmetric voltage pro ﬁle. Figure 5 examines the in ﬂuence of di ﬀerent voltage wave- forms on multiferroic nanomagnetic oscillators (2 nm Ni disk)response. Figure 5(a) shows the results from a square wave with a 1.8 V amplitude oscillated at 1.1 GHz producing steady oscilla- tion. As shown in Fig. 5(b) , changing the square wave to a sinus- oidal wave with a similar amplitude V ¼0:9þ0:9 sin(2 πf 0t) (f0¼1:1 GHz) produces a dramatically di ﬀerent magnetic response. To better evaluate this di ﬀerence, we use a zeroth- and ﬁrst-order components of the Fourier series expansion of the square wave to build the wave V ¼0:9þ4 π/C20:9 sin(2 πf0t). Using this input, Fig 5(c) shows that a steady magnetic oscillation is again achieved. In other words, modifying the sinusoidal wavecan achieve steady oscillations similar to the square wave. This is important because it is easier to create sinusoidal waves as compared to square waves. FIG. 3. T emporal evolution of perpendicular magnetization m zfor magnets with different thicknesses. (a) 2 nm-thick magnet with 4 V applied voltage at 3.2 GHz. (b) 1.8 nm-thick magnet with 4 V applied voltage at 3.2 GHz. (c) 1.8 nm-thick magnet with 4.3 V applied voltage at 3.2 GHz. FIG. 4. T emporal evolution of perpendicular magnetization m zunder alternate applied voltage with (a) symmetric square wave at 0.55 GHz and (b) asymmetricsquare wave at 0.55 GHz.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 163903 (2019); doi: 10.1063/1.5116748 126, 163903-5 Published under license by AIP Publishing.CONCLUSION In conclusion, a new magnetic oscillator mechanism is pro- posed using an alternating voltage applied to the piezoelectric sub-strate to excite sinusoidal magnetic oscillation. The oscillation frequency can be tuned by changing the amplitude of the alternate voltage or by changing the thickness of the magnet to dynamicallyadjust ferromagnetic resonance. The frequency range achieved inthis study is from 275 MHz to 1.6 GHz (note the magnetic oscilla-tion frequency is half of the voltage frequency). Using an asymmet- ric voltage pro ﬁle adds additional tunability to the system and further extends the lower bound of the oscillation frequency. Abasic analytical equation is derived to link the voltage amplitude tothe oscillation frequency. This work helps understand the opera-tional principles of the voltage-driven magnetic oscillator and guide future design of the oscillator to speci ﬁc frequency ranges. Furthermore, the proposed device can change the operating fre-quency by increasing the voltage amplitude and frequency. Thiscan be used for a number of applications such as driving/producing spin waves in a nearby spin bus or writing a bit of memory in the magnetization.ACKNOWLEDGMENTS This work was supported by NSF Nanosystems Engineering Research Center for Translational Applications of NanoscaleMultiferroic Systems (TANMS) Cooperative Agreement Award (No. EEC-1160504) and EFRI NewLaw with Award No. 1641128. REFERENCES 1A. Mourachkine, O. V. Yazyev, C. Ducati, and J. P. Ansermet, Nano Lett. 8, 3683 (2008). 2S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature 425, 380 (2003). 3F. Badets, L. Lagae, S. Cornelissen, T. Devolder, and C. Chappert, in Proceedings of 15th IEEE International Conference on Electronics, Circuits and Systems, ICECS 2008 (IEEE, 2008), pp. 190 –193. 4P. M. Braganca, B. A. Gurney, B. A. Wilson, J. A. Katine, S. Maat, and J. R. Childress, Nanotechnology 21, 235202 (2010). 5J. Torrejon, M. Riou, F. A. Araujo, S. Tsunegi, G. Khalsa, D. Querlioz, P. Bortolotti, V. Cros, K. Yakushiji, A. Fukushima, H. Kubota, S. Yuasa, M. D. Stiles, and J. Grollier, Nature 547, 428 (2017). 6Z. Zeng, G. Finocchio, B. Zhang, P. K. Amiri, J. A. Katine, I. N. Krivorotov, Y. Huai, J. Langer, B. Azzerboni, K. L. Wang, and H. Jiang, Sci. Rep. 3, 1426 (2013). 7Y. Zhou, H. Zhang, Y. Liu, and J. Åkerman, J. Appl. Phys. 112, 063903 (2012). 8M. D ’Aquino, C. Serpico, R. Bonin, G. Bertotti, and I. D. Mayergoyz, Phys. Rev. B Condens. Matter Mater. Phys. 82, 1 (2010). 9Y. Zhou, C. L. Zha, S. Bonetti, J. Persson, and J. Åkerman, Appl. Phys. Lett. 92, 262508 (2008). 10C. H. Sim, M. Moneck, T. Liew, and J.-G. Zhu, J. Appl. Phys. 111, 07C914 (2012). 11J. Cui, J. L. Hockel, P. K. Nordeen, D. M. Pisani, C. Y. Liang, G. P. Carman, and C. S. Lynch, Appl. Phys. Lett. 103, 232905 (2013). 12C. Y. Liang, S. M. Keller, A. E. Sepulveda, A. Bur, W. Y. Sun, K. Wetzlar, and G. P. Carman, Nanotechnology 25, 435701 (2014). 13Z. Zhao, M. Jamali, N. D ’Souza, D. Zhang, S. Bandyopadhyay, J. Atulasimha, and J. P. Wang, Appl. Phys. Lett. 109, 092403 (2016). 14Q. Wang, X. Li, C.-Y. Y. Liang, A. Barra, J. Domann, C. Lynch, A. Sepulveda, and G. Carman, Appl. Phys. Lett. 110, 102903 (2017). 15Q. Wang, J. Domann, G. Yu, A. Barra, K. L. Wang, and G. P. Carman, Phys. Rev. Appl. 10, 034052 (2018). 16V. Iurchuk, B. Doudin, and B. Kundys, J. Phys. Condens. Matter 26, 292202 (2014). 17V. Iurchuk, B. Doudin, J. Bran, and B. Kundys, Phys. Procedia 75, 956 (2015). 18N. Lei, T. Devolder, G. Agnus, P. Aubert, L. Daniel, J.-V. Kim, W. Zhao, T. Trypiniotis, R. P. Cowburn, C. Chappert, D. Ravelosona, and P. Lecoeur, Nat. Commun. 4, 1378 (2013). 19S. Cherepov, P. Khalili Amiri, J. G. Alzate, K. Wong, M. Lewis, P. Upadhyaya, J. Nath, M. Bao, A. Bur, T. Wu, G. P. Carman, A. Khitun, and K. L. Wang, Appl. Phys. Lett. 104, 82403 (2014). 20C. Chen, A. Barra, A. Mal, G. Carman, and A. Sepulveda, Appl. Phys. Lett. 111, 072401 (2017). 21D. Labanowski, A. Jung, and S. Salahuddin, Appl. Phys. Lett. 108, 22905 (2016). 22X. Li, D. Labanowski, S. Salahuddin, and C. S. Lynch, J. Appl. Phys. 122, 43904 (2017). 23M. Manteghi and A. A. Y. Ibraheem, IEEE Trans. Antennas Propag. 62, 6491 (2014). 24G. Yu, H. Zhang, Y. Li, J. Li, D. Zhang, and N. Sun, Mater. Res. Express 5, 045021 (2018). 25C.-Y. Y. Liang, S. M. Keller, A. E. Sepulveda, W.-Y. Y. Sun, J. Cui, C. S. Lynch, and G. P. Carman, J. Appl. Phys. 116, 123909 (2014). 26X. Li, D. Carka, C. Liang, A. E. Sepulveda, S. M. Keller, P. K. Amiri, G. P. Carman, and C. S. Lynch, J. Appl. Phys. 118, 14101 (2015). 27R. C. O ’handley, Modern Magnetic Materials (Wiley, 2000). FIG. 5. T emporal evolution of perpendicular magnetization m zfor applied voltage in different waveforms. (a) Square wave with 1.8 V amplitude at 1.1 GHz frequency. (b) Sinusoidal wave V ¼0:9þ0:9 sin(2 πf0t). (c) Sinusoidal wave V¼0:9þ4 π/C20:9 sin(2 πf0t), where f0¼1:1 GHz.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 163903 (2019); doi: 10.1063/1.5116748 126, 163903-6 Published under license by AIP Publishing.28D. Wang, Y. Fotinich, and G. P. Carman, J. Appl. Phys. 83, 5342 (1998). 29S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nat. Mater. 9, 721 (2010). 30A. Chaturvedi, H. Sepehri-Amin, T. Ohkubo, K. Hono, and T. Suzuki, J. Magn. Magn. Mater. 401, 144 (2016). 31H. Sato, M. Yamanouchi, K. Miura, S. Ikeda, R. Koizumi, F. Matsukura, and H. Ohno, IEEE Magn. Lett. 3, 3000204 (2012).32D. Wang, C. Nordman, Z. Qian, J. M. Daughton, and J. Myers, J. Appl. Phys. 97, 10C906 (2005). 33J.-V. Kim, in Solid State Physics , edited by R. E. Camley and R. L. Stamps (Academic Press, 2012), pp. 217 –294. 34T. J. Silva and W. H. Rippard, J. Magn. Magn. Mater. 320, 1260 (2008). 35C. Kittel, Phys. Rev. 73, 155 (1948). 36J. Shieh, J. E. Huber, and N. A. Fleck, Acta Mater. 51, 6123 (2003).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 163903 (2019); doi: 10.1063/1.5116748 126, 163903-7 Published under license by AIP Publishing.
1.44703.pdf	AIP Conference Proceedings 286, 309 (1992); https://doi.org/10.1063/1.44703 286, 309 © 1992 American Institute of Physics.Scaling behavior of magnetization in the critical region of a-Fe90−xCoxZr10 alloys Cite as: AIP Conference Proceedings 286, 309 (1992); https:// doi.org/10.1063/1.44703 Published Online: 29 May 2008 V. Siruguri , B. D. Babu , and S. N. Kaul SCALING BEHAVIOR OF MAGNETIZATION IN THE CRITICAL REGION OF a-Fego-xCox%rlo ALLOYS V.Sirugnri, P.D. Babu and S~N. Kaul Scho61 of Physics, University of Hyderabad, Hyderabad-500 134, INDIA Contrar I to earlier reports, a detailed ]erromafnetic resonance (FMR) stldy in the critical region of Feeo-ffiCozZrlo alloys with 0 <_ z < 10 shows that the critical ez- ponents ~ and ~/ for spontaneous mafnetization and initial nsceptibilitl, respectieeil, which characterize the ferromaonetic (FM) - paramaonetic (PM) phase transition at the Csrie tentperatere, To, possess ealnes which are close to the three-dimensional Heisen- berf ealnes and are independent o~ composition. The ~raction of spins that participates in the FM-PM phase transition possesses a ealne of llfG for the alloy with x = 0 and increases with Co concentration x. The peak-to-peak FMR linewidth is foeerned by the Landau-Lifshitz-Gilbert relazation mechanism in the critical region and the Land~ splitting factor g and the Gilbert dampin# parameter ~ are independent of temperatere. WTtile A decreases with x, g has a constant value of ~.07 -I. 0.0~. In this paper, we present the results of the ferromagnetic resonance (FMR) study in the critical region of ~-Fego_~Co~Zrlo alloys with 0 < x < 10 undertaken to resolve the controversy 1 surrounding the nature of pareanagnetic (PM) to ferromagnetic (FM) phase transition in these alloys. This t~hnique has been used earlier ~ to deter- mine the critical exponents for some of the concentrations in the above-mentioned series. In the present study, extra efforts have been made to determine the critical exponents and 7 for spontaneous magnetization and initial susceptibility, respectively, more accu- rately by t~king FMR data at temperatures much closer to the Curie temperature, To, than in the previous set of measurements 2 through an improved temperature control and to extend such FMR measurements to higher Co concentrations. In addition an at- tempt has been made to investigate the temperature dependence of the FMR linewidth in the critical region and deduce a reliable estimate of the Gilbert damping parameter, ~, for a given alloy configuration. Amorphous Feeo-sCo, Zrlo alloys were prepared by the single-roller melt-quenching technique in Ar atmosphere and their amorphous structure was confirmed by x-ray diffraction. FMR measurements were carried out using an X-band ESR spectrome- ter operating at a fixed microwave frequency of -~9.23 GHz in the temperature range (To - 15K) ~ T _< (To + 15K). The sample temperature was measured by a copper- constantan thermocouple and kept fixed to within + 50mK at a given temperature setting. The microwave power absorption derivatives, dP/dI'I, were measured as a func- tion of the external static magnetic field (H) using horizontal- parallel (in which H lies in the ribbon plane and is directed along the length of the sample) and vertical-parallel (in which H is directed along the breadth within the sample plane) orientations. Since these alloys are extremely sensitive to stress, adequate care was taken to ensure a stress- free mounting of the samples. Accurate values of saturation magnetization, M,, at different temperatures are de- duced from a detailed lineshape analysis s of the FMR spectra. With M,(Hr6,, T) -= @ 1994 American Institute of Physics 309 310 Scaling Behavior of Magnetization M(H,T) and H~, identified with the ordering field H conjugate to M (-ffi M,), the M(H,T) data are found 2 to satidy the waling equation of state (SES) m = f (1) where plus and mln~ls Slanmt refer to T > Tv and T < Tc and rn ffi M/[e[ p and h ffi H/lelP+~ are the ,~ded masnet~atlon and scaled field, respectively, and e -- (T- Tc)/Tv. A more rigorous test of whether or not the critical exponents and Tv are accurately determined is provided by the SF_~ of the form m' = :~=~ + b, (hl~) (~) which also allows determination of the a'itical amplitudes mo = a~/2 and ho/mo = =+/b+, defined by u.(d = ~o(-d p, ~ < 0 (3) and X2(d = (h./'~-) ~-~, ~ > 0 (4) The m 2 versus h/m sca~ag plots for a few representative concentrations are shown in Fig. 1. The intercepts on the ordinate and the abscissa of the universal curves give the critical amplitudes. The values of the critical exponents fl (---0.384- 0.02) and ~f (= 1.38 4- 0.03) obtained from the SES analysis are dose to the 3D Heisenberg values and are found to be isdcpendelt of the composition. If he is an =~erafe eMective elementary moment involved in the FM-PM phase transition, the ratio #,Hho/kBTv = 1.58, the 3D Hekeuberg value, since the exponents pmsess 3D Hekenberg values. The concentration of such effective moments equals the fraction of spins participating in the FM-PM phase transition and is give~x by where/to is the average magnetic moment per alloy atom at OK. The variation of #ey! and c with Co concentration strong]y indicates that only a small fraction of the moments (11% for x=O) partidpates in the PM-FM transition and the variation of c with Co concentration = can be described by an empirical relation c(=) - r -~ az =. The FMR linewidth, AHM, , has two main contributions given by/XH0, which is independent of the microwave frequency u, and/XHLr.o, which lure a linear dependeace on U. Thus, ZkH~,(v,T) is given by ZkH~,0,, T) = ~T/o(T) + LkHz~o(v,T) (6) The first term arises from magnetic inhomoge~eities and multi-magnon scattering mech- anlmms, which is weaJdy temperature-dependent, whereas the second term = 1.45)~0~/72 [M,(T)] -~ re~t. from the Landau-L~h't~Gabert (LLG) ~meon m~,-~--. In the critical region, the LLG term dominates (Fig. 2) over the other terms. The LLG dampi~ parameter, ~, and the Land6 splitting factor, 9, turn out to be independent of temperature in the critical region. While ~ decreases with = from 5.0xl0Sser -~ for = = 0 to 2.1xl0Ssec -z for = = 10, g remains constant at a value g = 2.074-0.03. V. Siruguri et al. 311 5 3 x-lO 2 -- ~-- "~5 2 ~_ 31" x='6 _~gJ ja~-'~ I o 4 go' x= ,o "r'51- ~ .---12 ~ r'~ I ..s t _ r 45i l- i , il 9 2 3 4 5 6 2 M'-I (10-= G-I) xffiO 0 1 2 3 4 5 Fig. 1. FMRlinewidth, &Hm,, plotted ,g&inst h/m (10 4) hwegse saturation m~et~fion in the ~m~rs- ture int~ -0.0\|<_ ~ <_0~\| for s-~_.~.Zrt0 ~. ~. m= - ~e - A/m pbt, for ~F~_.Co.gno ~s. The st~_~t ~es ~ t~u~ the d~a d~ constru~ runs Ms(T) d~a d~u~ ~m ~in~ ~reseat ~usree fit to the &H~T) the PMR s~rs reco~ M ~ffet-ent data ~ on ~. (6). tures in the ~ti~ resin. To conclude, the critical exponemts for &Fem..=Co=Zrlo alloys are r eitios-isdepeBde~t and possess values that are close to the 3D Heisenl>erg values. The fraction of spins participating in the FM-PM phase transition is very small (~11% for = = 0) and increases with z. The LLG relAY~tion mecbanlmm dominantly contributes to the FMR iinewidth in the critical region because of its M7 ~ dependence. Both the Land~ factor g and Gilbert parameter )~ are tentperatlre-isdepeBde~t but A decreases 1. S.NJ(anl, J. Phys. F18, 2089 (1988). 2. S2~J(anl and PJ)J~abu, Phys Rev. B45, 295 (1992). 3. S~J~au] and V.Sirugurl, J. Phys.: Condens. Matter 4, 505 (1992),
1.5050916.pdf	J. Appl. Phys. 125, 103902 (2019); https://doi.org/10.1063/1.5050916 125, 103902 © 2019 Author(s).Definition of the interlayer interaction type in magnetic multilayers analyzing the shape of the ferromagnetic resonance peaks Cite as: J. Appl. Phys. 125, 103902 (2019); https://doi.org/10.1063/1.5050916 Submitted: 03 August 2018 . Accepted: 18 February 2019 . Published Online: 08 March 2019 O. G. Udalov , A. A. Fraerman , and E. S. Demidov ARTICLES YOU MAY BE INTERESTED IN XFEM analysis of the fracture behavior of bulk superconductor in high magnetic field Journal of Applied Physics 125, 103901 (2019); https://doi.org/10.1063/1.5063893 General coupling model for electromigration and one-dimensional numerical solutions Journal of Applied Physics 125, 105101 (2019); https://doi.org/10.1063/1.5065376 Magnetic-field-induced incommensurate to collinear spin order transition in Journal of Applied Physics 125, 093902 (2019); https://doi.org/10.1063/1.5066625Deﬁnition of the interlayer interaction type in magnetic multilayers analyzing the shape of the ferromagnetic resonance peaks Cite as: J. Appl. Phys. 125, 103902 (2019); doi: 10.1063/1.5050916 View Online Export Citation CrossMar k Submitted: 3 August 2018 · Accepted: 18 February 2019 · Published Online: 8 March 2019 O. G. Udalov,1,2,a) A. A. Fraerman,2and E. S. Demidov3 AFFILIATIONS 1Department of Physics and Astronomy, California State University, Northridge, California 91330, USA 2Institute for Physics of Microstructures RAS, Nizhny Novgorod 603950, Russia 3Physics Department, Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod 603950, Russia a)oleg.udalov@csun.edu ABSTRACT We present a theoretical study of the ferromagnetic resonance in a system of two coupled magnetic layers. We show that an interaction between the layers leads to the occurrence of the so-called Fano resonance. The Fano resonance changes the shape of the ferromagnetic res- onance peak. It introduces a peak asymmetry. The asymmetry type is de ﬁned by the sign of the interaction between the magnetic layers. Therefore, by studying the shape of the ferromagnetic resonance peaks, one can de ﬁne the type of the interlayer coupling (ferromagnetic or antiferromagnetic). Published under license by AIP Publishing. https://doi.org/10.1063/1.5050916 I. INTRODUCTION Ferromagnetic resonance (FMR) is a powerful tool for study- ing the magnetic multilayer structures.1–10The FMR method allows one to obtain the information on the magnetization magnitude andmagnetic anisotropy of each layer. It can be used for studying theinterlayer coupling. A lot of e ﬀorts were spent on investigation of the coupling in the systems with magnetic layers separated by a metallic non-magnetic spacer. 1,8,11–15In this case, the interlayer coupling is strong enough. This makes it relatively easy to de ﬁne the coupling sign and magnitude studying shifts of FMR peaks. The situation is di ﬀerent for systems where ferromagnetic ﬁlms are separated by an insulating spacer. In this case, the inter- layer coupling is much weaker,16–19leading to a small shift of the FMR peak which is of the same order or even less than the reso-nance linewidth. While the FMR method is applicable in this case, 4 people mostly use the magneto-optical Kerr e ﬀect to study the cou- pling in multilayers with an insulator spacer.16–19It is important to mention that both methods require several samples to register the shift and to study the interaction. Note that there are at least two types of interlayer couplings in multilayers. The ﬁrst one is the exchange coupling16–19which isisotropic. This kind of coupling can be studied using the in-plane orientation of an external magnetic ﬁeld. The second interaction type is the dipole-dipole “orange-peel ”eﬀect.20,21This coupling is anisotropic. When magnetization of the layers is in-plane, the“orange-peel ”eﬀect favors a ferromagnetic arrangement. For the out-of-plane orientation of the magnetization, the dipole-dipole coupling is antiferromagnetic. To study the anisotropic interaction(with Magneto-optical Kerr e ﬀect (MOKE) or FMR method) one needs to perform measurements for both the orientations of themagnetic ﬁeld. The magnitude of the ﬁeld in the out-of-plane geometry is much higher than that in the in-plane. Since the e ﬀect of the interlayer interaction ~Jis of the order of ~J=H ext/C281 (where Hextis the external ﬁeld), it is hard to observe it in the out-of-plane geometry. Therefore, an alternative method is desirable. In the present work, we propose a novel approach for deﬁning the interlayer interaction sign and magnitude. The approach is based on studying the FMR peak shape rather thanthe shift. We will show that the interaction induces an FMR peakasymmetry. Such an asymmetry can be considered as the Fanoresonance 22in a magnetic multilayer. Studying the shape of this asymmetry, one can de ﬁne the interaction sign and magnitude. Such a method is particularly useful when resonance frequenciesJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 125, 103902 (2019); doi: 10.1063/1.5050916 125, 103902-1 Published under license by AIP Publishing.of two interacting layers are close to each other. As we will show, it also allows studying the interlayer coupling in the case when the external magnetic ﬁeld and magnetization are out of the sample plane. Studying the interaction sign and magnitude with the conven- tional method based on the FMR peaks shift requires a reference sample without the interlayer interaction. This allows us to measure the peak shift. The approach based on the peak shape does not havesuch a disadvantage. One can de ﬁne the interaction sign and magni- tude using a single sample. The paper is organized as follows. In Sec. II, we analyze a sim- pliﬁed model in which two magnetic moments are placed into a strong magnetic ﬁeld. Such a model allows analytical consideration providing insight into the physics behind the FMR peak shape(asymmetry). In Sec. III, we study the numerically magnetic bilayer system (NiFe/Co) with an arbitrary orientation of the external mag- netic ﬁeld. II. SIMPLIFIED MODEL In this section, we consider a simpli ﬁed model of two coupled magnetic moments. We calculate dissipation (FMR signal) in this system and demonstrate how the asymmetric peak of absorption appears. Consider two ferromagnetic (FM) ﬁlms with uniform magnetizations M 1,2(seeFig. 1 ). For simplicity, we assume that the magnetic moments of both layers are the same jM1,2j¼M0. There is a uniaxial anisotropy in each ﬁlm along the z-axis. It can be induced by a demagnetizing ﬁeld or by an internal anisotropy. The anisotropy constants are λ1,2. An external magnetic ﬁeld Hext¼H0z0is applied to the system. There is also a weak high- frequency alternating ﬁeld along the x-axis h(t)¼h(t)x0. Magnetic ﬁlms interact with each other. The interaction energy is given by the expression Eint¼/C0 ~J(M1M2): (1) We linearize the Landau-Lifshitz-Gilbert (LLG) equations for both magnetic moments M1,2in the vicinity of equilibriumpositions M1,2¼M0z0. The equations take the form _m1x¼/C0H1m1y/C0J(m1y/C0m2y)/C0~α1_m1y, _m1y¼H1m1x/C0J(m2x/C0m1x)þ~α1_m1x/C0h, _m2x¼/C0H2m2yþJ(m1y/C0m2y)/C0~α2_m2y, _m2y¼H2m2xþJ(m2x/C0m1x)þ~α2_m2x/C0h:8 >>< >>:(2) Here, m1,2are the corrections to the equilibrium magnetizations normalized by M0, the magnitude of the e ﬀective ﬁeld acting on the layers are H1,2¼γ(H0þ2λ1,2M0),J¼γ~Jis the interaction constant multiplied by the gyromagnetic ratio γ. The renormalized damping constants are ~α1,2. The system equation (2)can be trans- formed into two second order equations of the form €m1xþα1_m1xþω2 1m1x¼A1m2xþD1_m2xþh1, €m2xþα2_m2xþω2 2m2x¼A2m1xþD2_m1xþh2,/C26 (3) where we introduced the following notations: α1,2¼2~α1,2(H1,2þJ) 1þ~α2 1,2/C252~α1,2(H1,2þJ), ω2 1,2¼(H1,2þJ)2þJ2 1þ~α2 1,2/C25H2 1,2þ2JH1,2, A1,2¼(H1þH2)Jþ2J2 1þ~α2 1,2/C25(H1þH2)J, D1,2¼J(~α1þ~α2) 1þ~α2 1,2/C250, h1,2¼H1,2hþ~α1,2_h 1þ~α2 1,2/C25H1,2h:(4) Equation (3)describes the system of two coupled oscillators with the resonant frequencies ω1,2. There are two types of cou- pling between the oscillators. We assume that the damping is weak ( ~α1,2/C281, or α1,2/C28ω1,2for the case of weak coupling), which is often the case for ferromagnets. In this limit, one canneglect the dissipative coupling terms D 1,2_m1,2x.A l s o ,t h e retarded external excitation ~α1,2_hcan be omitted. For our pur- poses, we can also neglect ~α2 1,2in denominators in Eqs. (4).W e assume that the coupling between the ﬁlms Jis weak compar- ing to the e ﬀective ﬁelds H1,2. Therefore, we keep only the t e r m sl i n e a ri n J. A response of the system to a periodic external ﬁeldh1,2¼ h(0) 1,2eiωtcan be represented as m1,2x(t)¼m1,2eiωt. The complex amplitudes m1,2are given by m1¼(ω2 2/C0ω2þiα2ω)h(0) 1þA1h(0) 2 (ω2 2/C0ω2þiα2ω)(ω2 1/C0ω2þiα1ω)/C0A1A2, m2¼(ω2 1/C0ω2þiα1ω)h(0) 2þA2h(0) 1 (ω2 2/C0ω2þiα2ω)(ω2 1/C0ω2þiα1ω)/C0A1A2:8 >>>< >>>:(5) FIG. 1. A model system. Two magnetic moments placed in an external mag- netic ﬁeldHext. An alternating magnetic ﬁeldhis applied perpendicular to Hext. M1,2shows equilibrium orientation of the magnetic moments.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 125, 103902 (2019); doi: 10.1063/1.5050916 125, 103902-2 Published under license by AIP Publishing.A. Layers with essentially different damping, but the same resonant frequencies Let us now further simplify our consideration assuming that α2¼0 and ω1¼ω2. This means that H1¼H2,h(0) 1¼h(0) 2, and A1¼A2¼A. Next, we assume that the interaction is weak com- paring to the damping ( α1/C29A=ω1or~α1/C29γ~J=ω1). In this case, the oscillation amplitude of the ﬁrst layer magnetization is given by jm1j2¼[(ω2 2/C0ω2)þA]2(h(0) 2)2 (ω2 2/C0ω2þA)2(ω2 2/C0ω2/C0A)2þω2α2 1(ω2 2/C0ω2)2:(6) In the case of no interaction ( A¼0), we have an ordinarily reso- nance peak with the frequency ω2/C0α2 1=(4ω2)¼ω2(1/C0~α2). Introduction of the ﬁnite interaction Aleads to an additional shift of the peak, but we can neglect it when α1/C29A=ω1(~α1/C29γ~J=ω1). Theﬁnite interaction is also responsible for the appearance of two peculiar points at ω¼ω2+A=(2ω2)¼ω2+γ~J. At the point ω¼ω2/C0A=(2ω2)¼ω2/C0γ~J, the amplitude reaches its maximum. Oppositely, the oscillation amplitude goes to zero at the frequencyω¼ω 2þA=(2ω2)¼ω2þγ~J. Such a reduction of the oscillation amplitude is called the dynamical damping and is very well known in the oscillation theory.23–25Two periodic forces act on the the magnetic moment m1. The ﬁrst one is due to the external ﬁeld and the second one is due to the interaction with the second magneticlayer. Phases of the forces depend on frequency. When the phase diﬀerence is π, the forces cancel each other. Such a cancellation appears at ω¼ω 2þA=(2ω2) and therefore, m1does not oscillate at this frequency. At ω¼ω2/C0A=(2ω2), these two forces are in phase leading to enhancement of oscillations. Finally, the shape ofthe resonance peak is distorted and the peak asymmetry appears. Such a peculiarity in the frequency dependence of the oscillation amplitude is well known as the Fano resonance. 22 When we take ﬁniteα2into account, there is no full damping and the amplitude is not zero, but one still has the minimum at ω¼ω2þA=(2ω2) and the maximum at ω¼ω2/C0A=(2ω2). Important feature here is that if one changes the interaction sign the minimum and maximum switch their positions. For A,0 (antiferromagnetic (AFM) interaction), the dynamical dampingappears below ω 2.F o r A.0 (FM interaction), the dynamical damping appears above ω2. This feature can be used for de ﬁning the interaction sign. Figure 2 demonstrates behavior of jm1j2as a function of nor- malized frequency ( ω=ω2). Damping constant is ~α1¼0:05. The solid red curve shows the case of zero interaction, ~J¼0. In this case, there are no peculiarities in the amplitude behavior. Blue dashed curve in Fig. 2 shows jm1j2forﬁnite AFM interaction γ~J=ω2¼/C01:5/C110/C04. These parameters are within the limitations of the present simpli ﬁed model. One can easily see the asymmetry of the resonant peak. According to our consideration, the dynami- cal damping occurs in this case below ω2. Note that the curve is plotted for ﬁnite α2and therefore instead of zero amplitude at ω¼ω2(1/C0~α2 1), we have ﬁnite oscillations. The dynamical enhancement appears at ω¼ω2(1þ~α2 1). Dash-dotted green line shows jm1j2for positive FM interaction γ~J=ω2¼1:5/C110/C04. One can see that the Fano resonance (asymmetry) is re ﬂected withrespect to ω¼ω2in this case. So, the shape of the peak is clearly diﬀerent for di ﬀerent signs of the interlayer interaction. Closing this section, we have to mention that the Fano reso- nance disappears if the dissipation is the same in both layers. This happens because the coupled oscillators become the same and theirmutual in ﬂuence is the same. In the simpli ﬁed model (the case considered in this section), we ﬁxed the same magnetization and anisotropy of the layers. Therefore, the only di ﬀerence between the oscillators occurs due to di ﬀerent damping constants. In real systems, the coupled magnetic ﬁlms may have di ﬀerent magnetiza- tions or thicknesses or other parameters. This should also inducediﬀerence of the oscillators and lead to appearance of the peak asymmetry. B. Layers with essentially different resonant frequencies Similar behavior occurs when the resonant frequencies of two layers are not the same. The Fano resonance appears around the resonant frequency of the layer with lower dissipation. Again, thesign of the interlayer interaction de ﬁnes the shape ( “direction ”)o f the Fano resonance. Figure 3 shows the amplitude jm 1j2as a func- tion of normalized frequency ( ω=ω1)a tω2=ω1¼1:01,~α1¼0:015 and ~α2¼5/C110/C04,γ~J=ω1¼0,+1:5/C110/C04. It is important to note that the Fano peculiarity disappears as the resonance frequencies become far from each other and there isno overlap between the FMR peaks. C. Absorption In the FMR experiment, the measured quantity Wis the absorption or imaginary part of the system response W=ω¼M 0h(0)Im(m1xþm2x)/differenceα1jm1xj2þα2jm2xj2:(7) Figure 4 shows the absorption as a function of normalized fre- quency ( ω=ω1) for two interacting magnetic moments, where ω1¼ ω2are the resonance frequencies, dissipation constants are FIG. 2. Amplitude of the magnetization of the ﬁrst layer jm1j2as a function of frequency ω. The red line is for the zero interlayer coupling ( ~J¼0). The blue dashed line is for the ﬁnite AFM interaction ( ~J,0). The green dash-dotted line corresponds to ~J.0. The black line shows the amplitude of the second layer oscillation jm2j2(reduced 10 times to make it comparable to jm1j2).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 125, 103902 (2019); doi: 10.1063/1.5050916 125, 103902-3 Published under license by AIP Publishing.~α1¼5/C110/C04,~α2¼0:01,γ~J=ω1¼0,+2:5/C110/C03. One can see that at zero interaction, the absorption peak is symmetric, while forﬁnite interaction, the peak asymmetry appears. Here, the asymme- try is de ﬁned by the interlayer interaction sign. III. NUMERICAL SIMULATIONS In Sec. II, on the basis of the simpli ﬁed model, it was shown that the FMR peak asymmetry arises due to a weak interaction of the magnetic layers. The frequency dependencies of FMR signal were studied which was relevant for comparison of magnetic multi-layer systems with other systems showing the Fano resonances. Inthe FMR experiment, the ﬁeld dependence is ordinarily measured at aﬁxed frequency of alternating ﬁeld. Besides, in the model, a limit of strong ﬁeld was considered in which magnetizations M 1,2were co-directed with each other and with the external ﬁeld. In a real FMR experiment, the magnitude of the external ﬁeld is limited. Therefore, the coincidence of resonanceﬁelds of the magnetic layers ( Hr1/C25Hr2) may appear in the situa- tion when the external magnetic ﬁeld and the equilibrium magnetic moments of the layers are not co-directed. The analytical solutionof the problem in this situation is not feasible. Therefore, here wepresent numerical demonstration of the FMR peak asymmetry in arealistic situation. We use a well known numerical algorithm to solve the LLG equations for magnetic ﬁlms. 1,26The system energy is given by E¼EZþEDþEAþEint, (8) where the Zeeman energy is EZ¼/C0X i¼1,2di(MiHext), (9) the magneto-dipole shape anisotropy is ED¼X i¼1,22πdiM2 icos2(θi), (10) the uniaxial anisotropy is EA¼X i¼1,2diKicos2(θi): (11) Here, θ1,2are the polar angles of magnetizations (see Fig. 5 ). Note that the FMR spectrum is obtained at a strong magnetic ﬁeld. Therefore, there are no domains in the system and we can treat themagnetization as uniform. The external magnetic ﬁeldH extis inclined by an angle θHwith respect to the sample normal. Kis the anisotropy constant. Equilibrium angles of magnetizations θ(0) 1,2are deﬁned by minimization of system energy equation (8). We use the parameters approximately corresponding to the NiFe/I/Co FIG. 4. Absorption W[Eq. (7)] as a function of frequency ω. The case of equal resonant frequencies of the magnetic layers is shown. The red line is for the zero interlayer interaction ( ~J¼0). The blue solid line is for ﬁnite FM interaction (~J.0). The green dash-dotted line corresponds to ﬁnite AFM interaction (~J,0). FIG. 5. System geometry used in our numerical modeling. Two magnetic layers (NiFe and Co) with thicknesses d1,2are placed in an external magnetic ﬁeld Hext. The ﬁeld makes angle θHwith the layers normal. The alternating magnetic ﬁeldhis applied perpendicular to Hext. Equilibrium magnetic moments M1,2 make angles θ(0) 1,2with the normal. FIG. 3. Amplitude of the magnetization of the ﬁrst layer jm1j2as a function of frequency ω. The case when the resonant frequencies of the layers are different. The red line is for the zero interlayer coupling ( ~J¼0). The blue dashed line is for the ﬁnite AFM interaction ( ~J,0). The green dash-dotted line corresponds to the ﬁnite FM interaction ( ~J.0).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 125, 103902 (2019); doi: 10.1063/1.5050916 125, 103902-4 Published under license by AIP Publishing.magnetic bilayer. Such a system is a good candidate for veri ﬁcation of the method that we propose in the present work, since it can also be studied by conventional methods (shift-based FMR andMOKE). The thickness of NiFe and Co is the same d¼1 nm. The g-factors are g 1,2¼2, the frequency of the alternating ﬁeld isω¼9:5 GHz, the saturation magnetizations are M1¼325 G and M2¼1420 G, the uniaxial anisotropy constants are K1¼/C07:5/C1105G/C1Oe and K2¼4/C1106G/C1Oe, and the damping parameters are α1¼0:006 and α2¼0:04. Figure 6 shows the behaviour of equilibrium magnetization angles as a function of the external ﬁeld magnitude at θH¼5:8/C14. Theﬁeld magnitude and angle are chosen in the region where we will observe the FMR peak asymmetry. One can easily see that theequilibrium magnetic moments are not co-directed with each otherand with the magnetic ﬁeld. Figure 7 shows the dependence of the FMR signal as a func- tion of the external magnetic ﬁeld magnitude [ W(H ext)] at a ﬁxed frequency of the alternating ﬁeld. The upper and lower panels cor- respond to di ﬀerent signs of the exchange interaction ~J¼+0:001 J=m2.E a c h ﬁgure shows several plots for di ﬀerent angles θHof the applied ﬁeld. We consider two ﬁlms made of di ﬀerent materials. Therefore, the FMR peaks may occur at di ﬀerent resonant ﬁelds. One sees two separate resonances corresponding to NiFe and Colayers at the angles θ H.6:5/C14andθH,5:5/C14. The NiFe peak is the narrow one and the Co peak is the wide one. There is no peak asymmetry when the NiFe and Co peaks are far from each other. This is in agreement with our analytical model. However, changing the angle of the applied ﬁeld one shifts the resonance ﬁeld of NiFe and Co ﬁlms Hr1,2. Since the magnetic anisotropy of these ﬁlms is quite di ﬀerent Hr1,2(θH) the dependen- cies are not the same and intersect with each other at a certain angle θH. One can see that the peaks overlap at the angle θH/C255:9/C14. In this case, the asymmetry appears. Comparing upper an lower panel, one can see that the peak asymmetry is di ﬀerent for FM and AFM interaction. Therefore, one can de ﬁne the interac- tion sign by measuring FMR spectrum at conditions of intersection of peaks. If the slope of the narrow peak is higher on the left side, FIG. 6. Equilibrium angles θ(0)for Co and NiFe layers as a function of external ﬁeld magnitude. The external ﬁeld is applied by the angle θH¼5:8 deg with respect to the sample normal. FIG. 7. FMR spectrum (absorbed power Was a function of the external ﬁeld magnitude Hext) obtained numerically for NiFe/Co system. (a) FM interlayer interaction ~J¼0:001 J =m2. (b) AFM interlayer interaction ~J¼/C0 0:001 J =m2. Different curves in the same plot correspond to different inclination angles of the external magnetic ﬁeldθH. The curves for different θHare shifted with respect to each other for better visibility. FIG. 8. FMR spectrum (the derivative of absorbed power dW=dHextas a func- tion of the external ﬁeld magnitude Hext) obtained numerically for the NiFe/Co system for different exchange coupling constants ~J¼/C0 0:001 J =m2(black solid line), ~J¼0J=m2(green dashed line), ~J¼0:0005 J =m2(blue solid line), ~J¼0:001 J =m2(red solid line and blue squares), ~J¼0:002 J =m2( purple dash-dotted line).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 125, 103902 (2019); doi: 10.1063/1.5050916 125, 103902-5 Published under license by AIP Publishing.the interaction is FM. If the slope is higher on the right side, the interaction is AFM. The extent of the asymmetry can be used fordeﬁning the magnitude of the interaction. A. The procedure de ﬁning the interlayer coupling Generally, the procedure for studying the interaction can be the following: (1) one calculates the resonance ﬁelds for both mag- netic layers as a function of the angle of the external ﬁeld, and ﬁnds the orientation at which these ﬁelds are the same. At this step one can consider the ﬁlms as non-interacting. (2) One measures the FMR spectrum applying the external ﬁeld at the angle found at the previous step. (3) One ﬁts the experimental data using the numerical simulations taking the interlayer interaction into account, and estimates the sign and the magnitude of the coupling. Figure 8 illustrates the third step of the procedure. It shows “experimental ”(which in our case is a result of numerical model- ing) FMR signal for NiFe/Co interacting system (open squares, J¼0:001 J/m 2). This time we plot not the absorption itself, but its derivative with respect to magnetic ﬁeld magnitude. This plot is more easy to use for ﬁtting of experimental data. There is an asym- metry of the FMR spectrum. The asymmetry shows itself as a diﬀerence in the height of the peak and deep of the narrow line (left peak amplitude is higher than the right deep amplitude). Thisdiﬀerence corresponds to di ﬀerent left and right slopes of the narrow peak shown, for example, in Fig. 7 . Solid lines in Fig. 8 show FMR spectrum calculated for di ﬀerent exchange interactions between the ﬁlms. If we take zero interaction (dashed green line), the peak and deep heights are the same, meaning that there is nocoupling. If we take negative interaction (black line), we get oppo-site asymmetry (peak amplitude is lower than the amplitude of thedeep). If one takes positive small (twice smaller, J¼0:0005 J/m 2) interaction, the asymmetry is small comparing to the “experiment ” (see solid blue line). If one takes twice stronger interaction (dot-dashed purple line, J¼0:002 J/m 2), the asymmetry is too high. Finally, taking the right magnitude of the coupling, one gets good ﬁt of the data (solid red line) and de ﬁne the sign and the magni- tude of the coupling.B. Discussion We consider here the ﬁlms with the same thickness. The dependence of the asymmetry on the ratio of the thickness requires a more detailed investigation. However, some trends can be easily understood without a modeling. If one ﬁlm is much thicker than the other, this ﬁlm mostly contributes to the FMR signal. Here, the interaction does not in ﬂuence this ﬁlm. Therefore, one can not observe the coupling (asymmetry) in this case. So, to observe the coupling, it is better to study the ﬁlms with similar thickness. As we mentioned previously, there should be di ﬀerent dissi- pations in ﬁlms to observe the asymmetry. If the materials of which the ﬁlms are made have similar dissipation constants, one can tune the damping constant in one of the ﬁlms by adding a Pt layer on top of it. It was demonstrated that the interfacialspin-orbit interaction in this case enhances the dissipation in amagnetic ﬁlm. 27,28 We investigate numerically the dependence of the FMR spectra on the damping constant of magnetic ﬁlms. Figure 9 dem- onstrates the FMR absorption of the NiFe/Co magnetic bilayer withthe FM interlayer interaction, ~J¼0:001 J/m 2. Three di ﬀerent angles of the external ﬁeld are studied. At these angles, the reso- nance ﬁelds of both NiFe and Co layers are approximately the same. We change the ratio of damping constants of these ﬁlmsα1 andα2. One can see that when the ratio α2=α1is 2 or 3, the asym- metry is seen for all angles. Interestingly, that even in the case of equal damping constant, there is some asymmetry at angle θH¼5:9/C14andθH¼5:95/C14. IV. CONCLUSION We considered the FMR resonance in two coupled magnetic layers. We showed that the interaction between these layers leads tothe occurrence of the so-called Fano resonance. The Fano reso-nance shows as a peculiarity in the absorption spectrum of the coupled system. In particular, the resonance peak becomes asym- metric. The asymmetry type is de ﬁned by the sign of the FIG. 9. FMR spectrum (absorbed power Was a function of the external ﬁeld magnitude Hext) obtained numerically for NiFe/Co system for different damping constants α1 andα2. There is a positive (FM) interaction between the ﬁlms ( ~J¼0:001 J =m2). (a) The external ﬁeld angle is θH¼5:85/C14, (b)θH¼5:9/C14, (c)θH¼5:95/C14.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 125, 103902 (2019); doi: 10.1063/1.5050916 125, 103902-6 Published under license by AIP Publishing.interaction between the layers. One can use the asymmetry to dis- tinguish between FM and AFM interlayer coupling. Generally, using numerical simulations, one can even estimate a magnitude ofthe interaction ﬁtting the asymmetric FMR peak. As a ﬁnal remark, we would like to mention that in our work, we considered the isotropic interaction equation (1). Such an equa- tion describes the exchange coupling. However, many experiments evidence that in magnetic multilayer systems, there is also themagneto-dipole coupling called the orange-peel e ﬀect. In contrast to the exchange coupling, the orange-peel e ﬀect is anisotropic and described by a di ﬀerent equation. 20The anisotropy will lead to the angular dependence of the coupling constant J¼J(θH). This pecu- liarity can be used for distinguishing between the exchange cou-pling and the orange-peel e ﬀect. This opportunity requires further investigation. ACKNOWLEDGMENTS This research was supported by the State Program 0035-2018-0022 and the RAS program “Electronic spin resonance, spin-dependent phenomena and spin technologies. ” REFERENCES 1J. Lindner and K. Baberschke, J. Phys. Condense Matter 15, S465 (2003). 2V. P. Nascimento, E. B. Saitovitch, F. Pelegrinia, L. C. 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1.3671632.pdf	Hysteretic spin-wave excitation in spin-torque oscillators as a function of the in-plane field angle: A micromagnetic description G. Finocchio, A. Prattella, G. Consolo, E. Martinez, A. Giordano et al. Citation: J. Appl. Phys. 110, 123913 (2011); doi: 10.1063/1.3671632 View online: http://dx.doi.org/10.1063/1.3671632 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v110/i12 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 03 Oct 2013 to 128.233.210.97. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsHysteretic spin-wave excitation in spin-torque oscillators as a function of the in-plane field angle: A micromagnetic description G. Finocchio,1,a)A. Prattella,1G. Consolo,2E. Martinez,3A. Giordano,1 and B. Azzerboni1 1Dipartimento di Fisica della Materia e Ingegneria Elettronica, University of Messina, Salita Sperone 31, 98166 Messina, Italy 2Dipartimento di Scienze per l’Ingegneria e l’Architettura, University of Messina, C.da di Dio, 98166 Messina, Italy 3Departamento de Fisica Aplicada, University of Salamanca, Plaza de la Merced s/n, 37008 Salamanca, Spain (Received 8 September 2011; accepted 18 November 2011; published online 23 December 2011) This paper describes a full micromagnetic characterization of the magnetization dynamics driven by spin-polarized current in anisotropic spin-torque oscillators (STOs). For ﬁeld angles approaching the hard in-plane axis, the excited mode is uniform and a super-critical Hopf-bifurcation takes placeat the critical current density J C. For ﬁeld angles close to the easy axis of the free layer, the excited mode is localized (non-uniform) and a sub-critical Hopf-bifurcation occurs at JC. In this latter region, a hysteretic behaviour is, therefore, found. We demonstrate numerically that thenon-linearities of the STO are strongly reduced when the oscillation frequency at the critical current is near the ferromagnetic resonance (FMR) frequency computed at zero bias current, and in particular, this condition corresponds to the ﬁeld orientation at which a minimum in theFMR-frequency is achieved. VC2011 American Institute of Physics . [doi: 10.1063/1.3671632 ] I. INTRODUCTION The discovery that the spin-transfer torque can drive per- sistent oscillations of the magnetization of a nanomagnet in the GHz range1,2has been giving the possibility to use spin- valves,3point-contact geometries,4and magnetic tunnel junc- tions5as high frequency tuneable oscillators (spin-torque oscillators (STOs)). The behaviour of these devices has been extensively studied in the last decade experimentally,6–8ana- lytically,9,10and numerically.11,12The STOs are strongly non-linear because of the coupling between the oscillation fre- quency x¼2pfand the oscillation power ( p)(xðpÞ).13,14 This coupling results in an enhancement of the STO-linewidth in the presence of thermal ﬂuctuations since the power noise in turn induces a frequency noise.15Experimentally, it has been demonstrated that most of the dynamical properties of a STO can be controlled via an external magnetic ﬁeld. For example, in point contact geometries, the direction and theamplitude of the external ﬁeld determines the nature of the spin-wave mode which is excited by a spin-polarized cur- rent: 16propagating Slonczewski mode for out of plane bias ﬁelds17and evanescent bullet mode for in-plane bias ﬁelds.18 In different geometries, such as spin-valves or magnetic tun-nel junctions having elliptical cross section or with uniaxialmagneto-crystalline anisotropy (anisotropic STOs), a strong dependence of the linewidth as function of the direction of an in-plane bias ﬁeld has been observed. 19,20In general, the line- width of a typical STO ( Dx) can be expressed as15 Dx¼C/C0ðpÞkBT bp1þtðpÞ2/C16/C17 ; (1)where C/C0ðpÞis the negative damping due to the spin polar- ized current and tis the ratio between the non-linear frequency shift N¼dxðpÞ dpand the derivative of the effective dampingdðCþ/C0C/C0Þ dpwith respect to the power being CþðpÞthe positive damping. Tand kBare the temperature and the Boltzmann constant, respectively, while b¼xðpÞMSV0=cis the power-energy proportionality coefﬁcient. MSis the satu- ration magnetization, V0is the free layer volume where the oscillation takes place, and cis the gyromagnetic ratio. If the applied current density Japproaches the critical current den- sityJCneeded to excite persistent magnetization oscillations, the relationship xðpÞcan be simply written as 2 pfðpÞ ¼2pf0þNpwhere f0is the oscillation frequency at the criti- cal current. Analytical calculations21and measurements22 showed that the origin of the minimum linewidth corre-sponds to the situation at which the non-linear frequencyshift vanishes ( N¼0). In particular, as the ﬁeld angle approaches the in-plane hard axis direction, the physical coupling between oscillation frequency and power tends todisappear. Previous micromagnetic simulations and computa- tions based on complex Ginzburg-Landau equation showed that this variation is related to a transition of the excitedmode from a non uniform to an uniform mode with a conse- quent change of the magnetic volume V 0where the oscillation takes place.19,23Here, we performed a complete numerical experiment to fully understand the dynamical behaviour of an anisotropic STOs as function of the bias ﬁeld angle by consid- ering the same experimental framework of Ref. 19. Our main results can be summarized as follows. Simi- larly to what observed in point contact geometries,16we found a range of ﬁeld angles where uniform and non uniformmodes are both excited and they are non-stationary in time. For ﬁeld angles approaching the hard in-plane axis, the excited mode is uniform and a super-critical Hopf-bifurcationa)Author to whom correspondence should be addressed. Electronic mail: gﬁnocchio@ingegneria.unime.it. 0021-8979/2011/110(12)/123913/6/$30.00 VC2011 American Institute of Physics 110, 123913-1JOURNAL OF APPLIED PHYSICS 110, 123913 (2011) Downloaded 03 Oct 2013 to 128.233.210.97. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionstakes place at JC. For ﬁeld angles close to the easy axis of the free layer, the excited mode is localized (non-uniform) and a sub-critical Hopf-bifurcation occurs at JC. In this latter region, a hysteretic behaviour is, therefore, found. We also demonstrate numerically for the ﬁrst time that in proximity of the condition N/C250 (null tunability), the oscillation frequency coincides with the ferromagnetic reso- nance frequency ( fFMR) as predicted by analytical theory.10 II. NUMERICAL MODEL We studied the magnetization dynamics driven by spin- transfer-torque in exchange biased spin-valves, the active part of spin-valve is composed by Py(4)/Cu(8)/Py(4)/IrMn(8)(Py¼Ni 81Fe19) (the thicknesses are in nm). The cross section is elliptical with axes of 150 nm and 50 nm. In the following, we refer to the exchange biased Py-layer acting as pinnedlayer by using the index pand to the single Py-layer (free layer) with the index f. Our computations take also into account the coupled magnetization dynamics ( m fandmpare the normalized magnetization vectors) of the two nanomag- nets due to both magnetostatic ﬁeld and spin-transfer-torque. Our results are based on the numerical solution of theLandau-Lifshitz-Gilbert- Slonczewski (LLGS) equation 24,25 dmf ds¼/C0 ð mf/C2heff/C0fÞþaGfmf/C2dmf ds/C0Tðmp;mfÞ dmp ds¼/C0 ð mp/C2heff/C0pÞþaGpmp/C2dmp ds/C0Tðmp;mfÞ8 >< >:; (2) where aGfandaGpare the damping parameters for the free and pinned layer, respectively, ds¼cMSdtis the dimension- less time step. We implemented the 3-dimentional general- ization of the spin-torque formulation Tðmp;mfÞcomputed by Slonczewski in 2002 for symmetric spin-valves,26in which the free layer magnetization acts as the polarizer of the pinned layer and vice versa Tðmp;mfÞ¼glBjj jejcM2 sJ Lfeðmf;mpÞmf/C2ðmf/C2mpÞ /C0J Lpeðmp;mfÞmp/C2ðmp/C2mfÞ8 >>< >>:;(3) where gis the gyromagnetic splitting factor, lBis the Bohr magneton, eis the electron charge, Jis the current density, LfandLpare the thicknesses of the free and pinned layer, respectively, eðmp;mfÞ(eðmp;mfÞ¼eðmf;mpÞ) is the polarization function given by eðmp;mfÞ¼0:5PK2=1þK2þð1/C0K2Þmp/C15mf/C0/C1 ;(4) where P and K2are the torque parameters. The material pa- rameters are: exchange constant A¼1.3 10/C011J/m and MS¼650/C2103A/m for both the free and pinned layer and damping parameters aGf¼0.025 and aGp¼0.2 for the free and the pinned layer, respectively. The presence of the exchange bias with an antiferromagnet increases the damp-ing by an order of magnitude. 27The torque parameters are P¼0.38 and K2¼2.5, respectively (see Ref. 24for moredetails about the parameter values). The initial conﬁguration of the magnetization for each ﬁeld value has been computed by solving the Brown equation (with a residual of 10/C07) mf/C2heff/C0f¼0 mp/C2heff/C0p¼0/C26 : (5) For sub-threshold current densities (smaller than those required to excite magnetization self-oscillation), the static conﬁguration is computed by solving a generalized expres- sion of the Eq. (5) mf/C2ðheff/C0f/C0rJeðmf;mpÞðmf/C2mpÞÞ ¼ 0 mp/C2ðheff/C0p/C0rJeðmf;mpÞðmp/C2mfÞÞ ¼ 0/C26 ;(6) where r¼glBjj jejc0M2sd(Ref. 28). III. RESULTS AND DISCUSSIONS Here, we will focus on the description of the computa- tional results due to an applied ﬁeld with an amplitude of 100 mT. Similar qualitative results have been also observed for ﬁelds from 90 mT to 140 mT. Fig. 1(a) summarizes the value of the average x-component of the free layer (black ‘þ’) and the pinned layer (red ‘o’) as function of the in- plane ﬁeld angle b(see Fig. 1(a)). It is possible to identify three different regions: (1) “collinear” for 0/C14<b/C2050/C14, where the two magnetizations mpandmfat the equilibrium are near to the parallel conﬁguration, (2) “non-collinear” for 50/C14<b/C2095/C14where the offset angle between the equilib- rium position of mpandmfis larger than 4/C14, and (3) “anti- parallel” for b>95/C14(we simulated ﬁeld angle up to b¼125/C14) where the initial conﬁguration of the magnetiza- tions is close to the anti-parallel state. For positive currents,the magnetization dynamics is observed in the ﬁeld angle regions (1) and (2) only. First of all, we studied the dependence of the critical current (J C)as function of the ﬁeld angle ( b). Fig. 1(b) sum- marizes the computations of the critical currents obtained sweeping back and forth the current density from J¼0t o J¼1.0/C2108A/cm2. For current densities J<JON, the mag- netization is found to be in a static state S computed by solv- ing the Eq. (6). When the current reaches the critical value J¼JON, the magnetization is no longer static and a dynami- cal state (D) is excited. The D state is also stable for sub- critical values J<JONup to the value JOFF. For J<JOFF, the D state disappears and the stable conﬁguration S-state is achieved. Our results show the presence of an hysteretic region S/D.29Analytical computations based on Melnikov theory point out for collinear conﬁguration of the magnetiza- tion, the coexistence of a limit cycle, and a ﬁxed point of dy- namics in the phase diagram for some range of ﬁeld andcurrent amplitude. 9The bi-stability region S/D becomes smaller as the ﬁeld angle approaches 90/C14. The inset of Fig. 1(b) displays an example of frequency vs current density hysteresis loop computed for b¼45/C14. We characterized the dynamical properties of the STO by also studying the power spectrum of the magnetization dynam-ics and the spatial distribution of the oscillation power within the cross sectional area of the spin-valve. Figs. 2(a)–2(c) show123913-2 Finocchio et al. J. Appl. Phys. 110, 123913 (2011) Downloaded 03 Oct 2013 to 128.233.210.97. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsthe power spectra computed at JONby means of the micromag- netic spectral mapping technique30,31for different ﬁeld angles: (a)b¼45/C14,( b ) b¼60/C14, and (c) b¼87.5/C14. As found in previ- ous micromagnetic simulations19and in computations based on the complex Ginzburg-Landau equation,23also our numeri- cal computations showed that, as the ﬁeld angle bincreases, the spatial distribution of the excited dynamical mode changes from non-uniform to uniform (inset of Fig. 2(a)compared to the inset of Fig. 2(c), the oscillation power increases from white to black). The reason for the change of the spatial proﬁle of the excited mode can be understood qualitatively by consid- ering the magnitude and direction of the spin-transfer torquesas a function of the ﬁeld angle bin the case of collinear (orquasi-collinear) and non-collinear initial conﬁguration of the magnetization of the two ferromagnets. Since for each compu- tational cell, the magnitude of spin torque is approximatelyproportional to sin h i(where hiis the angle between the mag- netization of the free and the pinned layer computed at the i- micromagnetic cell), and in the non-collinear conﬁguration,the spin torque acting on each spin of the free layer is relatively large and the variations from one micromagnetic cell to another are small. This situation of, a large, nearly uniformspin torque leads to a nearly coherent rotation of all spins of the free layer and gives rise to the excitation of the uniform mode. For the case of collinear conﬁguration, the excitation ofa non uniform mode is due to the fact that the spin torque FIG. 1. (Color online) (a) Average x- component of the magnetization for the two ferromagnets as function of the ﬁeld angle ( H¼100 mT). (b) Critical current densities JONandJOFFfor the transition from the S !D state and vice versa com- puted as function of the ﬁeld angle. Inset: Example of frequency vs current densityhysteresis loop computed for b¼45 /C14. (c) Trajectories of the average magnetiza- tion of the free layer computed at JONfor b¼45/C14and for b¼87.5/C14. (d) and (e) projection of the trajectories of average magnetization plotted in (c) in the x-z plane and x-y plane, respectively. FIG. 2. (Color online) Power spectra at JON computed by mean of the micromagnetic spectral mapping technique for (a) b¼45/C14, (b)b¼60/C14,a n d( c ) b¼90/C14. The insets repre- sent the power distribution of the excited mode related, (the distributed power is nor- malized (the maximum value in the power spectrum coincides to one, see the colour scale indicated in (a)) and increases from white to black. (d) A comparison among the oscillationfrequency computed at J ONandJOFFand the ferromagnetic resonance frequency.123913-3 Finocchio et al. J. Appl. Phys. 110, 123913 (2011) Downloaded 03 Oct 2013 to 128.233.210.97. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsacting on each micromagnetic cell is small and can vary in direction because during the dynamical transient the micro- magnetic state of the free magnetic layer can differ from anuniform conﬁguration. Our ﬁrst result points out that the transition in term of ﬁeld angle between the excitation of a non-uniform mode toan uniform mode is not abrupt but there exists a narrow range of ﬁeld orientation where both uniform and non- uniform mode are excited as displayed in the power spectraof Fig. 2(b) which was computed at J ONforb¼60/C14(two Lorentzian functions have been used to ﬁt the peaks). A time domain study based on wavelet analysis32points out that the excitation of the two modes is non-stationary (not shown) similarly to the results published in Ref. 16for point contact geometries. Fig. 2(d) summarizes the oscillation frequency atJONand JOFF as a function of the ﬁeld angle, where the frequency of the excited mode with largest power is displayed. The different qualitative behaviour of the magnetization dynamics at 45/C14and 87.5/C14can be attributed to the different Hopf bifurcation at JON. In particular, we identify, as sub- critical Hopf bifurcation, the transition from S to D when the excited mode is non-uniform.33This aspect is conﬁrmed by the ﬁnite oscillation power observed at the critical currentdensity (see Fig. 1(c)for the average normalized magnetiza- tion trajectory at b¼45 /C14). From the theoretical point of view, the presence of a sub-critical Hopf bifurcation givesrise in the bifurcation phase diagram to an hysteretic behav- ior. 33As the ﬁeld angle approaches the in-plane hard axis, we identify the transition from S to D state as a super-criticalHopf bifurcation. The oscillation power at J ONis close to zero (see Fig. 1(c)for the average normalized magnetization trajectory at b¼87.5/C14). Figs. 1(d) and1(e) display the pro- jection of the two trajectories for b¼45/C14andb¼87.5/C14at JONin the x-z and x-y plane, respectively. From physical point of view, the key difference between the two Hopfbifurcation is related to the oscillation axis. While for sub- critical Hopf bifurcations (non-uniform mode), the oscilla- tion axis of the magnetization at J ON(in the cell where the mode is excited) is different from the equilibrium axis of the magnetization and for super-critical Hopf bifurcations (uni- form mode), the oscillation axis at JONcoincides with the equilibrium conﬁguration. To take into account the presence of sub-critical Hopf bifurcation in the usual spin-torque oscillator theory, theamplitude of the “dimensionless power” cannot be inter- preted as the power of the spin wave mode. 10 In order to qualitative understand whether the spin- torque-driven modes observed in our full micromagnetic sim- ulations can be related to the normal modes (eigenmodes) of the system, we apply the technique recently developed inRef. 34(compared to the full micromagnetic simulations, the Oersted ﬁeld, the dipolar coupling between pinned and free layer, the spatial-time dependence of the polarization function,and the back spin-torque on the pinned layer have not been taken into account). That technique allows to identify the nor- mal modes of the ferromagnets which become unstable underthe action of non-conservative contributions, such as damping and spin-transfer torque. It should be pointed out that can becaptured the only dynamics which takes place very close to the excitation threshold, where the magnetization conﬁgura- tion only slightly differ from that at equilibrium within a linearapproximation (very small magnetization trajectory). By applying that technique to the setup of this paper, our ﬁnding is that none of the sub-critical modes (modeswhich exhibit a ﬁnite power at the excitation threshold and whose precessional trajectory is rather large) can be related to the normal modes, these are non-linear modes and cannotbe classiﬁed as eigenmodes. In the supercritical case, on the contrary, our computations conﬁrm that the mode which is excited at threshold is a normal mode and corresponds to theeigenmode which exhibits the lowest frequency (energy). In some experimental data, a low frequency tail in the measured power spectra is observed (see for example theFig.2(a)in the Ref. 19). Here, we demonstrate from our nu- merical computations that the origin of the low frequency tail is related to the co-existence of a limit cycle and a ﬁxedpoint of dynamics in the phase diagram. In other words, there exists an hopping of the magnetization between the S and the D state in the time domain near a sub-critical Hopf bifurca-tion. The presence of this hopping is clearly observed also at low temperature. Fig. 3summarizes our results computed for J¼0.6/C210 8A/cm2,ab¼45/C14, and a temperature T¼25 K. FIG. 3. Dynamical characterization of the x-component of the magnetiza- tion computed at JONforb¼45/C14in presence of the thermal ﬂuctuations (T¼25 K): (a) time domain trace; (b) wavelet scalogram (the colorbar is related to the amplitude of the wavelet transform); and (c) power spectrum.123913-4 Finocchio et al. J. Appl. Phys. 110, 123913 (2011) Downloaded 03 Oct 2013 to 128.233.210.97. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsFig3(a) shows the time trace of the hmXi(the most signiﬁ- cant component for the giant-magneto-resistance signal), while Fig. 3(b) displays the Wavelet transform of the time traces in Fig. 3(a)computed by means of the complex Morlet wavelet mother as described in Ref. 32, the color bar is related to the amplitude of the wavelet coefﬁcient (powerdensity increases from black to white). As can be clearly observed, from 68 to 73 and from 77 to 80, the oscillation output power (white) disappears indicating a static state, outof those time slots, a dynamical state is observed where the instantaneous frequency (as deﬁned for the Wavelet trans- form) 35is modulated from the power and frequency noise close to the zero-temperature oscillation frequency. Fig. 3(c) shows the power spectrum computed by means of the micro- magnetic spectral mapping where can be observed a low fre-quency tail also for a simulation long 500 ns at low temperature ( T¼25 K) (as example, compare Fig. 2(a) of Ref. 19to Fig. 3(a)). On the other hand, this hopping is not observed close to the in-plane hard axis, in agreement with the experimental data (not shown). 19We argue that this hop- ping of the magnetization gives rise to the shape distortionand the linewidth enhancement observed near the critical current in STOs with large non-linear frequency shift. 36–38 From the relationship between oscillation frequency and power (parametric plot as function of the current density), it is possible to identify the properties of STOs.10,39,40In those devices, strong non-linearities correspond to a large slope ofthe curve xðpÞcomputed near the critical current. Recently, we numerically demonstrated that for a ﬁxed ﬁeld angle, the non-linearities of the oscillator can be tuned by means of theamplitude of the applied ﬁeld, and an oscillation frequency independent of the applied current density (and consequently on the oscillation power) can be achieved. 41Fig.2(d) shows a comparison between the oscillation frequency computed at JON(fON) and JOFF(fOFF) and the fFMR. The fFMR (Ref. 42) has been computed as response of the system to a weak external ac current density of the form J¼JMcosð2pfACtÞ(JM¼0.25/C2108A/cm2) (no bias current density) computing the oscillation amplitude of theaverage x-component of the magnetization. Our numerical results prove that exists a ﬁeld angle region where the oscil- lation frequency coincides (or it is very close) to the f FMR computed at zero bias current in agreement with the predic- tion of analytical theory based on the universal model of non-linear oscillators. As well known when the condition atwhich the non-linear frequency shift change its sign ( N/C250) is achieved, a minimum in the linewidth is observed. 15,21 However, we point out that for the whole ﬁeld angle region where a super-critical Hopf-bifurcation is observed, the os- cillation frequency at the critical current is very close to the fFMReven if N/C290 being at JONthe oscillation power very small. For a given ﬁeld amplitude, our results indicate that the non-linearities of a STO is strongly reduced near a minimumin the f FMR computed as function of the ﬁeld angle “hard angle” (which in general does not coincide with the hard axis of the ellipse20) at zero bias current. This result could be used as a systematic pre-processing tool to identify an opti- mal STO conﬁguration bias point.Fig.4(a)shows typical FMR-spectra computed for three different angles b¼45/C14,b¼60/C14, and b¼87.5/C14. The larger area of the curve near the in-plane hard axis is related to the larger offset angle between the magnetization of the two fer- romagnets (see Fig. 1(a)). The reduced non-linearity gives also rise to a narrow linewidth FMR-spectra (see Fig. 4(b)). The effect of spin-torque driven FMR can be used for appli- cation such as microwave frequency detector43reaching very large sensitivity comparable with the Shottky diode,44our computations predict that the sensitivity at the hard angle due to the non-collinear conﬁguration of the pinned and freelayer is sensibly increased together to an improvement of the selectivity (FMR-spectra with narrow linewidth). These results can be used in the next generation of spin-torquediode based on anisotropic magnetic tunnel junction. In summary, we performed a multidomain micromag- netic study of the spin-wave properties excited in STOs asfunction of the in-plane ﬁeld angle in exchange biased spin-valves, characterizing their behaviour in term of self- oscillations and FMR-spectra. As the ﬁeld angle approaches the in-plane hard axis of the ellipse, the magnetization dynamics is characterized by a transition of the exited mode from a non-uniform spatial dis-tribution (sub-critical Hopf bifurcation) to an uniform spatial distribution (super-critical Hopf bifurcation). While the sub- critical modes are non-linear, the super-critical modes canbe related to the eigenmode of the free layer with the lowest frequency. We also ﬁnd that in the ﬁeld angle region where a non-uniform mode is excited, a hysteretic behaviour isobserved as a function of the current density, and in presence of thermal ﬂuctuation in that region, a hopping between a FIG. 4. (Color online) (a) Examples of FMR-spectra computed with no bias current for three different angles b¼45/C14,b¼60/C14, and b¼90/C14; (b) linewidth of the FMR spectrum as a function of the ﬁeld angle.123913-5 Finocchio et al. J. Appl. Phys. 110, 123913 (2011) Downloaded 03 Oct 2013 to 128.233.210.97. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsdynamical state and a static state can be achieve. We argue that the origin of the linewidth enhancement and distortion observed in some experimental data near the critical currentis related to the presence of this hysteretic region. At inter- mediate angles, the magnetization is characterized by the excitation of two modes in the power spectrum. When the super-critical modes are excited at the critical current, the oscillation frequency is very close to the f FMR even if the non-linear frequency shift is very large due to the low oscillation power. We demonstrated numerically that the non-linearities of the STO are strongly reduced, only when the oscillation frequency at the critical current is near thef FMRcomputed at zero bias current in the particular condition which corresponds to the ﬁeld orientation (hard angle) at which a minimum in the fFMRis achieved (the non-linear fre- quency shift change its sign). Finally, our computations pre- dict that for applications such as resonant microwave detection at the hard angle, the spin-torque diode present thelarger the sensitivity and the selectivity. ACKNOWLEDGMENTS This work was supported by Spanish Project under Contract Nos. MAT2008-04706/NAN and SA025A08. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 2L. Berger, Phys. Rev. B 54, 9353 (1996). 3S. I. Kiselev, J. C. 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1.3087748.pdf	Nonlinear electromagnetic response of ferromagnetic metals: Magnetoimpedance in microwires D. Seddaoui, D. Ménard, B. Movaghar, and A. Yelon Citation: Journal of Applied Physics 105, 083916 (2009); doi: 10.1063/1.3087748 View online: http://dx.doi.org/10.1063/1.3087748 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/105/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Superferromagnetism and coercivity in Co-Al2O3 granular films with perpendicular anisotropy J. Appl. Phys. 111, 123915 (2012); 10.1063/1.4730397 Magnetic structure and resonance properties of a hexagonal lattice of antidots Low Temp. Phys. 38, 157 (2012); 10.1063/1.3684279 Nanoscale magnetic structure and properties of solution-derived self-assembled La0.7Sr0.3MnO3 islands J. Appl. Phys. 111, 024307 (2012); 10.1063/1.3677985 Study of CoFeSiB glass-covered amorphous microwires under applied stress J. Appl. 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Yelon1 1Department of Engineering Physics and Regroupement Québécois sur les Matériaux de Pointe (RQMP), Ecole Polytechnique de Montréal, CP 6079, Succursale Centre-ville, Montréal, Québec H3C 3A7,Canada 2Center for Quantum Devices, Electrical Engineering, and Computer Science, Northwestern University, Evanston, Illinois 60208, USA /H20849Received 11 November 2008; accepted 22 January 2009; published online 23 April 2009 /H20850 Numerical calculations based on simultaneous solution of the Maxwell and Landau–Lifshitz equations were performed, in order to study the voltage response of ferromagnetic conductorscarrying ac current. Since no signiﬁcant approximations are made in the calculations, the modelyields both linear and nonlinear giant magnetoimpedance /H20849GMI /H20850behavior and low and high power ferromagnetic resonance. Application to nonlinear GMI in ideal wires, with regions of uniformanisotropy, allows us to understand many aspects of the observed behavior and to predictphenomena such as solitary-wave-like propagation of the magnetization at fairly high currentamplitude. Using appropriate magnetic structure, we were able to reproduce, with good agreement,the experimental observations for cobalt rich amorphous microwires. We have also found that evenharmonics of GMI signal are very sensitive to the domain structure of the wire, whereas the oddharmonics are not. © 2009 American Institute of Physics ./H20851DOI: 10.1063/1.3087748 /H20852 I. INTRODUCTION Calculation of the electromagnetic response of magnetic metals in the linear continuum limit has been well estab-lished for over 50 years. 1,2The methodology of simulta- neously solving Maxwell’s equations and the Landau–Lifschitz equation, and satisfying the appropriate boundaryconditions, was applied especially to ferromagnetic reso-nance /H20849FMR /H20850in ﬁlms 1,2and later in wires3and more recently to giant magnetoimpedance /H20849GMI /H20850/H20849see. Refs. 4–7and ref- erences therein /H20850. When a dc magnetic ﬁeld is applied parallel to the axis of a very soft magnetic conductor, the ac imped- ance Zis an extremely sensitive function of the ﬁeld. First reported in 1935 for NiFe alloy,8,9the effect was rediscov- ered about 60 years later for ultrasoft magneticmicrowires 10–12and is now referred to as GMI. Such calcu- lations are essentially analytic, even if numerical means areemployed for obtaining the solutions. Over the past 15 years, there have been a vast number of experimental studies of GMI, due to its promise for manyapplications, especially to low cost magnetic sensors. Whenthe applied sinusoidal current amplitude is relatively low, thevoltage response across the sample is sinusoidal, and theGMI is linear. Dynamic models for this regime are welldeveloped 4,6,7,13,14with various levels of approximation.15At high current amplitude, distortions appear in the voltage,16–18 leading to the appearance of higher harmonics in the re- sponse. In this case, the GMI is nonlinear /H20849NLGMI /H20850, and it is sometimes referred to as magnetoinductance in the low fre-quency regime. 16The second harmonic signal is of particular interest due to its extreme sensitivity to variations inﬁeld 19–22and stress.23,24It should also be mentioned that alarge fraction of experimental studies have been performed under conditions of NLGMI but have been analyzed usinglinear models. 15 NLGMI is not well understood at the present time. Ex- isting models for this regime are essentially quasistatic17,20,25 and are therefore limited to relatively low frequencies. While they yield useful insights concerning the second harmonicsignals, 20they cannot properly describe most of the existing experimental data in the NLGMI regime. This has motivatedus to develop nonlinear calculations, despite the great in-crease in difﬁculty which this produces, since these must betotally numerical. We present here the methodology whichwe have developed for such calculations and apply it to theanalysis of NLGMI in a commercial soft magnetic wire. Themethod is very general and has the potential for explainingmany phenomena in GMI and in FMR /H20849including parallel pumping 26/H20850which may be treated by a continuum model, that is, which do not require explicit quantum mechanicalexplanations. Ideal wires, consisting of symmetrical regionsof uniform anisotropy, are treated here for simplicity, but themethod can also be readily adapted to other sample shapes,such as microtubes, electroplated wires, and thin ﬁlms, andto more complicated magnetic structure. However, its appli-cation to a speciﬁc set of properties, such as magnetization, anisotropy, exchange, and conductivity, as a function of po-sition requires extensive calculations for variable static ﬁeld,current amplitude, and frequency. If we are to obtain detailedpredictions as a function of the various parameters, this willrequire a greater effort than has been expended on linearbehavior, over the past 50 years. Here, we treat ﬁrst the simplest situation which can pro- vide an idea of the evolution of the nonlinear voltage: uni- a/H20850Electronic mail: djamel.seddaoui@polymtl.ca.JOURNAL OF APPLIED PHYSICS 105, 083916 /H208492009 /H20850 0021-8979/2009/105 /H208498/H20850/083916/12/$25.00 © 2009 American Institute of Physics 105, 083916-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 19:07:19form properties and free surface spins. We then present the behavior of a real sample and introduce the least complicatedmodel which can reproduce this behavior. II. DYNAMIC MODEL AND NUMERICAL PROCEDURE A. Dynamic model The origin of the nonlinearity of GMI is the nonlinear dependence of the circumferential magnetic ﬂux on the cir-cumferential magnetic ﬁeld H /H9272induced by the ac current. The behavior of the magnetic ﬂux within the wire may beobtained by solving the nonlinear Landau–Lifshitz equationas a response to an effective ﬁeld dominated by the circum-ferential excitation ﬁeld. Since the magnetic ﬁeld is not ho-mogeneous in the wire, Maxwell’s equations includingOhm’s law need to be solved as well. In this conﬁguration,the skin effect combined with the exchange interaction resultin radial spin waves, greatly affecting the electromagneticresponse. 1–6Therefore, the exchange ﬁeld must be included in the Landau–Lifshitz equation, and boundary conditions onthe magnetization are required, in addition to the usual con-ditions on electromagnetic ﬁelds. In what follows, we assume that the microwire is a long cylinder, submitted to a uniform axial dc magnetic ﬁeld, andthat its behavior is uniform with length zand circumferential angle /H9272. Then, the dynamic electric E/H20849r,t/H20850and magnetic H/H20849r,t/H20850ﬁelds within the wire are only functions of time tand radial position r. With these assumptions, Maxwell’s equa- tions in cylindrical coordinates give r/H115092Hz /H11509r2+/H11509Hz /H11509r=r/H9268/H92620/H20873/H11509Hz /H11509t+/H11509Mz /H11509t/H20874, /H208491a/H20850 /H115092H/H9272 /H11509r2+1 r/H11509H/H9272 /H11509r−1 r2H/H9272=/H9268/H92620/H20873/H11509H/H9272 /H11509t+/H11509M/H9272 /H11509t/H20874, /H208491b/H20850 Hr=−Mr, /H208491c/H20850 where Mis the magnetization and /H9268is the conductivity. The radial spin waves result in a high demagnetizing dipolar ﬁeld/H20851Eq. /H208491c/H20850/H20852. As a consequence, the radial component of the magnetization is very small. Thus, the magnetization rotatesprimarily in the z- /H9272plan. The solution of Maxwell’s equa- tions imposes the discretization of the radial axis into Ncol- location points. Equations /H208491a/H20850and /H208491b/H20850are solved using the boundary conditions H/H9272/H20849r=0 ,t/H20850=0 , /H208492a/H20850 /H11509Hz /H11509r/H20849r=0 ,t/H20850=0 , /H208492b/H20850 H/H9272/H20849r=a,t/H20850=I0 2/H9266asin/H20849/H9275t/H20850, /H208492c/H20850 Hz/H20849r=a,t/H20850/H110150. /H208492d/H20850 Equations /H208492a/H20850and /H208492b/H20850result from continuity in cylin- drical symmetry of the circumferential components of themagnetic ﬁeld H/H9272and of the electric ﬁeld E/H9272=−/H9268−1/H11509Hz//H11509rat the cylindrical wire axis. Equation /H208492c/H20850results from Am- pere’s law. The approximation /H20851Eq. /H208492d/H20850/H20852requires justiﬁcation. The fundamental /H20849ﬁrst harmonic /H20850of the electromagnetic ﬁeld within the wire is coupled to an electromagnetic wave propa-gating outside of the wire, with amplitude given by the Han-kel function. 27At the wire surface, the dynamic axial mag- netic ﬁeld is assumed to be sinusoidal. Its amplitude is givenby 13 Hza=/H92550 /H92620E/H9272aEza H/H9272a, /H208493/H20850 where E/H9272a,Eza, and H/H9272aare the values, at the surface, of the amplitudes of the circumferential and axial electric ﬁelds,and circumferential magnetic ﬁeld, respectively. From Eq./H208493/H20850, given that the typical order of magnitude of E za/H/H9272ais 1/H9024and that E/H9272amay reach a maximum value of the order of 100 V /m, a maximum of Hzawill be of the order of magni- tude 10−3A/m. This justiﬁes the approximation, Hz/H110150o f Eq. /H208492d/H20850even when the nonlinear regime is reached /H20849the same analysis can be carried out for each higher harmonic /H20850. The solution of Eq. /H208491/H20850requires knowledge of the varia- tion in the magnetization with time, which is given by theLandau–Lifshitz equation /H11509M /H11509t=/H92620/H9253/H20849M/H11003Heff/H20850−/H92620/H9253/H9251 Ms/H20851M/H11003/H20849M/H11003Heff/H20850/H20852, /H208494/H20850 where /H9253is the gyromagnetic ratio, /H9251is the Gilbert damping coefﬁcient, Msis the saturation magnetization, and M =M/H20849r,t/H20850is the magnetization vector whose modulus is al- ways Ms. The effective magnetic ﬁeld Heffis given by Heff/H20849r,t/H20850=H/H20849r,t/H20850+Happ+Hani/H20849r,t/H20850+Hex/H20849r,t/H20850. /H208495/H20850 In Eq. /H208495/H20850,H/H20849r,t/H20850is the dynamic ﬁeld, Happis the applied static ﬁeld, Hani=2Ku//H92620Ms2/H20849M·nˆk/H20850nˆkis the anisotropy ﬁeld /H20849where Kuis the anisotropy constant and nˆkis a unit vector parallel to the easy axis /H20850andHex=2A//H92620Ms2/H20849/H116122M/H20850is the exchange ﬁeld /H20849where Ais the exchange constant /H20850. B. Numerical procedure Except for Happ, which is constant and homogeneous, all ﬁelds in Eq. /H208495/H20850are dependent on time and radial position. The radial variation in both the dynamic ﬁeld and the mag-netization, between successive collocation points nand n +1, is interpolated by polynomial functions of third orderwhich pass through points n−1 to n+2. Thus, the radial de- rivatives used in the expression of H exare easily determined. Figure 1is a diagram summarizing the method of calcu- lation. At t=0, the magnetization of the wire is set in a static conﬁguration, not necessarily the equilibrium state. All initialconﬁgurations with cylindrical symmetry are permitted.However, it is preferable to avoid abrupt radial variation inorder to reduce computation time. At t/H110220, a sinusoidal cur- rent begins to ﬂow in the wire, leading to a change in themagnetization. The response time of the magnetization isvery long in comparison with the time step of the calculation,/H9004t/H20849of the order of 1 ps /H20850. This allows us to assume that083916-2 Seddaoui et al. J. Appl. Phys. 105, 083916 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 19:07:19/H11509M//H11509t/H20849r,t/H20850remains approximately constant from instant t until t+/H9004t. Hence, /H11509M//H11509t/H20849r,t/H20850calculated from Eq. /H208494/H20850for instant tmay be used in Eq. /H208491/H20850for instant t+/H9004t. Equation /H208491/H20850gives the dynamic magnetic ﬁeld H/H20849r,t/H20850for instant t +/H9004twhich is then used in Eq. /H208494/H20850by means of the effective ﬁeld /H20851Eq. /H208495/H20850/H20852, for the same instant, and so on. The magne- tization M/H20849r,t+/H9004t/H20850at the instant t+/H9004tis calculated from M/H20849r,t/H20850using the Runge–Kutta method. After several periods of the current, the system reaches a stationary state. Convergence is achieved when the magneti-zation shows periodic behavior M/H20849r,t+T/H20850=M/H20849r,t/H20850, where T is the current period. After convergence, when the stationary state is reached, the voltage across the wire is obtained fromthe axial electric ﬁeld at the wire surface V/H20849t/H20850=lE z/H20849r=a,t/H20850. /H208496/H20850 The harmonics Vnfare given by the Fourier transform of V/H20849t/H20850 Vnf=2 T/H20879/H20885 0T V/H20849t/H20850ein/H9275tdt/H20879. /H208497/H20850 Within the wire, Ez/H20849r,t/H20850may be expressed as /H20849see the Appen- dix/H20850Ez/H20849r,t/H20850=RdcI/H20849t/H20850 l+/H20885 0r/H11509B/H9272/H20849r/H11032,t/H20850 /H11509tdr/H11032 −2 a2/H20885 0a r/H20885 0r/H11509B/H9272/H20849r/H11032,t/H20850 /H11509tdr/H11032dr, /H208498/H20850 where Rdcis the dc resistance, lis the wire length, and B/H9272 =/H92620/H20849H/H9272+M/H9272/H20850is the circumferential induction. The second term on the right hand side /H20849RHS /H20850of Eq. /H208498/H20850 is the inductive part of the impedance. The third term is atime dependent correction, which results from the adjustmentof the current I/H20849t/H20850to its imposed value at each time t.I nt h e linear regime, as the frequency increases, this term tends to cancel the dc resistance contribution term R dcI/H20849t/H20850/l, so that Ez/H20849r,t/H20850and the current density J/H20849r,t/H20850=/H9268Ez/H20849r,t/H20850tend to zero at the center of the wire. The radial proﬁle of the current density is then governed by the induction term, which isnegligible in the interior of the wire due to the smallness ofthe current /H20849weak H /H9272/H20850in this region. Thus, the two last terms of Eq. /H208498/H20850are interdependent. We can then say that the last term of Eq. /H208498/H20850is at the origin of the skin effect in the linear regime. In the nonlinear regime, because the time and radial variations in B/H9272are not simple, the skin effect is not well established. That is, the radial proﬁle of the current cannot bedeﬁned by a known function, as it is in the linear regime,where the radial dependence of the current density amplitudefollows a Bessel function with the skin depth in itsargument. 3,4,7However, even in the nonlinear regime, the root mean square /H20849rms /H20850value of the current density is lower in the interior of the wire than at the surface, especially athigh frequency. Here, we refer to this as the skin effect, andrefer to the last term in Eq. /H208498/H20850as the skin effect term. III. HOMOGENEOUS MAGNETIC PROPERTIES AND FREE SURFACE SPINS The model described so far is quite general. The only signiﬁcant assumption has been that the properties of thewire are independent of zand /H9272. In order to perform realistic calculations on ferromagnetic conductors, it is necessary tospecify the properties, especially the anisotropy distributionof the sample. It is well known that the magnetic propertiesof the sample, especially anisotropy, have a large inﬂuenceon GMI results, as demonstrated by the difference betweenthe GMI curves of wires with nearly axial and nearly trans-verse anisotropy. 13,28These properties are determined by the composition, fabrication, and subsequent treatment of thewire in question. For example, for amorphous wires, whichare typically studied in GMI experiments and used in GMIdevices, the anisotropy is strongly dependent on the magne-tostriction and stress distribution. 29However, as long as the shape and the magnetic properties of the sample can be as-sumed to exhibit cylindrical symmetry, all types of wires,amorphous or crystalline, can be treated using the presentmodel. FIG. 1. /H20849Color online /H20850Diagram of the method of calculation. In the ﬁrst iteration, the initial value of /H11509M//H11509t/H20849ri,t=0/H20850/H20849which is taken to be null for all collocation points /H20850is used for solving Maxwell’s equations at t=/H9004t/H20849/H9004tis the time increment /H20850. The value of the current at t=/H9004tis taken into account in the boundary conditions. /H11509M//H11509t/H20849ri,t=0/H20850andM/H20849ri,t=0/H20850are assumed to be constant from t=0 to t=/H9004t. The solution of Maxwell’s equations gives the dynamic ﬁeld, which is added to the effective ﬁeld. Knowing the effectiveﬁeld H eff/H20849ri,t=/H9004t/H20850andM/H20849ri,t=/H9004t/H20850, the Landau–Lifshitz equation gives a new value of /H11509M//H11509t/H20849ri,t=/H9004t/H20850/H20849which is assumed to be constant until t =2/H9004t/H20850and then the conﬁguration of the magnetization M/H20849ri,t=2/H9004t/H20850att =2/H9004tis obtained with the Runge–Kutta method. At this step, one iteration is completed. Hence, the system evolves, iteration after iteration until the in-stant t=nT/H20849nis an integer and Tis the current period /H20850when the condition of convergence M/H20849r i,t=nT/H20850=M/H20849ri,t=/H20849n−1/H20850T/H20850is observed. The voltage may be calculated in the last period.083916-3 Seddaoui et al. J. Appl. Phys. 105, 083916 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 19:07:19A. Simpliﬁed magnetic structure In this section, and the next, we treat the simplest pos- sible structure which can give an idea of the general lines ofnonlinear behavior: a sample with homogeneous anisotropyand spins free at the surface. In Sec. V , we present a morecomplex structure in order to reproduce experimental resultson a particular material. We assume that the easy axis is directed in a helical direction, making an angle /H9274with respect to the circumfer- ential direction and that the anisotropy ﬁeld amplitude isH k=2Ku//H92620Ms=40 A /m. The value of /H9274is assumed to be small /H20849near-circumferential anisotropy /H20850, as we believe that this case is more interesting than that of /H9274/H11011/H9266/2/H20849near-axial anisotropy /H20850. The method described in Sec. II generates a core shell equilibrium structure. Near the surface, the magnetiza-tion is along the easy axis. On the axis of the wire, cylindri-cal symmetry requires that the magnetization be in the axialdirection. This generates a vortex, as discussed in Sec. IV . The free surface spin condition means that these spins are not submitted to any additional ﬁelds related to the pres-ence of the surface, such as a surface anisotropy ﬁeld. 30The only difference between the effective ﬁeld at the surface andin the interior of the wire /H20851which is described by Eq. /H208495/H20850/H20852lies in the exchange ﬁeld which takes into account the fact thatthe surface spins have neighbors only on one side. At thewire surface, the free spin condition is given in cylindricalcoordinates by 4 /H20879/H20873/H11509M/H9272,r /H11509r+M/H9272,r r/H20874/H20879 r=a=0 . /H208499/H20850 B. Validation of the numerical procedure The results of our calculations have been compared, at different frequencies, with the analytical results of Melo et al.14for homogeneous properties, free surface spins, and /H9274 =10° in the linear regime. The results of the two methods forthe same parameters are in excellent agreement, as shown inFig. 2where we present Z/R dcversus Happ /Hk,a t1 0M H z and 1 GHz frequencies. The negligible discrepancies be-tween the two are due to the limitation of the accuracy of ourcalculation in order to reduce the computation time. Thiscomparison validates the methodology. In all calculations reported here, unless indicated other- wise, the following parameters are used: saturation magneti-zation, M s=660 kA /m; gyromagnetic ratio, /H9253=176 rad /sT /H20849/H9253/2/H9266=28 GHz /T/H20850; damping coefﬁcient, /H9251=10−2; wire ra- dius, a=15/H9262m; wire length, l=3 cm; wire conductivity, /H9268 =8/H11003105/H20849/H9024m/H20850−1; and exchange constant, A=10−11J/m. These parameters are typical for Co-rich amorphous wires.4 All GMI curves were obtained by sweeping Happfrom nega- tive to positive values. IV. NLGMI RESPONSE OF HOMOGENEOUS WIRE WITH FREE SURFACE SPINS In Fig. 3, the normalized circumferential component of the magnetization M/H9272/Msis plotted as a function of normal- ized time t/T/H20849Tis the current period /H20850and radial position r/a/H20849r/a=0 corresponds to the wire axis and r/a=1 corresponds to the wire surface /H20850for no applied ﬁeld /H20849Happ=0/H20850and for 1 MHz driving currents at three different amplitudes. Here, the wire diameter is taken to be 35 /H9262m. In Fig. 3/H20849a/H20850, the static magnetization shows a vortex at the wire center since itpasses from the axial direction at the wire axis /H20849r/a=0/H20850to a helical direction at r/a/H110220. The vortex results from compe- tition between the exchange ﬁeld, which tends to keep themagnetization in the axial direction /H20849parallel to the magneti- zation of the wire axis /H20850, and the anisotropy ﬁeld, which tends to rotate the magnetization in the helical direction. Differentvalues of /H9274and of Awill change the details of the vortex, but it is always present for /H9274/HS11005/H9266/2. A. Effect of driving current on magnetization dynamics Figure 3/H20849a/H20850, corresponding to 5 mA rmscurrent amplitude and 1 MHz frequency, shows that magnetization reversal oc-curs in the surface region /H20849about 30% of wire radius depth /H20850, whereas the inner region remains in the initial direction. Atthe surface, the circumferential ﬁeld is high enough to re-verse the magnetization from one circumferential direction tothe other. The occurrence of attenuated propagation from thesurface toward the wire axis is indicated by the small tail inFig. 3/H20849a/H20850. When the current is increased /H20851Fig. 3/H20849b/H20850/H20852, the re- gion of reversed magnetization increases. If the tail persistsuntil the next current period, it enlarges due to the favorablecircumferential ﬁeld during the next period. It will continueto propagate toward the wire axis, forming a magnetizationstructure which resembles a solitary-wave, as shown in Fig.3/H20849b/H20850. In this case, the magnetization alternates symmetrically between the two circumferential directions in the entire wire.-8 -4 0 4 82468Z/RDC Happ/Hknumerical calculation linear analytical calculationf=1 0M H z 0 5 10 15 200102030 numerical calculation linear analytical calculationZ/RDC Happ/Hkf=1G H z(a) (b) FIG. 2. Normalized static ﬁeld dependence of the normalized wire imped- ance in the linear regime for /H20849a/H2085010 MHz and /H20849b/H208501 GHz frequency. Com- parison between our results and analytical results /H20849Ref. 14/H20850.083916-4 Seddaoui et al. J. Appl. Phys. 105, 083916 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 19:07:19At even higher current amplitude, the magnetization in the entire wire reverses quasicoherently, as shown in Fig. 3/H20849c/H20850. The solitary-wave velocity, clearly higher in Fig. 3/H20849c/H20850than in Fig. 3/H20849b/H20850, depends upon whether the circumferential mag- netic ﬁeld is favorable to the propagating tail or not. For atypical amorphous GMI wire, the amplitude of the current in Fig.3/H20849c/H20850might be higher than that required to yield irrevers-ible structural changes in the sample due to heating, which is not accounted for here. The aim of this ﬁgure is to show thetrend in the magnetization behavior as the current is in-creased. B. Effect of applied ﬁeld on NLGMI response Figure 4/H20849a/H20850shows the total /H20849rms /H20850voltage across the wire as a function of the normalized applied ﬁeld at 1 MHz fordifferent current amplitudes. In the linear regime /H20849low cur- rent /H20850, the peaks increase in height without changing ﬁeld position. When the nonlinear regime is reached, the valleygradually disappears, as the peaks begin to shift towardlower ﬁeld until they merge. This shows that the optimaldriving current for maximum ﬁeld sensitivity is roughly atthe onset of nonlinear behavior. The ﬁeld dependence of the second harmonic signal, V 2f, is shown in Fig. 4/H20849b/H20850for/H9274=1°. We may observe that all of theV2fsignals are conﬁned to the low-ﬁeld region /H20849−Hkto Hk/H20850in which the wire is not saturated. At intermediate cur- rent amplitude /H20849limit between the linear and nonlinear re- gimes /H20850, the ﬁeld dependence of the second harmonic signal V2fshows a two-peak structure. The peak positions corre- spond to abrupt jumps of the total signal. As the currentamplitude increases, the two peaks show the same behavioras those of the total voltage. They increase in height and shiftto lower ﬁeld until they merge. A maximum V 2fsignal is then obtained for current amplitude at which the two characteris-tic GMI peaks merge. At higher current amplitude, V 2fde- creases and shows a four-peak structure, as observedexperimentally. 19,20It is important to note that in this calcu- lation, the anisotropy angle /H9274is neither zero nor /H9266/2/H20849helical0.2 0.4 0.6 0.8 1.00.20.40.60.81.0 r/at/T-1.01.0 5m A rms (a) M/Mφs 0.20 0.40 0.60 0.80 1.000.20.40.60.81.0 r/at/T8mA (b) rms Mφ/Ms 0.2 0.4 0.6 0.8 1.00.20.40.60.81.0 r/at/T(c) 30 mA rms M/Mφs FIG. 3. /H20849Color /H20850Circumferential component of normalized magnetization, M/H9272/Msas a function of normalized radial position and normalized time at 1 MHz for /H20849a/H208505m Arms,/H20849b/H208508m Arms,a n d /H20849c/H2085030 mArms. There is no applied ﬁeld, the anisotropy is characterized by Hk=0.5 Oe /H208491O e = 1 03/4/H9266A/m/H20850, /H9023=1° and the initial magnetization /H20849att=0/H20850is set in the easy axis with negative circumferential component. The wire radius is a/H1100517.5 µm.-4 -2 0 2 40.00.20.40.6Voltage (Vrms) Happ/Hk0.1mA0.5 mA1m A2m A3m Af= 1 MHz -1 0 10.00.10.20.30.5 mA 2m A 3m A 5m ASecon dharmon ic (V) Happ/Hkf= 1MHz -0.3 0.0 0.30.000.010.020.03(b)(a) FIG. 4. Calculated voltage responses as a function of normalized applied ﬁeld for various current amplitudes at 1 MHz frequency: /H20849a/H20850total signal and /H20849b/H20850second harmonic. The inset shows the second harmonic for 5 mArmsat low ﬁeld. The applied ﬁeld is swept from negative to positive. Hk=0.5 Oe /H208491O e = 1 03/4/H9266A/m/H20850,/H9023=1° and a/H1100515µm.083916-5 Seddaoui et al. J. Appl. Phys. 105, 083916 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 19:07:19anisotropy /H20850, otherwise the second harmonic disappears. However, even a very small angle /H9274is sufﬁcient to obtain a V2fsignal of several hundreds of mV , as in Fig. 4/H20849b/H20850where /H9274=1°. C. Skin effect in the nonlinear regime In order to gain more insight into the electromagnetic behavior in the nonlinear regime, we now consider the spaceand time evolution of the ﬁelds. The normalized radial andtime dependences of E z/H20849r,t/H20850andM/H9272/H20849r,t/H20850are shown in Figs. 5and6, respectively, for I=1 mA rms,f=1 MHz, and for sev- eral applied ﬁelds. For this set of current frequency and am-plitude, corresponding to the middle curve in Fig. 4/H20849a/H20850, the voltage is already nonlinear and the peak of the GMI curve issituated at H pk=0.6 Hk/H20849Hk=0.5 Oe=40 A /m/H20850instead of Hpk=Hk, as in the linear regime.10,11,31The Happ=0 behavior is not shown in Fig. 6due to the negligible variation in thecircumferential magnetic ﬂux. For this ﬁeld and driving cur- rent, the circumferential ﬁeld is too weak to cause magneti-zation reversal as in Fig. 3/H20849a/H20850. In these circumstances, the skin effect is weak /H20849 /H9254/H11271a/H20850, as shown in Fig. 5/H20849a/H20850, and Zis linear and nearly equal to Rdc. When an axial static ﬁeld is applied, for a given driving current, greater variation in the magnetization is induced,leading to increases both in the skin effect and in the induc-tion effect. A comparison between responses for H app=Hk /H20851Fig. 5/H20849c/H20850/H20852andHapp=Hpk/H20851Fig. 5/H20849b/H20850/H20852shows that the skin ef- fect is stronger at Hkthan at Hpk/H20849the value of Ezatr=0 is smaller /H20850. In contrast, in the linear regime, the attenuation is maximal /H20849skin depth minimal /H20850at the peak position, Hpk. This suggests that the GMI peak in the nonlinear regime is essen-tially due to the induction term rather than to the skin effectterm. This is conﬁrmed in Fig. 6where we may see that the0.20.40.60.8 1.0-30-20-100102030 0.20.40.60.81.0 Ez(V/m) t/T r/aHapp=0 0.20.40.60.81.0-30-20-100102030 0.20.40.60.81.0Happ=Hpk=0.6 Hk Ez(V/m) t/T r/a 0.20.40.60.81.0-30-20-100102030 0.20.40.60.81.0 Ez(V/m) t/T r/aHapp=Hk(a) (b) (c) FIG. 5. Normalized radial and time dependence of the axial electric ﬁeld for current of 1 mArmsamplitude and 1 MHz frequency. The static applied ﬁeld is/H20849a/H20850Hz=0, /H20849b/H20850Hz=0.6 Hk, and /H20849c/H20850Hz=Hk.0.2 0.4 0.6 0.8 1.0-1.0-0.50.00.51.0 0.20.40.60.81.0 Mϕ/M s t/T r/aHapp=Hpk=0.6Hk 0.2 0.4 0.6 0.8 1.0-1.0-0.50.00.51.0 0.20.40.60.81.0 Mϕ/M s t/T r/aHapp=Hk 0.20.40.60.81.0-0.04-0.020.000.020.04 0.20.40.60.81.0Mϕ/M s t/T r/aHapp=10 Hk(a) (b) (c) FIG. 6. Normalized radial and time dependence of the circumferential com- ponent of the normalized magnetization for current of 1 mArmsamplitude and 1 MHz frequency at applied ﬁeld /H20849a/H20850Hz=Hpk=0.6 Hk,/H20849b/H20850Hz=Hk,a n d /H20849c/H20850Hz=10Hk.083916-6 Seddaoui et al. J. Appl. Phys. 105, 083916 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 19:07:19rate of variation with time of the circumferential magnetiza- tion is higher at Hpk/H20851Fig. 6/H20849a/H20850/H20852than at Hk/H20851Fig. 6/H20849b/H20850/H20852, espe- cially in the surface region. For currents for which 0 /H11021Hpk/H11021Hk, the skin effect is weak for Happ/H11021Hk. This is due to the fact that the perme- ability is small in the interior of the wire, which behavesalmost like a nonmagnetic medium. When H appis increased /H20851Fig. 6/H20849b/H20850/H20852, the magnetization variation becomes smoother and extends toward the wire axis, that is, the abrupt time andradial variations in the magnetization gradually disappear,leading to higher skin effect /H20849lower penetration depth /H20850.A t higher H app, the magnetization variations diminish and tend to be linear with the circumferential ﬁeld as does the totalresponse. Figure 6/H20849c/H20850, obtained for H app=10Hk, shows that the wire magnetization behavior is similar to that of mag-netic wire in the linear regime with a relatively small perme-ability and a quasistaticlike linear variation in the circumfer-ential ﬁeld as a function of the radius. D. Critical switching ﬁeld The abrupt jumps at Hpk, shown in Fig. 4/H20849a/H20850, are due to abrupt increases in the circumferential permeability, as mag-netization reversal becomes possible. This is followed by agradual decrease in permeability at higher ﬁeld. The positionof these transitions /H11011H pkmay be estimated very approxi- mately using the astroid of Stoner–Wohlfarth critical switch-ing ﬁelds 32shown in Fig. 7for the case of circumferential anisotropy.20The magnetization is periodically reversed when the total ﬁeld Htot/H20849composed of the static applied ﬁeld Happand the dynamic circumferential ﬁeld H/H9272/H20850crosses the astroid in both directions. If not, the sign of the magnetiza-tion remains constant. In the linear regime, this never takesplace, and the peak in permeability is at H k, where the effec- tive internal ﬁeld reaches a minimum. In the nonlinear re-gime but with H /H9272small, this happens only when Happis close toHk. Thus, Hpk/H11015Hk.A s H/H9272increases, Hpkdecreases. It should be noted that this representation works best in thequasistatic regime 20where the magnetization may be as- sumed to always be in its equilibrium state. E. Effect of frequency Increasing frequency has almost the same effect on the magnetization behavior as decreasing current amplitude.That is, at constant large current /H20849for which Hpk=0/H20850, with increasing frequency, the depth in which quasicoherent re-versal of the magnetization occurs at H app=0 decreases and then disappears. The peak position moves to higher ﬁeld un-til it reaches H kand then stabilizes. The system has returned to the linear regime, in which the peak remains at Hk, until resonance occurs at high ﬁeld.4,33Higher current amplitude is needed to reach the nonlinear regime. Despite the fact thatthe spatial region of quasicoherent reversal of magnetizationdecreases with frequency, the voltage signal increases due tomore rapidly increasing time variation in the circumferentialinduction. AtH z=100 Hk, even at 10 mA rms /H20849H/H9272/H11015100 A /m/H20850, the voltage is linear with current at 10 MHz and higher. The characteristic of FMR is shown in Fig. 8/H20849a/H20850where the real and imaginary parts of the voltage are plotted as a functionof the frequency. The radial dependence of the amplitude ofE zis shown for various frequencies in Fig. 8/H20849b/H20850. We note that, in these circumstances, the skin effect is large /H20849penetra- tion depth is small /H20850compared to that of the nonmagnetic metal only near resonance /H208491.9 GHz /H20850. At higher frequency /H20849beyond resonance frequency /H20850, the penetration depth in- creases, tending to that of the nonmagnetic metal of the sameresistivity. Preliminary calculations indicate that at extremely high current, and for small H app, bifurcation and chaotic behavior should occur, provided there are no signiﬁcant changes in thematerial properties due to the driving current. A possible wayto investigate this behavior is by using high power FMRmeasurements in cavities, in a pulsed regime, for which it is easier to drive the wire with high amplitude hf signals thanH//H⊥⊥⊥⊥ HkHk -HkφZ Htot1 Htot2Hz1 Hz2 FIG. 7. Astroid of magnetization reversal H/H110362/3+H/H206482/3=Hk2/3in the case of circumferential anisotropy. The astroid axes coincide with the wire axes.The total ﬁeld H totseen by the astroid is composed of the static axial ﬁeld Hz and dynamic circumferential ﬁeld H/H9272. When the amplitude of H/H9272increases, the static ﬁeld Hzneeded to fully cross the astroid decreases.0.1 1 10-10-505101520Voltage (V) Frequency (GHz)Re Im 0 . 00 . 20 . 40 . 60 . 81 . 0050100200400600Ez(V/m) r/a10 MHz 100 MHz 1G H z 1.9 GHz 10GHz(a) (b) FIG. 8. For Happ=100 Hkand 10 mArmscurrent, /H20849a/H20850Real /H20849Re/H20850and imaginary /H20849Im/H20850parts of the voltage signal as a function of the frequency and /H20849b/H20850the radial dependence of the axial electric ﬁeld amplitude at various frequencies.083916-7 Seddaoui et al. J. Appl. Phys. 105, 083916 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 19:07:19with usual GMI measurement setups. This needs to be ex- plored in detail, and will be discussed elsewhere.34 V. COMPARISON WITH EXPERIMENT In this section, we compare the results of the model with experimental results obtained on typical microwires used instudies of GMI. We shall show that the spin free magneticstructure, assumed in the previous section, needs to be modi-ﬁed in order to reproduce the experimental results ad-equately. Nevertheless, the general trends discussed in Sec.IV remain valid. A. Experimental procedure The sample of interest was an amorphous melt-extracted microwire of composition Co 80.89Fe4.38Si8.69B1.52Nb4.52 /H20849wt % /H20850manufactured by MXT Inc. of Montreal. The micro- wire of 35 /H110065/H9262m diameter and 25 mm length had satura- tion magnetization of 660 kA /m and very weak helical an- isotropy, with an anisotropy ﬁeld, known from the linearregime, of about 0.5 Oe /H208491O e = 1 0 3/4/H9266A/m/H20850. The sample was placed on the axis of a long coil, which provides a dc axial magnetic ﬁeld of maximum value of5.8 Oe. The sample was connected, in series with a largeresistor, to a function generator. The purpose of the resistor isto keep the distortion of the current generated by the samplenegligible in comparison with the total load /H20849resistor +microwire /H20850of the generator. Thus, the sinusoidal current I ﬂowing in the sample remains undistorted. The amplitude of the current Iwas measured by an oscilloscope using a cur- rent probe and was kept constant by adjusting the amplitudeof the generator signal. The voltage across the microwire wasFourier analyzed by the same oscilloscope. B. Experimental results The ﬁeld dependence of the amplitude of the total signal is shown in Fig. 9for a range of values of current amplitude at 1 MHz frequency. These curves were obtained by varyingthe dc magnetic ﬁeld, step by step, and measuring, at eachstep, the total voltage across the wire. At low current, thecurves are similar to those observed for linear GMI despitethe fact that the voltage was already nonlinear at 3 mA rms.A s the current increased, so did the amplitude, while the char-acteristic peaks of GMI shifted to lower ﬁeld, until they merged at the ﬁeld origin, in qualitative agreement with themodel. Figure 10shows a comparison between calculated and measured total voltage amplitude across the microwire as afunction of applied ﬁeld for 5 mA rmscurrent amplitude and 1 MHz frequency. As we may see, despite the fact that thegeneral behavior of the calculated GMI signal agrees withexperiment when the current amplitude is varied, the twocurves in Fig. 10are very different, both in shape and in height. The calculation overestimates the effect of currentamplitude, that is, the numerical curve is similar to that ob-tained experimentally for higher current /H20849see Fig. 9/H20850. This indicates that the effective permeability is overestimated, es-pecially in the surface region where most of the contributionto the GMI effect is produced. C. Hard surface condition Introducing a surface spin pinned condition, /H11509M/H20849a,t/H20850//H11509t=0, as was done elsewhere14for the linear re- gime, reduces the peak somewhat, but not enough. To further reduce the calculated total signal, we have included a hardsurface of thickness bin the structure of the wire. The easy axis direction at the surface is assumed to remain the same asin the interior of the wire and the anisotropy ﬁeld is assumedhigh enough to keep surface spins insensitive to currentvariation in the range of study. This appears justiﬁed by ex-periment. In Fig. 11, axial hysteresis loops measured for two different ﬁeld ranges /H20849the ﬁrst from −500 to 500 Oe and the second from −20 to 20 Oe /H20850are shown. These loops are ob- tained on microwire of 2.7 mm length using a vibratingsample magnetometer /H20849VSM /H20850. In this ﬁgure, the second loop shows a very small coercivity and corresponds to one branchof the ﬁrst loop. This suggests that about 95% of the wirevolume has a very weak anisotropy /H20849we assume this to be the interior of the wire /H20850and the rest /H20849which we assume to be the surface /H20850is hard enough to be almost insensitive in the ﬁeld range from −20 to 20 Oe. Figure 12shows a comparison at 5 and 10 mA rmscurrent amplitude and 1 MHz frequency between the measured ﬁelddependence of the total signal and that calculated including ahard layer. The hard surface magnetic structure results in adecrease in GMI signal and a separation between the two-6 -4 -2 0 2 4 60.00.20.40.60.81.0 II= 14mARMS I=3 m ARMSVoltage (V) Applied field (Oe) (1 Oe ~ 80 A/m) FIG. 9. Field dependence of the voltage response of the wire at f=1 MHz andIvarying in 1 mArmssteps between 3 and 14 mArms.-4 -2 0 2 40.20.40.60.8calculation with free spin condition experimentVoltage (V) Applied field (Oe) (1 Oe ~ 80 A/m) FIG. 10. Comparison between total signal amplitude measured and calcu- lated with the free spins condition, as a function of applied ﬁeld, for5m A rmsand 1 MHz.083916-8 Seddaoui et al. J. Appl. Phys. 105, 083916 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 19:07:19peaks, without abrupt variation between them. A good ﬁt is obtained when the shell thickness is in the vicinity of750 nm, the wire radius is a=15 /H9262m, and the anisotropy ﬁeld in the surface is 50 Oe, 100 times higher than that in theinterior of the wire. At low ﬁeld, the exchange interactionwith the immobile spins of the surface region reduces themotion of the inner spins. This results in a minimum in theGMI curve at H z=0. Higher current is required to reverse magnetization and then merge the peaks. D. Second harmonic signals Figure 13shows a comparison between measured and calculated V2fas a function of Happin same conditions as in Fig. 12. In contrast to the total signal, the predicted second harmonic disagrees with experiment. The ﬁeld dependenceof the calculated V 2fshows one peak of about 250 mV in height at the ﬁeld origin. The measured V2fshows a four- peak structure of a maximum value of 20 mV. The reason forthis two order of magnitude difference is that the inﬂuence ofthe hard surface magnetization creates an asymmetry in thecircumferential hysteresis loop, which leads to an increase inthe second harmonic signal. It is evident that the presence ofa single-domain hard surface is not sufﬁcient to entirely ex-plain the measurements.While the evidence for a magnetically hard component is strong, and placing it at the surface is plausible, its single-domain structure is far less obvious. We interpret the verylow value of the second harmonic as an indication of thedivision of the hard surface into two types of domain /H20849D1 and D2 /H20850. In this way, under the inﬂuence of the surface, the behavior of the magnetization of the core is different, de-pending on whether it is situated under D1 or D2. Due to theopposite directions of the magnetizations of the two do-mains, the asymmetry in circumferential hysteresis loops,upon which the second harmonic depends, 19,25is opposite for the core under D1 from that under D2. Figure 14shows the normalized ﬁeld dependence of the real and imaginary parts of the fundamental signal generatedby each domain. All parameters of the calculation are thesame as in Figs. 12and13and any interaction between the-100 -50 0 50 100-1.0-0.50.00.51.0 -500 to 500 Oe cycle - 2 0t o2 0O ec y c l eAxial reduced magnetization Applied field (Oe) (1 Oe ~ 80 A/m) FIG. 11. Normalized axial hysteresis loops for two different ranges of ap- plied ﬁeld. Line: from −500 to 500 Oe. Line+point: from −20 to 20 Oe/H208491O e = 1 0 3/4/H9266A/m/H20850. -6 -4 -2 0 2 4 60.20.40.60.8Voltage (V rms) Applied field (Oe) (1 Oe ~ 80 A/m)experiment theory10 mArms 5m Arms FIG. 12. Comparison between the calculated and the measured total signal amplitude as a function of applied ﬁeld for 5 mArmsand 10 mArmscurrent amplitude and 1 MHz frequency. Hard surface condition is used in calcula-tion. /H9274=1°, b/a=6%, Hksurface =100 Oe, Hkcore=0.5 Oe /H208491O e =103/4/H9266A/m/H20850.-2 0 20.000.050.100.150.200.25 experiment calculation with hard surface conditionSecond harmonic (V) Applied field (Oe) (1 Oe ~ 80 A/m) FIG. 13. Comparison between the calculated and the measured second har- monic signal as a function of applied ﬁeld for current amplitude of 5 mArms and frequency of 1 MHz. The parameters of the calculation are the same asin Fig. 12. -12 -8 -4 0 4 8 12-0.2-0.10.00.10.2Re {V2f} Happ/HkD2 D1Mean value -12 -8 -4 0 4 8 1 2-0.10.00.1Im {V2f} Happ/HkD2D1 Mean value(a) (b) FIG. 14. Normalized applied ﬁeld dependence of real and imaginary parts of V2fin D1 and D2 domains, and total wire.083916-9 Seddaoui et al. J. Appl. Phys. 105, 083916 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 19:07:19magnetizations of the two domains is neglected. It is also assumed that the domain walls are pinned in the workingfrequency range. In this ﬁgure, it may be seen that both thereal and the imaginary parts of the second harmonic signalsof the two domains are of opposite sign whereas the funda-mental /H20849not shown here /H20850does not show any signiﬁcant dif- ference since it is not sensitive to the hysteresis loop asym-metry. Because the asymmetries in the two domains are in op- posite directions, the signs of the real and imaginary parts ofV 2fare different in the two domains. Thus, both the real and imaginary parts across the wire, which are given by a linearcombination of those obtained for each domain, decreasedrastically. The resulting total second harmonic modulus isthen reduced, while the fundamental remains almost un-changed. Due to the fact that the fundamental is the predomi-nant signal, the change in the total voltage is very small/H20849/H110215% /H20850. Here, the anisotropy angle is taken to be very small /H208491°/H20850but not zero because if it were, the even harmonic sig- nals of the two domains would be exactly opposite and thenall total even harmonics disappear when the lengths of thetwo domains are equal. When the relative lengths of the twodomains are changed, the V 2f/H20849Happ/H20850curve changes sensi- tively and shows complicated structure. Figure 15shows a comparison of the resulting V2fwith experiment. The calculation yields not only the right order ofmagnitude but also the four-peak structure with the rightasymmetry. Despite the discrepancies between the twocurves, we consider the result of the calculation satisfactory,considering the complexity of the second harmonic signal, itshigh sensitivity to impurities and shape defects of the wire,and to the fact that the interaction between domains is ne-glected. However, the signiﬁcance of the assumed structureis not easy to interpret. Since we use a linear combination ofresponses of two independent regions, there are no speciﬁcassumptions on the structure of D1 and D2. For instance,they can be well separated into distinct regions or inter-mixed. Because the proposed magnetic structure is not unique, it is evident that reproducing one of the experimental curvescannot demonstrate the validity of our model. However, thefact that the same magnetic structure ﬁts both the secondharmonic and the total signal for different current amplitudesand frequencies increases our conﬁdence concerning thischoice.The presence of a magnetically hard single-domain sur- face reduces the GMI signal and makes the calculated data ﬁtthe measured one. The presence of the domains at the surfacedrastically reduces the even harmonics without affecting theodd harmonics. Modeling the total signal is useful for obtain-ing information about sample parameters but the second har-monic, due to its complexity, gives more details about do-main structure or the variation in the surface anisotropy ofthe sample. E. Inﬂuence of the axial dynamic ﬁeld The axial current produces a circumferential ﬁeld. As- suming locally coherent rotation of the magnetization, as im-plicit from Eq. /H208494/H20850which preserves the modulus of the mag- netization, applied ﬁelds below the anisotropy ﬁeld producea signiﬁcant rotation of the magnetization. That is, we mayhave a signiﬁcant term in /H11509Mz//H11509t/H20849r,t/H20850. We may see from Eq. /H208491a/H20850that/H11509Mz//H11509t/H20849r,t/H20850is coupled to the axial dynamic mag- netic ﬁeld Hz/H20849r,t/H20850, which we have shown is negligible at r =a, but may not be so elsewhere. In other words, through Maxwell’s equations, this ﬁeld is coupled to the circumfer-ential electric ﬁeld, which causes the current to follow ahelical trajectory rather than a straight one. The circumferen-tial component of current is, in its turn, coupled to the axialmagnetic ﬁeld. All of these quantities must be obtained in aself-consistent manner. This is assured by the method de-scribed in Fig. 1. In Fig. 16, the axial dynamical ﬁelds H zin the two types-3 0 302040Secon dharmon ic( m V ) Applied field (Oe) (1 Oe ~ 80 A/m)experiment theoryf=1M H z I=5m Arms ψ=1o Hk=0 . 5 O e a=1 5 µm b/a=6 % FIG. 15. Applied ﬁeld dependence of second harmonic, calculated with two-domain hard surface condition, compared with experiment.0.2 0.4 0.6 0.8 1.0-0.4-0.20.00.20.4 0.20.40.60.81.0 Hz(Oe) t/Tr/a(a) 0.2 0.4 0.6 0.8 10-0.4-0.20.00.20.4 0.20.40.60.81.0 Hz(Oe) t/Tr/a(b) FIG. 16. Axial dynamical ﬁeld as a function of time and radial position inthe two-domain hard surface magnetic structure. /H20849a/H20850domain D1; /H20849b/H20850domain D2. The applied ﬁeld is 0.5 Oe /H208491O e = 1 0 3/4/H9266A/m/H20850.083916-10 Seddaoui et al. J. Appl. Phys. 105, 083916 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 19:07:19of domain are shown as a function of time and radial position for the applied ﬁeld Happof 0.5 Oe /H2084940 A /m/H20850. It can be seen thatHz/H20849r/H20850is almost uniform in the interior of the wire due to the fact that the current ﬂows only in the vicinity of the surface, where Hzchanges from its value in the interior to zero at r=a. The values of Hz/H20849r,t/H20850in D1 and D2 are of opposite signs at any time t. The calculated dependence of total voltage on Happis different in the low-ﬁeld region depending on whether or notH zis taken into account, as shown in Fig. 17where the experimental curve is also shown. The parameters of the cal-culation are the same as in Figs. 12–16. When H zis ignored in the calculation, the result ﬁts experiment better than whenH zis taken into account. A likely explanation for this is that in the two-domain structure, the axial dynamic ﬁelds createdin the two domains are in opposite directions. This suggeststhat the ﬁelds created in a given domain are largely canceledby the axial ﬁelds generated by the two neighboring domainsor by dipole generated at the domain wall. Another possibleexplanation is that the magnetization does not rotate coher-ently, even locally. All of the calculations reported above, concerning the comparison with experiment, do not include an H zcontribu- tion. We note from Figs. 12and17that the ﬁt of total signal to the model is then excellent at all currents and frequencies,except at the lowest applied ﬁelds, where the sample is un-saturated. A possible reason for this discrepancy is that theaxial ﬁeld which is created at the domain wall and whichmay cancel H zmay then be greater than Hzso that the re- sulting total axial ﬁeld is opposite to Hz. This would explain the observation that the results of the calculation excludingH zare closer to experiment than those including Hz. F. Extension to other GMI wires Here, we have presented a comparison of the model with experimental results for one type of wire, as an example ofapplication of the model and method. The calculation hasbeen based on coaxial regions of uniform anisotropy. Theanalysis suggests that magnetically hard regions, possiblyclose to the surface of the wires, have a signiﬁcant role inreducing the GMI signals as compared to an ideal soft wire.It is expected that these conclusions would remain valid forother wires with different magnetic properties. Indeed, theassumption of cylindrical symmetry is generally reasonable for all types of microwires and the use of coaxial regions ofuniform anisotropy, rather than a true vortex structure,should not lead to signiﬁcant change in the outcome of thecalculation. While different values of the anisotropy ﬁelds inother wires would change the peak position and thresholdcurrent for nonlinear behavior, the general behavior wouldremain essentially unchanged. In particular, signiﬁcant modi-ﬁcations of the GMI response due to magnetically hard re-gions are to be expected in other types of wires as well. VI. CONCLUSION Dynamic calculations, solving the nonlinear Landau– Lifshitz and Maxwell equations including Ohm’s law havebeen performed numerically for magnetic wire with simpli-ﬁed coaxial magnetic structure, without introducing any sig-niﬁcant approximations. This allows us to model both thespace and time variation in the magnetization for linear andfor nonlinear behavior. This, in turn, permits us to obtaininformation about the electromagnetic ﬁelds within the wirein NLGMI conditions and to understand the mechanism ofvariation in the circumferential induction. The model and themethod are quite general. For example, we have found that,at relatively high current, circumferential ﬂux variation oc-curs by means of solitary-wave-like propagation of magneti-zation. We have also found that a maximum of skin effectoccurs when the applied ﬁeld is in the vicinity of the aniso-tropy ﬁeld even if the voltage maximum in the nonlinearregime appears at lower ﬁeld. These conclusions should begenerally valid for all types of GMI microwires. Comparison with experimental results for a GMI wire shows that the calculations based on a homogeneous wire,with a spin free condition at the surface, overestimate thevoltage response. The presence of a magnetically hard sur-face reduces the voltage response as observed. The secondharmonic gives additional information concerning the mag-netic structure. However, due to its high sensitivity to de-tailed domain structure of the wire, the second harmonic isvery difﬁcult to reproduce with accuracy. Detailed investigation of the effect of varying the aniso- tropy skew angle /H9274, as well as varying each of A,M,Hk, and /H9268while leaving the other properties ﬁxed as will applying it to parallel pumping in FMR, and to other sample geometries,will require lengthy, if straightforward, application of themethod. More complex, inhomogeneous, models will requirefurther effort. We have begun work on studying the nonlinearproperties of other GMI wires and hope to report results ofthese in the near future. We expect that the behavior of thetotal signal will remain qualitatively as reported here even ifquantitatively quite different. We also intend to pursue studyof the rich structure of higher harmonics such as we havepreviously reported. 20,23It is possible that higher, especially even, harmonics may vary greatly from sample to sample. ACKNOWLEDGMENTS We are grateful to Dr. A. Rochefort for giving us access to his multiprocessor computer. Thanks are due to L. G. C.Melo for providing analytical calculation results and for0120.20.40.60.8 experiment calculation including Hz calculation excluding HzVoltage (V) Applied field (Oe) (1 Oe ~ 80 A/m)f=1 0M H z I=5m A FIG. 17. Wire voltage vs applied ﬁeld. Comparison between experiment and calculation with and without including Hz. The current amplitude is 5 mArms and the frequency is 10 MHz.083916-11 Seddaoui et al. J. Appl. Phys. 105, 083916 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 19:07:19helpful discussion. This work was supported by the Natural Sciences and Engineering Research Council of Canada, andby CSI Inc. APPENDIX: EXPRESSION OF THE VOLTAGE ACROSS THE WIRE Let us assume that there is no signiﬁcant space variation in the ﬁelds along the wire axis, that is, the ﬁelds are inde-pendent of z. The voltage across the wire is proportional to the axial electric ﬁeld at the surface, V/H20849t/H20850=lE z/H20849r=a,t/H20850, where lis the wire length. The Maxwell–Faraday equation written in cylindrical coordinates gives the axial electric ﬁeld Ez/H20849r,t/H20850=Ez/H20849r=0 ,t/H20850+/H20885 0r/H11509B/H9272/H20849r/H11032,t/H20850 /H11509tdr/H11032. /H20849A1/H20850 The last term on the RHS is the contribution of the time variation in the circumferential magnetic ﬂux. The current ﬂowing in the wire is I/H20849t/H20850=/H20885/H20885 SJds=/H20885/H20885 S/H9268Eds=2/H9266/H9268/H20885 0a rEz/H20849r,t/H20850dr,/H20849A2/H20850 where Jis the current density, Sis the wire cross section, and /H9268is the conductivity. Substituting Eq. /H20849A1/H20850into Eq. /H20849A2/H20850 I/H20849t/H20850=/H9266/H9268a2Ez/H20849r=0 ,t/H20850+2/H9266/H9268/H20885 0a r/H20885 0r/H11509B/H9272/H20849r/H11032,t/H20850 /H11509tdr/H11032dr. /H20849A3/H20850 From Eq. /H20849A3/H20850, the axial electric ﬁeld at the wire axis is Ez/H20849r=0 ,t/H20850=I/H20849t/H20850 /H9266/H9268a2−2 a2/H20885 0a r/H20885 0r/H11509B/H9272/H20849r/H11032,t/H20850 /H11509tdr/H11032dr. /H20849A4/H20850 The ﬁrst term of the right member corresponds to the electric ﬁeld in the case of homogeneous ﬂow of the currentin the wire and may be written in terms of dc resistance, I/H20849t/H20850 /H9266/H9268a2=RdcI/H20849t/H20850 l. /H20849A5/H20850 The second is due to the fact that the current is imposed. 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Phys. 105, 083916 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 19:07:19
5.0029284.pdf	J. Appl. Phys. 128, 191102 (2020); https://doi.org/10.1063/5.0029284 128, 191102 © 2020 Author(s).Characterization of ferroelectric domain walls by scanning electron microscopy Cite as: J. Appl. Phys. 128, 191102 (2020); https://doi.org/10.1063/5.0029284 Submitted: 11 September 2020 . Accepted: 21 October 2020 . Published Online: 17 November 2020 K. A. Hunnestad , E. D. Roede , A. T. J. van Helvoort , and D. Meier COLLECTIONS Paper published as part of the special topic on Domains and Domain Walls in Ferroic Materials DDWFM2021 This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Contributions to polarization and polarization switching in antiphase boundaries of SrTiO 3 and PbZrO 3 Journal of Applied Physics 128, 194101 (2020); https://doi.org/10.1063/5.0030038 Tracking ferroelectric domain formation during epitaxial growth of PbTiO 3 films Applied Physics Letters 117, 132901 (2020); https://doi.org/10.1063/5.0021434 Intrinsic and extrinsic conduction contributions at nominally neutral domain walls in hexagonal manganites Applied Physics Letters 116, 262903 (2020); https://doi.org/10.1063/5.0009185Characterization of ferroelectric domain walls by scanning electron microscopy Cite as: J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 View Online Export Citation CrossMar k Submitted: 11 September 2020 · Accepted: 21 October 2020 · Published Online: 17 November 2020 K. A. Hunnestad,1 E. D. Roede,1 A. T. J. van Helvoort,2 and D. Meier1,a) AFFILIATIONS 1Department of Materials Science and Engineering, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway 2Department of Physics, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway Note: This paper is part of the Special Topic on Domains and Domain Walls in Ferroic Materials. a)Author to whom correspondence should be addressed: dennis.meier@ntnu.no ABSTRACT Ferroelectric domain walls are a completely new type of functional interface, which have the potential to revolutionize nanotechnology. In addition to the emergent phenomena at domain walls, they are spatially mobile and can be injected, positioned, and deleted on demand,giving a new degree of flexibility that is not available at conventional interfaces. Progress in the field is closely linked to the development ofmodern microscopy methods, which are essential for studying their physical properties at the nanoscale. In this article, we discuss scanningelectron microscopy (SEM) as a powerful and highly flexible imaging technique for scale-bridging studies on domain walls, continuously covering nano- to mesoscopic length scales. We review seminal SEM experiments on ferroelectric domains and domain walls, provide practical information on how to visualize them in modern SEMs, and provide a comprehensive overview of the models that have beenproposed to explain the contrast formation in SEM. Going beyond basic imaging experiments, recent examples for nano-structuring andcorrelated microscopy work on ferroelectric domain walls are presented. Other techniques, such as 3D atom probe tomography, areparticularly promising and may be combined with SEM in the future to investigate individual domain walls, providing new opportunities for tackling the complex nanoscale physics and defect chemistry at ferroelectric domain walls. © 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/ ).https://doi.org/10.1063/5.0029284 I. INTRODUCTION The research on ferroelectric materials and phenomena has matured significantly since the discovery of ferroelectricity inRochelle salt in 1920. 1Today, ferroelectrics are used in different fields of technology, for instance, finding application in active damping units, capacitors, and random-access memories.2Despite the tremendous progress that has been made in understanding fer-roelectrics, this class of materials keeps attracting attention as a richsource for new emergent properties, representing a fascinating playground for both fundamental and applied sciences. Recent examples include spin-driven ferroelectrics, 3which facilitate a unique coupling between spin and lattice degrees of freedom,4as well as ferroelectric skyrmions and vortices,5–7representing new and intriguing states of matter. In this article, we will focus on the rapidly growing field of research that studies ferroelectric domain walls and their functionality.8–11Due to the small length scales associated with ferroelectric domain walls, which usually have a width in the order of 1–10 nm,11progress in this field is closely related to advances in spa- tially resolved characterization methods.12Transmission electron microscopy (TEM) is nowadays readily applied to study theatomic-scale structure at ferroelectric domain walls, 13–15and electron energy loss spectroscopy (EELS) provides insight into the local elec-tronic and chemical properties. 16–19At the nanoscale, scanning probe microscopy (SPM) methods, such as piezoresponse force microscopy(PFM) 20,21and conductive atomic force microscopy (cAFM),22,23are routinely used to determine domain wall charge states and studytheir electronic transport behavior, respectively. 14,24–31In addition, photoemission electron microscopy (PEEM)32–35and low-energy electron microscopy (LEEM)36–39have been explored to widen the accessible parameter space, mapping transport phenomena andelectrostatic potentials with nanoscale spatial resolution. 40Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-1 ©A u t h o r ( s )2 0 2 0One of the most common characterization techniques that allows for spatially resolved measurements with nanometer scale resolution is scanning electron microscopy (SEM). As such, it isnot surprising that SEM also plays a special role among theimaging techniques that have been applied in the research on ferro-electric domain walls. In the 1970s, SEM was used to image ferro- electric domain walls and gain insight into their unusual nanoscale physics. 41–43Since then, the SEM technology has evolved consider- ably and, together with SPM, has turned into a mainstream techni-que for surface analysis. SEM has an outstanding —yet not fully exploited —potential for domain-wall related investigations, offering contact-free and non-destructive high-speed imaging, nanoscale spatial resolution, and a high flexibility in terms of specimen prepa-ration and geometry that allows, for example, to combine micros-copy with nano-structuring or in situ /in operando transport measurements. In this Tutorial, we discuss the practical use of the SEM technique in connection with visualizing ferroelectric domains anddomain walls. Mastering the ferroelect ric contrast enables possibilities for combining SEM with other complementary techniques, such asSPM, TEM, and FIB (focused ion beam), opening new pathways for the investigation of the complex physics at domain walls and property monitoring in device-relevant geometries. We begin the Tutorial withan introduction to domain walls in ferroelectrics (Sec. IA)a n dt y p i c a l characterization techniques applied to investigate them (Sec. IB). In Sec.II, the SEM technique is introduced; we begin with a history of SEM-based domain and domain wall imaging experiments (Sec. II A), followed by the basic operating principles (Sec. II B)a n ds o m e practical advice for imaging ferroelectric domain walls (Sec. II C). In Sec. II D, we provide an overview of the different models used to explain the SEM contrast at neutral and charged domain walls. Sections IIIandIVare devoted to correlated techniques, considering SEM in combination with FIB. We discuss how SEM can be inte-grated/essential/correlated to other techniques such as TEM, SPM,and atom probe tomography (APT), respectively, with a focus on new possibilities for future domain wall research. A. Domain walls in ferroelectrics Ferroelectric materials exhibit a spontaneous polarization that can be switched by an external electric field. 44Depending on thesymmetry of the unit cell, ferroelectrics have at least two symmetri- cally equivalent directions for polarization. A region with a constant direction of the polarization is called a domain, and thedomains are separated by a natural type of interface, that is, the“domain wall ”(see Fig. 1 for a schematic illustration). Depending on the possible domain states, anisotropy and dipole –dipole inter- actions, there are different ways for the polar order to changeacross domain walls as discussed in detail in the comprehensivetextbook by Tagantsev et al. 44In BiFeO 3, for example, the polariza- tion can point along any of the ⟨111⟩directions in the rhombohe- dral unit cell, forming 71°, 90°, and 180° domain walls.45In prototypical ferroelectrics, such domain walls are pre-dominantlyIsing-type walls, but more complex mixed structures are also possi- ble, involving Bloch- or Néel-like rotations of the polarization vector. 46Here, for simplicity, we will focus on 180° domain walls, where the polarization changes by 180° from one domain to thenext, without discussing further details of the inner domainwall structure. Within the category of 180° domain walls, we can further sep- arate between three fundamental cases: The polarization P at the wall can either be in side-by-side ( ↑↓), head-to-head ( →←), or tail-to-tail ( ←→) configuration as depicted in Figs. 1(a) –1(c). At head-to-head and tail-to-tail domain walls, the polarization has a component normal to the wall (div P ≠0), which leads to the for- mation of bound charges. 47–49These bound charges require screen- ing, driving a redistribution of mobile charge carriers (ionic and/orelectronic). Electrons, for example, are attracted by the positiveelectrostatic potential at head-to-head walls and repelled by thenegative electrostatic potential at tail-to-tail domain walls. As aconsequence of this redistribution, increased and reduced con-ductivities can be observed at charged ferroelectric domainwalls. 26,29However, the charge-driven mechanism described here is only one of the established mechanisms causing conduc-tion. Other mechanisms include a reduction in the bandgap or formation of intra-bandgap states caused by defects at the walls (see, e.g., Refs. 50and 51for a review). In short, domain walls represent naturally occurring interfaces, which exhibit uniqueelectronic properties different from the surrounding bulk, offer-ing great potential as 2D systems for the development of thenext-generation nanotechnology. 52 FIG. 1. Schematic illustration of the three fundamental types of ferroelectric 180° domain walls. (a) Neutral side-by-side domain wall (purple). Black arr ows indicate the direction of the spontaneous polarization P in the adjacent domains (yellow and green). (b) Positively charged head-to-head domain wall (red). The a ssociated domain wall bound charges (+) are screened by accumulating mobile charge carriers, which can either be electrons, negatively charges ions, or a combination of bo th. (c) Negatively charged tail-to-tail domain wall (blue). Negative bound charges ( −) are screened by mobile positively charged carriers (electron holes and/or positively charged ions).Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-2 ©A u t h o r ( s )2 0 2 0B. Spatially resolved measurements To access all relevant length scales from atomically sharp domain walls to mesoscopic domains, a variety of microscopy tech-niques has been applied, as illustrated in Fig. 2 . Historically, optical techniques with a resolution of about 1 μm were the primary option for ferroelectric domain imaging, nowadays reaching downto 200 nm in near-field optical microscopy. 53,54Optical imaging was used, for example, to resolve domains in optically active mate-rials, such as lead germanate (Pb 5Ge3O11)55and in combination with preferential chemical etching in hexagonal manganites (YMnO 3).56With the advent of modern microscopy methods, fer- roelectric domains and domain walls in a much wider range ofmaterials became accessible, facilitating a more comprehensiveanalysis. Figure 2 presents selected microscopy studies performed on the family of hexagonal manganites, which has evolved into one of the most intensively studied model systems in the field ofdomain wall nanoelectronics. Going beyond the resolution limit ofclassical optical microscopy measurements, the domain formationin hexagonal manganites and other ferroelectrics has been investi- gated by piezoresponse force microscopy (PFM). 57,58The func- tional properties of domain walls, such as conductivity andmagnetism, have been investigated by conductive-atomic forcemicroscopy (c-AFM) and magnetic force microscopy (MFM),respectively. 26,59–61At the atomic scale, transmission electron microscopy (TEM) allows for the direct observation of the atomic positions and can visualize the internal domain wall structure.62–65 Thus, it is possible to cover all relevant length scales in spatially resolved measurements, ranging from exact atomic positions to the macroscopic correlation phenomena associated with domain walls. It is important to note, however, that the specimen requirementsfor the different microscopy methods are completely different, which can make correlated studies challenging. SEM is a surface sensitive imaging technique that is highly flexible and has been proven to be a very powerful technique forthe visualization of ferroelectric domains and domain walls.Compared to the other methods, SEM sticks out because of the remarkably large range of length scales it can cover as illustrated in Fig. 2 . Due to this continuous coverage of nano- and macroscopic length scales, SEM is an explicitly promising tool for domain-wallrelated research, adding value regarding accessibility and the inte-gration of other advanced characterization and nanostructuring methods as we will discuss in Secs. II–IV. II. SCANNING ELECTRON MICROSCOPY —FUNDAMENTALS AND APPLICATION OPPORTUNITIES A. Domain and domain wall imaging by SEM —A short history The SEM as we know it today was invented by Zworykin et al. in 1942. 66In 1965, SEMs became commercially available67and just two years later, the first paper on ferroelectric domain imaging in BaTiO 3by SEM was published [ Fig. 3(a) ].41Since then, SEM has become a standard tool for many fields of surfacescience, as it can provide significantly higher resolution ( ≈1n m ) than optical measurements and can exploit diverse contrast mechanisms, such as topographic contrast, elemental contrast, and grain contrast (see the book on SEM by Reimer 68for a detailed description). An important breakthrough regarding the investigation of ferroelectrics by SEM was made by Le Bihan et al. in 1972.85 FIG. 2. Upper part: length scales covered by different characterization techniques that are regularly applied to investigate ferroelectric domains and do main walls. Lower part: examples of spatially resolved measurements of the ferroelectric domain structure in the hexagonal manganites, including optical micr oscopy, SEM, PFM, and TEM. In the two images on the left, bright and dark regions correspond to +P and −P domains with P oriented normal to the surface plane (out-of-plane polarization). In the two images on the right, P lies in the surface plane pointing in the directions indicated by the arrows (in-plane polarization). PFM and TEM image s are adapted from Refs. 26and 64. Optical image is reproduced with permission from Šafránková et al., Czech. J. Phys. B 17, 559 (1967). Copyright 1967 Springer Nature. SEM image is reproduced with permission from Li et al., Appl. Phys. Lett. 100, 152903 (2012). Copyright 2012 AIP Publishing LLC.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-3 ©A u t h o r ( s )2 0 2 0In pioneering experiments, the team achieved domain contrast in ferroelectric BaTiO 3using low acceleration voltages. The result is remarkable, as it removed the need for domain-selective-etching for creating topographic features, providing an opportu-nity to study ferroelectric domains and related phenomena in amuch wider range of materials. In the following years, SEM was applied to study different ferroelectrics, including triglycine sulfate (TGS), 70Gd2(MoO 4)3,42LiNH 4SO4,71and LiNbO 343[see also Figs. 3(c) –3(e)]. As advances in electron sources, optics and stability greatly improved the low voltage capabilities of SEM,high-resolution imaging became possible giving the SEM the unique ability to image everything from macroscopic ferroelectric and ferroelastic domain structures to nanometer sized domainwalls (see Fig. 3 ). Originally, however, the primary focus of SEM studies in the field of ferroelectrics was the imaging and characterization of domain structures with little attention being paid to the domainwalls. The majority of SEM studies were performed on samples with out-of-plane polarization and neutral domain walls. Charged ferroelectric domain walls were first investigated by SEM in 1984by Aristov et al. in LiNbO 343[Figs. 3(d) and 3(e)]. Although the exact contrast mechanism and physical origin of the anomalousresponse at the domain walls was not known at that time (see Sec. II D), the work already foreshadowed many aspects discussed in modern domain wall research as we will discuss later on. B. Basic operation To understand the image formation and emergent domain and domain wall contrasts in SEM, we begin by discussing thebasic structure of modern SEMs. A typical SEM consists of (1) theelectron source and electron optics column, (2) the specimen chamber with a goniometric stage, and (3) one or more detectors for recording of secondary electrons and backscattered electrons as FIG. 3. Overview of seminal SEM-based studies on ferroelectric domains and domain walls. (a) First observation of SEM domain contrast in BaTiO 3, realized by selective etching and resulting topographic contrast. Bright and dark areas correspond to 90° domains as illustrated on the left. Black arrows indicate the pol arization direction. Although such 90° domains and other variants are common in ferroelectrics, most SEM imaging studies have focused on 180° domains as presented in (c) –(e). Reproduced with permission from Robinson and White, Appl. Phys. Lett. 10, 320 (1967). Copyright 1967 AIP Publishing LLC. (b) Observation of SEM contrast from ferro- electric domains (black and gray regions) and neutral domain walls (bright stripes) in BaTiO 3. The SEM data was recorded with low acceleration voltage (3 kV), removing the need for selective etching or specific coatings for imaging the domain distribution in ferroelectrics. Reproduced with permission from Le Bihan , Ferroelectrics 97,1 9 (1989). Copyright 1989 Taylor & Francis Ltd. (c) Observation of domain walls (bright stripes) in improper ferroelectric Gd 2(MoO 4)3. Reproduced with permission from Meyer et al., Ultramicroscopy 6, 67 (1981). Copyright 1981 Elsevier. (d) and (e) show SEM images of charged domain walls in periodically poled LiNbO 3. In (d), the negatively charged tail-to-tail walls are visible as black stripes under negative surface charging (3 kV); in (e) positively charged head-to-head domain walls are visible as bright stripes under positive surface charging (1 kV). The SEM data in (d) and (e) are adapted with permission from Aristov et al., Phys. Status Solidi A 86, 133 (1984). Copyright 1984 John Wiley & Sons, Inc.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-4 ©A u t h o r ( s )2 0 2 0shown in Fig. 4 .67,68,72In addition, optional optical cameras are available for monitoring the specimen chamber. The electron source is based on thermionic or field emission; the source in thelatter case is called a field emission gun (FEG). After the electronsare emitted from the source under a certain acceleration voltage(V), they enter the column where a series of electromagnetic lenses and apertures focuses the beam onto the sample surface. Both the column and the chamber are kept in vacuum (<10 −6Pa). Modern SEMs often use an in-lens detector (ILD) where the detector ispositioned inside the electron column for immersion modeimaging. The beam diameter, or spot size, is closely tied to the res- olution of the microscope and for modern SEMs it can even be sub-nanometer when using a FEG. As the electrons from the beam (primary electrons, PEs) reach the sample, a variety of complex interactions occurs; for example, thePE can either interact with the negative electron clouds of the atoms or the positive nuclei. The interaction with the nuclei causes the PE to be scattered approximately elastically and, in some cases, electronsare reflected with an energy close to the incident energy (E 0). These reflected electrons are called back-scattered electrons (BSEs). Theprobability of backscattering increases with atomic number of the material under investigation, which implies that detection of BSEs will contain elemental contrast. Interactions of PEs with the electroncloud of the target atoms are typically inelastic and associated withthe generation of secondary electrons (SEs). Generated SEs have a low kinetic energy (<50 eV 73) and thus quickly recombine with holes so that only SEs generated close to the surface ( ≲30 nm depth) can escape the specimen. SEs can also be generated from BSEs withinthe material, as well as on surfaces inside the specimen chamber, blurring the imaging results and mixing the SE signal with BSE con- trast. The angle of the sample surface with respect to the beam direc-tion, as well as the surface topography, co-determine the region fromwhich SEs can escape, generating topographic contrast which iscommon when imaging with SEs. Combined with selective etching, such topographic contrast was originally exploited for domain visual- ization (see Sec. II D 1 ). Note that the contributing volume of the BSEs, and thus also that of the BSE induced SEs, is highly dependenton the acceleration voltage (V). Thus , lowering the acceleration voltage means the signal is generated from a smaller volume, which can be exploited to increase the resoluti on. However, V also affects the spot size, with higher V giving the smaller spot size due to reduced aberra-tions. Thus, a compromise has to be found between probe size andinteraction volume to optimize resolution. To obtain spatially resolved data, the electron beam is raster- scanned across the specimen. At evenly spaced points, the beam stops for a time interval (dwell time) in the range of 1 μs, while the detectors record a SE or BSE signal. The most common detectorused is the Everhardt –Thornley detector (ETD); a more detailed explanation can be found elsewhere. 68The ETD is surrounded by a metallic grid that can be biased positively to attract the low energy SEs for imaging, or negatively to repel them and only detect BSEsinstead. Imaging in the immersion mode (immersing the sample inthe magnetic field of the objective lens) improves resolution as a smaller probe can be formed but requires the use of a detector mounted in the electron column (ILD). While ETDs can display aso-called shadowing effect due to the positioning on the side of the FIG. 4. (a) Schematic of a dual-beam FIB with major components added and labeled. The setup of classical SEMs is similar but without the additional ion beam. (b ) Representative example of a domain image gained in SIM mode (SEs generated by ion beams) gained on ErMnO 3with out-of-plane polarization. Bright and dark areas indicate ferroelectric 180° domains with opposite polarization orientation. (c) SEM image recorded on an ErMnO 3sample with the same surface orientation as in (b) with bright and dark areas corresponding to ±P domains.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-5 ©A u t h o r ( s )2 0 2 0chamber, the ILD provides a more homogeneous signal and typically has a better signal to noise ratio. However, even when operating both detectors in the SE imaging mode, there will be adifference in the ratio of BSEs and SEs that reach the detectorsbecause of the geometry and the contrast might differ. 74 A specific challenge arises for ferroelectric domain and domain wall imaging as ferroelectrics usually exhibit insulating or semi-conducting properties. Thus, irradiation with charged parti-cles can generate a significant surface charge, which affects theimaging conditions. The ratio between emitted electrons and inci-dent electron is called the electron yield ( δ). This yield is highly material dependent and varies with the acceleration voltage (V) as sketched in Fig. 5 . 69The figure shows that whenever the yield is not equal to unity, the sample gets charged: for δ>1( V 1<V<V 2), the sample charges positively, and for δ<1( V>V 2and V < V 1), it charges negatively. As a result, the PEs are accelerated or deceler- ated on the way toward the surface, while the emitted SEs are attracted or repelled from the surface, respectively, dynamicallychanging the imaging conditions. The charging will proceed until adynamic equilibrium is reached where δ= 1. For conductive materi- als such charging can readily be avoided by grounding the sample so that excess charge can dissipate. For ferroelectrics, or insulating materials in general, however, the excess charges accumulate at thesurface. This can cause distortions and imaging artefacts. Thus, it isoften necessary to adjust and fine-tune the acceleration voltage around δ= 1 to achieve stable imaging conditions. This is particu- larly challenging when imaging areas with spatially varying con-ductivity as different areas charge differently, requiring special care. C. Practical considerations for optimizing domain and domain wall contrasts 1. Acceleration voltage The acceleration voltage is arguably the most important parameter when imaging ferroelectric domains and domain wallsby SEM, becoming increasingly important the more insulating thesample is. The impact on domain and domain wall visualization iswell demonstrated in Fig. 6 , showing an early example from the seminal work of Le Bihan et al. 69As shown in Fig. 5 , for insulating materials, the surface will become positively charged for accelera-tion voltages between V 1and V 2, and negatively charged for accel- eration voltages higher than V 2or lower than V 1. For domain contrast, working close to the equilibrium voltage V 1or V 2is favor- able, because otherwise the deposited surface charge can screen the polarization charges responsible for the contrast. We note that thesecond equilibrium point V 2is usually preferred as V 1is typically so low in energy that aberrations will dominate the final resolution.In contrast, for domain wall imaging off-equilibrium voltages are favorable, because here moderate charging can be useful as dis- cussed in more detail in connection with different models proposedfor contrast formation in Sec. II D. In general, strong charging of the material can create pro- nounced distortions in the SEM images. The distortions usually manifest as large variations in SE emission or drift within a single scan. 75In principle, finding the equilibrium or optimal voltage is not too difficult. It can be achieved by starting at a low voltage(e.g., 1 kV), progressively increasing the value. If domain ordomain wall contrast becomes visible, the voltage can then be fine- tuned until maximum domain/domain wall contrast is reached. In practice, however, dynamical charging effects often occur whileimaging, leading to variations in the surface potential and, hence,unstable imaging conditions. In cases where no contrast is seen from the domains or domain walls after quickly surveying the accessible range of acceler-ation voltages, or charging is too severe, other basic charging prin-ciples may be considered to find the equilibrium voltage wheredomain contrast is most pronounced. Below V 2, the sample should become positively charged and the scanned area will appear darker than the neutral surface. When the equilibrium point V 2has been reached, this contrast should invert: the scanned area becomesnegatively charged and thus brighter than the neutral surface(see Ref. 68for more rigorous methods). 2. Secondary parameters (beam current, dwell time, detector, and specimen) Adjusting the acceleration voltage is not always enough to get good domain and domain wall contrast and even when working at ideal acceleration voltage the contrast can be very subtle, requir- ing further optimization of secondary parameters. In general,increasing the beam current decreases noise and improves contrastsin SEM, and this also applies for the imaging of domain structures,possibly even more so if the contrast mechanism at play originates either from heating or charging effects (see Sec. I ID1 for details). Note, however, that as the beam current is increased, dynamicalcharging effects may get more pronounced. In addition, the risk ofcarbon contamination increases, 76as well as the risk of poling the ferroelectric sample under the beam while imaging. Both these effects are amplified by using longer dwell times, i.e., slower scans. Faster scanning may be preferred for strongly charging samples,and if excessive charging cannot be avoided, a compromise must bemade in terms of beam current and scan speed. In the case of very insulating specimens or multi-component systems, it is helpful to touch a grounded micromanipulator to the sample surface near to FIG. 5. Schematic of the secondary electron (SE) yield for varying acceleration voltages V .68At the equilibrium points, V 1and V 2, the yield is at unity and there is no surface charging. For V 1<V<V 2, the sample is slightly positively charged, while for V > V 2(and V < V 1), the sample becomes negatively charged. When working with voltages V > V 2, the dynamical negative charging caused by the electron beam leads to a deceleration of the primary electrons (referred to ascharging direction) until V 2is reached.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-6 ©A u t h o r ( s )2 0 2 0the region of interest, improving the path to ground. For SEMs without micromanipulators, conductive paint or tape can be placednear the region of interest as an alternative. Trivially, but very important, clean and flat surfaces with a root mean square roughness in the order of 1 nm, as obtained fromproper polishing, eliminate topographic contrast that can dominateover weak domain and domain wall contrast. Finally, it should benoted that both SE 42,69,77and BSE78,79detection modes have been reported to give domain and domain wall contrast, but not neces- sarily work equally well on the same material. From our experience,the best contrast is achieved using an in-lens detector biased for SEimaging. This can be explained by its high sensitivity to low energySEs and more effective discrimination of BSE induced SEs thus making it better suited to distinguish between small differences in the SE yield. 74The discussion in Sec. II D will thus focus on SE detection. 3. Biasing with the electron beam To verify that SEM contrast originates from ferroelectric domains, the domains may be switched with the electron beam,analogous to PFM studies, where the switching is realized usingan electrically biased AFM tip. 70,80,81If the detected contrast inverts along with the polarization orientation, this is a strong indi- cation that the SEM contrast is of ferroelectric origin. Focusing theelectron beam with a high current onto a small region has beendemonstrated to create a sufficiently high electric field that canswitch the polarization direction. 70,82,83While the sub-surface domain structure remains unknown, which is a limitation of SEM and surface analysis techniques in general, the surface can be modi-fied with high spatial precision, although one has to be careful notto confuse domain switching with charging effects. As demon-strated, for example, for LiTaO 380and LiNbO 3,82,83the electron beam at higher acceleration voltages (15 kV) induces negative charging at the surface, which can be used to locally flip the polari-zation from −P to +P. In general, with a small contact-free electron probe any specimen of any geometry can be manipulated and subsequently imaged, reflecting the high flexibility of SEM-based experiments.D. Contrast mechanisms 1. Out-of-plane polarization Using SEM, ferroelectric domains and domain walls have been visualized in many ferroelectric materials over the years. Although the technique offers a quick way to image domain structures, acareful analysis is required to make statements about the local ferro-electric properties, because emergent contrast mechanisms can gowell beyond the basic description of SE and BSE contrast as observed in common materials as described in Sec. II B. The original and probably most simple approach for achieving domain contrast is through domain-related topographical varia- tions that arise due to surface treatment, such as chemical polishingand etching processes. 84In their early work on BaTiO 3, Robinson and White exploited that ferroelectric domains with and without surface bound charges etch differently leading to variations in surface roughness, which was used to image 90° domains by SEM[Fig. 3(a) ] 41(note that although 90° domains and other variants are common in ferroelectrics, most of the SEM studies havefocused on 180° domains). Due to the higher surface roughnessand thus larger escape area, the SE yield was found to be enhancedfor the positive domains with P normal to the surface so that they are brighter in SEM measurements. Detrimental charging effects were suppressed by depositing a conductive coating onto thesample surface. Alternatively, height differences can also occur, forexample, when the domains polish at a different speed, leading tovisible domain-related steps in surface topography (i.e., Ref. 41). While this approach allows for studying the domains in out-of-planepolarized specimens, the topography transition from one domain to the next is not necessarily correlated to the existing domain wall structure as walls can, in principle, move after a topographic patternhas been imprinted. Furthermore, in situ experiments are not possi- ble and domain switching cannot be captured, and the added con-ductive layer can limit further investigations with other microscopytechniques, such as conductive or electrostatic force microscopy.Finally, the resolution is limited by the etching/polishing rather than the SEM instrumentation. However, it was later found that using low acceleration voltages, charging can be avoided and a conductive FIG. 6. SEM domain and domain wall images obtained on TGS. The images show how SEM contrast in ferroelectrics with out-of-plane polarization can vary dependi ng on the acceleration voltage V . (a) Domain contrast is achieved for V = V 2with +P domains appearing darker than −P domains (see Fig. 5 for an illustration of the imaging conditions). (b) and (c) show SEM contrast at neutral domain walls gained at V < V 2(bright walls) and V > V 2(dark walls), respectively. Figures are adapted with permission from Le Bihan, Ferroelectrics 97, 19 (1989). Copyright 1989 Taylor & Francis Ltd.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-7 ©A u t h o r ( s )2 0 2 0coating could be omitted.85This discovery was important as it enabled the visualization of domains and domain walls on the polar surface without selective etching. Aside from topographic contrast, polarization charges and electrostatics associated with the ferroelectric order can beexploited for domain imaging in SEM. Since the first observationof electrostatic contrast variations by Le Bihan in 1972, different theories have been proposed to explain the mechanisms behind SEM domain contrast on polar surfaces. 69,77,86Three main mecha- nisms for domain contrast that have been discussed in the litera-ture are illustrated in Figs. 7(a) –7(c), that is, (i) polarization contrast, 69(ii) pyroelectric contrast,87and (iii) emission contrast.69 FIG. 7. Main mechanisms that can lead to domain contrast (top row) and domain wall contrast (bottom row) in SEM. (a) Uncompensated negative (blue) and positiv e (red) bound charges (leading to an electric potential V pol)a tt h es u r f a c eo f −P and +P domains ( −and +) can repel and attract secondary electrons (SEs), respectively. This leads to variations in the SE yield and, hence, the detection of domain contrast, with more intensity for −P domains (light gray) compared to +P domains (dark gray).69(b) Due to local heating from the electron beam and the pyroelectric effect, the spontaneous polarization can decrease. Assuming that the bound charges at the surfac e were initially fully screened, the decrease in polarization can lead to excess screening charges, giving rise to a domain-dependent surface potential (V pyro) that is detectable by SEM. Note that the pyroelectric contrast is inverted compared to the polarization contrast in (a).87(c) Physical properties such as work function (W ±) and penetration depth (R ±) can be different from one domain to another leading to changes in SE emission and, in turn, to domain contrast.69(d) Because of the bound charges at the surface of −P and +P domains, a built-in electric field E exists (in-plane), where the domain wall intersects with the sample surface. This built-in field forces PEs and SEs into adj acent domains, keeping the wall neutral. Under positive surface charging, a potential V deparises. The wall region which has remained less charged than the bulk will have a smaller attraction of SEs and more detected yield.69(e) Deposited charge and the accompanying converse piezoelectric effect cause contraction and expansion of domains leading to a change in topograph y. When using a side-mounted detector (ETD), half the domain walls will have a larger exposed surface and the detector will detect more SEs.87(f) Due to local beam-induced switching, domain walls can tilt away from their ideal charge-neutral position. Under positive charging, −P domains expand and domain walls can tilt as sketched in (f), changing the domain wall configuration and the local surface charge state. Assuming that surface charges in the switched area are not instantaneously screene d, the emergent negative surface bound charges in the inverted area will create a negative surface potential relative to the screened regions, which increases the SE yield.77In addition, the emergent domain wall bound charges and related changes in local conductivity can result in SEM contrast (see Fig. 8 ).Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-8 ©A u t h o r ( s )2 0 2 0Mechanism (i) was discussed in connection with BaTiO 3, where the domain contrast has been attributed to uncompensated surface bound charges that arise from the bulk polarization [ Fig. 7(a) ]. The bound charges either repel or attract the SEs, reducing theyield for domains with positive surface charges compared todomains with negative surface charges. This effect was also used to explain the domain contrast observed in TGS [ Fig. 6(a) ], where domains with positive surface charges appeared to be darker thanthe domains with negative surface charges. While it is clear that asurface potential will alter the SE yield, the origin of the surfacepotential is not always clear and much more difficult to deter- mine. This is because surface bound charges can be screened to a varying degree by free charge carriers (ionic or electronic); thischarging may further vary dynamically while imaging, and thecontribution and impact of potentially charged adsorbents on thesurface are often unknown. The potential impact of adsorbents is reflected, e.g., by x-ray photoemission electron microscopy (X-PEEM), low-energy electron microscopy (LEEM), and x-rayphotoelectron spectroscopy (XPS) studies, 88,89and analogous sys- tematic investigations on ferroelectric domains and domain wallsby SEM are desirable. Aside from polarization contrast, the pyroelectric effect can lead to domain contrast in SEM as a result of heating bythe electron beam. 87A pyroelectric potential is formed as the polarization value changes due to the heating with opposite value for opposite domains [ Fig. 7(b) ], given that emergent changes ΔP = P(T) –P(T + ΔT) are not screened instantaneously. The result- ing domain-dependent surface potential modifies the total numberof SEs reaching the detector in a similar way as in the model con-sidering polarization charges at a constant temperature. As a conse- quence, different equilibrium voltages V 2exist for the two domain states (see Fig. 5 ). It is important to note that the domain contrast evolving from the pyroelectric effect [ Fig. 7(b) ] is inverted com- pared to the contrast that arises from surface bound charges[Fig. 7(a) ], highlighting the importance of a careful analysis to identify the correct polarization orientation and mechanism at play. Furthermore, variations in other physical properties can causeor contribute to the domain contrast, such as differences in electronpenetration depth (R ±) and work function (W ±)[Fig. 7(c) ]. Compared to the phenomena observed for 180° domains with out-of-plane polarization we discussed so far [ Figs. 7(a) –7(c)], the interpretation of effects related to the associated neutral domainwalls is even more challenging. This is because the domain wallscan have completely different intrinsic properties than the bulk (see Sec. IA), leading to a different interaction with the PEs and SEs. In particular, their electronic and thermal conductivity canbe different, which is known to be crucial for the contrast forma-tion in SEM. However, while it is clear that this correlationenables new research opportunities, systematic, and comprehen- sive SEM-based investigations of local transport phenomena at ferroelectric domain walls remain to be realized. An early model from Le Bihan 69(charging contrast model) explained the contrast at neutral 180° domain walls in out-of-planepolarized samples based on a built-in electric field. The built-in electric field arises from the bound surface charge next to the wall, pushing the PEs into the adjacent domains so that the neutralwalls do not charge up as shown in Fig. 7(d) . Thus, the yield forthe walls will be higher than for the bulk when the surface is posi- tively charged, owing to less attraction of SEs, and lower than for the bulk when the surface is negatively charged [see also Figs. 6(b) and 6(c)]. Similar effects are also expected if the neutral domain wall exhibits a higher conductivity than the domains, which locallyreduces the charging under the electron beam. This becomes increasingly important for thinner samples and films grown on conducting back-electrodes, where it becomes more likely thatconducting domain walls directly connect surface to ground,leading to substantially reduced resistivity relative to the domains.A second model (piezoelectric contrast) is presented in Fig. 7(e) , where domain wall contrast is attributed to topographical varia- tions caused by the converse piezoelectric effect: as the sample charges, a surface potential builds up, which leads to a contractionor expansion of the ferroelectric domains, depending on the accel-eration voltage and the respective polarization orientation. The latter translates into domain dependent variations in surface top- ography, which can be resolved in SEM [ Fig. 7(e) ]. When using a side-mounted detector (ETD), half of the domain walls will have alarger exposed surface and the detector will detect more SEs so thatthese walls appear brighter in SEM. 87Alternatively, SEM contrast can arise as domain walls are bent away from their ideal charge- neutral configuration (switching contrast) due to local heating orsurface charging by the electron beam. 77,86When negative domains expand, as it is the case for positive surface charging, the switched area will develop a negative surface potential, assuming that the bound charges of the newly switched area are not screened instantaneously.The surface potential then deacceler ates PEs and repels SEs, increasing the yield so that the switched area appears brighter [ Fig. 7(f) ]. Aside from the emergent surface potential due to domain switching, the domain walls bend away from their charge-neutral position, which leads to domain wall bound charges. The latter can cause additionalcontrast contributions as explained in detail in Sec. I ID2 ,w h e r ew e address SEM imaging of charged domain walls. The overview presented in Figs. 7(d) –7(f) shows that the interpretation of domain wall contrast in SEM can be highly non- trivial and the exact mechanisms are still far from being fullyunderstood. The development of a more comprehensive modelexplaining the SEM contrast formation at nominally neutral ferro-electric domain walls is therefore highly desirable to enable quanti- tative measurements and benefit from the SEM`s nanometer spatial resolution and high sensitivity to electronic and electrostaticdomain wall properties. 2. In-plane polarization Compared to surfaces with out-of-plane polarization, surfaces where the polarization lies in-plane are far less studied with SEM.Domain contrast has been observed on surfaces with in-planepolarization, 43and the contrast has been explained by the emission model [ Fig. 7(c) ], assuming that oppositely polarized domains have a difference in emission at equilibrium conditions. However, addi- tional work is desirable to clarify the origin of ferroelectric domaincontrast in SEM on non-polar surfaces. Importantly, it was clearearly on that emergent domain wall contrasts can provide valuable information about the physical properties at charged ferroelectric domain walls. In addition to the phenomena that arise at neutralJournal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-9 ©A u t h o r ( s )2 0 2 0walls, the charged domain walls exhibit bound charges and diverg- ing electrostatic potentials that add to their complexity, as well aspronounced variations in electronic conductivity. The first observation of charged domain walls by SEM was made in 1984 by Aristov et al . 43By mapping the ferroelectric domain structure in LiNbO 3under negative charging conditions (Fig. 5 ), it was found that tail-to-tail domain walls became visible due to a lower SE yield than the bulk [ Fig. 3(d) ]. For positive surface charging, the positively charged head-to-head domain walls became visible because of a higher SE yield than the bulk [ Fig. 3(e) ]. The authors explained this effect based on electrostatics, arguing thatthe domain wall represents a potential barrier which preventscharges to accumulate. For instance, under negative surface charging [Fig. 8(a) ] a negative domain wall (tail-to-tail) would be kept neutral and thus exhibit a lower yield than the bulk due to less repulsion ofSEs and PEs. The opposite happens at the head-to-head walls, whichremain neutral under positive charging so that more SEs reach thedetector. Alternatively, differences in the surface potential can arise at charged domain walls due to their distinct electronic transport properties [conductivity contrast, Fig. 8(b) ]. X-ray photoemission electron microscopy (X-PEEM) measurements demonstrated thatsurface charging is suppressed at charged domain walls withenhanced conductivity. 32The correlation between domain wall con- ductivity and SEM contrast is evident from recent measurements on ErMnO 3, revealing a direct connection between SEM domain wall contrast and the local transport properties90[seeFigs. 9(b) –9(d)]. In 2007,91further investigations on poled LiNbO 3with charged domain walls showed that the SE yield at negative tail-to-tail domain walls is higher than for the bulk under positive surface charging and lower under negative surface charging(head-to-head domain walls were not reported). It was suggestedthat the SEM contrast could be due to increased recombination activities, resulting from an accumulation of point defects and impurities at the negatively charged domain walls. This couldreduce the negative charging at the wall and thus lead to a lower yield [ Fig. 8(c) ]. Although the results are not fully consistent with the aforementioned SEM study on charged domain walls inLiNbO 3,43the work is intriguing as it foreshadows the possibility to apply SEM to explore the local defect chemistry at charged ferro- electric domain walls.92In general, it is very likely that multiple effects are present, contributing simultaneously to the SEM contrastat charged domain walls. Thus, especially with fixed imaging con-ditions, it is challenging to unambiguously identify the physical origin of the contrast as reflected by the work on charged domain walls in LiNbO 3. In conclusion, although SEM attracted much attention for imaging ferroelectrics early on, the reported observations have shown that we still do not have a good enough understanding of the underlying contrast mechanisms. In this sense, it is an over-looked technique with more potential, considering its great flexi-bility and speed in visualizing both nanoscale and macroscopicdomain structures. However, it is also clear that a better under- standing of the contrast formation process and the development of a comprehensive theory is highly desirable to deconvolutecompeting contrast-formation mechanisms and ultimately facili-tate quantitative SEM-based measurements at domain walls inferroelectrics. III. DUAL-BEAM FOCUSED ION BEAM Seen as a surface analysis technique for ferroelectric domains and domain walls, SEM combines several key aspects, offering non- destructive and contact-free imaging with high spatial resolution. Furthermore, SEM is much faster than comparable domain analysistechniques such as PFM. Possibly the biggest advantage of SEM isthe opportunity to combine domain wall imaging with other nano- characterization and fabrication tools. Most notably, the dual-beam FIB (focused ion beam, Fig. 4 ) combines nano-structuring with FIG. 8. SEM contrast at charged domain walls. (a) Under negative surface charging, the potential barrier at a negative tail-to-tail domain wall prevents neg ative charges to accumulate, keeping it more neutral with less yield (same as in Fig. 6 ). (b) Enhanced conductivity at domain walls can locally reduce charging effects,32causing a potential difference relative to the more insulating domains. Analogous to (a), this potential difference influences the SE yield, leading to domain wall cont rast in SEM. (c) Recombination contrast illustrated for the case of a negatively charged tail-to-tail domain wall (blue line). One possibility to screen the negativ ely bound charges at tail-to-tail domain walls is to accumulate positively charged ionic defects (white circles). It has been proposed that such ionic defects may locall y increase the recombination activity for the primary electrons (PE), which reduces the SE yield at the position of the domain wall so that they are darker than the in-plane polarize d domains in the SEM image.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-10 ©A u t h o r ( s )2 0 2 0high-resolution imaging within one setup, enabling preparation of ferroelectric specimens with varying shapes and dimensions fordomain engineering, 93–96as well as correlated microscopy studies of domain walls in device-relevant geometries.97Here, we will focus on these two topics and give recent examples related todomain wall research. For a comprehensive introduction to theFIB, we refer the reader to, e.g., the textbook by Yao. 95 A. Ions vs electrons The basic setup of FIB optics is very similar to the SEM optics (Fig. 4 ), but instead of an electron beam, FIB uses an ion beam.95 Replacing the electron source by a liquid-metal ion source, typically Ga+, results in completely new functionality. In general, FIB has four basic applications: milling, deposition, implantation, andimaging. In addition, supplementary features, such as micro-manipulators, energy dispersive x-ray spectroscopy (EDX) and compatible probe stations are often found on FIB instruments, making it a highly useful toolkit for nanotechnology-relatedresearch. Originally, the FIB was developed for the semiconductorindustry to do microfabrication and failure analysis, but today it isalso extensively used in research laboratories for characterization and specimen preparation at nano- to macroscopic length scales. Many modern FIBs include an SEM, referred to as dual-beam FIB. Here, both microscopes use the same vacuum chamber anddetector system, and the two beams (electrons and Ga +) are coinci- dent at the sample surface ( Fig. 4 ). In contrast to the electron beam, the ion beam is focused with electrostatic lenses (notelectromagnetic) due to the higher ion mass compared to an elec- tron. When reaching the sample, the heavy Ga+ions strongly inter- act with the surface atoms. This interaction can be an elastic collision with the nucleus and/or inelastic processes with the elec- tron cloud generating SEs. Due to the large mass of the incidention, the former interaction can transfer significant momentum tothe target atoms, knocking them out of position. This creates a cascade of collisions in the targeted specimen and can lead to sputtering of surface atoms, which is the basis for ion beammilling. Aside from milling, the incident Ga +beam can cause amorphization and implantation of Ga+. Related artefacts can be minimized by adequately adjusting the acceleration voltage of the Ga+beam. Lower acceleration voltages typically lead to more implantation, but also a smaller penetration depth and thinnerdamage layers. Should the sputtered atoms also be ionized andemitted from the surface as SI (secondary ions), they can bedetected and reveal strong elemental contrast. Due to ion channel- ing, the penetration depth of the Ga +ions depends on the crystal orientation. The likelihood for SEs to escape and be detectedincreases and thus contrast due to variations in crystallographicorientation is possible for ion induced SE imaging. In general, the FIB retains the basic imaging functions, complementary to the electron imaging of the SEM, referred to as scanning ion microscopy (SIM, see Fig. 4 ). The biggest advantage, however, arises from the added functionalities beyond just imaging.For instance, sputtering can be used to remove the surface layer-by-layer and create cross sections of the material to reveal sub-surface structures. Furthermore, by introducing gases into the FIG. 9. Applications of dual-beam FIB to ferroelectrics. (a) SEM image (BSE mode) of an ErMnO 3lamella with in-plane polarization, extracted from a bulk sample using a dual-beam FIB.90The red dashed square marks the region where the data shown in (b) –(d) was recorded. (b) SEM image (SE mode) obtained in the area marked in (a). Ferroelectric domain walls are visible as bright lines on a homogeneous gray background. (c) PFM image (in-plane contrast) of the same location as in ( b), revealing the polarization direction in the domains (bright: −P; dark: +P). (d) cAFM data showing enhanced conductance at the position of the domain walls. Consistent with Ref. 17, tail-to-tail domain walls are observed to exhibit higher conductance than the head-to-head domain walls, reproducing the contrast levels observed in the SEM scan in (b). (a)–(d) are reproduced with permission from Mosberg et al., Appl. Phys. Lett. 115, 122901 (2019). Copyright 2019 AIP Publishing LLC. (e) Example of a TEM lamella pre- pared by FIB. The lamella is attached to a half TEM grid (left side) in a “flagpole ”geometry, where the outermost part of the lamella has a thickness below 40 nm.100 Image is adapted with permission from Schaffer et al., Ultramicroscopy 114, 62 (2012). Copyright 2012 Elsevier.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-11 ©A u t h o r ( s )2 0 2 0specimen chamber, the ion beam can be used for depositing elec- trical surface contacts or protecting capping layers with nanoscale spatial precision. Typically, this would be tungsten or platinumwhen high conduction is required, or carbon and silicon dioxidefor a more resistive material. Please note that the deposits containhigh amounts of carbon remains for the carrier molecules and Ga with ion-assisted deposition. 98,99Combining the different deposi- tion possibilities with milling al lows for complete 3D nanostruc- turing. Another essential feature of modern FIBs are integratedmicromanipulators, which allow for extracting specimens with thedesired shape and dimension from bulk samples. Other microma- nipulator setups include probe stations that enable in situ charac- terization of local transport properties, making the FIB awell-equipped toolbox for studying ferroelectric domain walls. Inthe following, we will give two examples that highlight how theapplication of dual-beam FIB pushes the frontiers of domain wall research, addressing the extraction and study of individual domain walls ( Fig. 9 ) and creation, and testing of device-relevant geometries ( Fig. 10 ). B. Specimen preparation and nanostructuring FIB-SEM is widely used to prepare specimen for the TEM, 93,100which otherwise can be a time-consuming process and less site-specific. The FIB can readily extract lamellas (typical dimensions: 5 to 10 μm squared and 1 μm thick) from a site-specific region of interest (ROI) with 10 nm precision and thinthe lamella down to the desired thickness and shape [ Fig. 9(e) ]. As a result, the FIB has become a standard tool for TEM specimen preparation. The preparation of thin lamella-shaped specimen, however, is no longer just of interest to enable high-resolutionTEM studies. Nanostructuring by FIB-SEM has evolved into aresearch field by itself, enabling preparation and manipulation ofmaterials to study confinement effects, emergent phenomena at the nanoscale and more. 97For example, FIB-SEM has been used totailor ferroelectrics leading to different breakthroughs, including the creation of exotic domain states and controlled injection of domain walls with nanoscale spatial precision.96,101 As we discussed in Sec. I, conductive domain walls have been intensively studied and great progress has been made in recentyears. Yet, the investigations mostly focused on the application of surface sensitive techniques, while more detailed information on the 3D structure is needed for a better understanding of theunusual electronic transport phenomena at ferroelectric domainwalls. Using FIB-SEM, it becomes possible to study individualdomain walls with a well-defined geometry as recently demon- strated by measurements on the hexagonal manganite ErMnO 3.90 Using a micromanipulator, a lamella was extracted from a bulk out-of-plane polarized sample. After extraction, the lamella wasfurther thinned down to 700 nm and then polished with the ionbeam at low acceleration voltages to remove the surface damage layer and improve contrast of surface sensitive techniques. This approach made it possible to image the lamella from both sides inSEM, giving an estimate of the orientation of the domain walls in3D via linear extrapolation. The lamella was then placed on anMgO substrate for correlated PFM and cAFM studies as shown in Figs. 9(b) –9(d). The correlated investigations, combined with the knowledge about the 3D structure, enabled a refined understandingof the conducting domain walls. In particular, the work explainedwhy deviations from the expected transport behavior occur when considering only the domain wall state at the surface, highlighting the importance of the domain wall orientation hidden withinthe bulk. Going beyond the advanced imaging capabilities, 3D nano- structuring by FIB has been applied to control domain wall motions exploiting size-effects. Although ferroelectric domains can be controlled with electric fields, they will only shrink and expanddepending on the direction of the electric field. Domain wallsenclosing one domain will then necessarily move in oppositedirections. In order to achieve a unidirectional movement, the surface of the ferroelectric material can be altered, creating an asymmetric potential landscape for the domain walls, facilitatingthe design of domain-wall based shift registers as demonstratedby Whyte et al. 102 Figure 10 shows a domain wall diode extracted from ferroelec- tric KTiOPO 4using FIB, representing an intriguing example for 3D nanostructuring. A lamella was extracted from a bulk sample andplaced between two electrodes as shown in Fig. 10(a) . While the backside was kept flat, the surface of the lamella was milled into a wedge shape followed by a step in the topography, as sketched in Fig. 10(b) .G a +-induced damage at the surface was partially removed by thermal annealing and subsequent acid etching.103A strong electric pulse of +100 V was then applied to remove anyexisting domains, followed by two electric pulses of −55 V and +32 V, respectively. PFM images were acquired after each pulse as shown in Fig. 10(b) , where the domain walls are the interfaces between up (yellow) and down (purple) polarized domains. Thefirst pulse nucleated a domain in the right part of the lamella,which progressively moved to the left and over the surface step. During the second pulse, the surface step prevents the existing domain wall from moving to the right and instead a seconddomain wall was nucleated and moved toward the left. Thus, the FIG. 10. FIB-SEM nano-structuring of ferroelectric devices. (a) SEM image of a ferroelectric domain wall diode. (b) In the top, a schematic of the topography is shown and below are two PFM images taken after a voltage pulse has been applied to the lamella in single domain state. First, a voltage pulse of −55 V is applied, creating a domain wall that is moved from the right to the left, stoppingat the surface step. Then, a second voltage pulse of +32 V is applied, creating a new domain wall and moving it to the base of the wedge. Dark arrows in the PFM image indicate the polarization direction, and bright arrows indicate domainwall movement. Adapted with permission from Whyte and Gregg, Nat. Commun.6, 7361 (2015). Copyright 2015 Springer Nature.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-12 ©A u t h o r ( s )2 0 2 0domain walls can only move in one specific direction, analogously to the operating principle of a conventional diode that only allows current to flow from the anode to cathode. In conclusion, the examples in Figs. 9 and 10reflect the diverse application opportunities of combining FIB nanostructur-ing and SEM imaging in the field of domain wall nanoelectronics. Applications range from the characterization of the transport behavior at individual domain walls to the creation of device-relevant geometries that allow for precise control of domain wallmotions. At present, however, we are just beginning to explore allthe different nanofabrication capabilities and related opportunities for the research on functional domain walls. We anticipate that FIB-SEM will play an important role in the future for facilitatingproof-of-concept devices such as memory cells, domain wall enabledmemristors, and FE-RAM with domain-wall based read-out. IV. CORRELATED SEM AND ATOM PROBE TOMOGRAPHY INVESTIGATIONS OF INTERFACES In this last part of the Tutorial, we will go beyond conven- tional and rather well-established research directions and discusspossible future opportunities for the studies of ferroelectric domainwalls, arising from the combination of FIB-SEM and atom probetomography (APT). We consider this combination a particularly promising example as it allows for characterizing domain walls in 3D down to the atomic scale and with highly sensitive element-specific compositional analysis. 104–108This capability facilitates unprecedented insight into the chemical composition of domainwalls and their interaction with point defects, which may lead to important breakthroughs in the field of domain wall nanoelec- tronics, likely pushing the state of the art in the years to come. A. Atom probe tomography The basic function of an APT instrument is to field evaporate materials atom-by-atom, which are then detected using a time-of-flight method so that the atoms can be identified (i.e., chemically labelled) and back projected onto a virtual specimen to build a 3D model. For details about the setup and the general working princi-ple of APT we refer the reader to, for example, the textbook byGault et al. 106Here, we will restrict ourselves to a short summary of key parameters, focusing on the added value of combined FIB-SEM and APT for the study of ferroelectric domain walls. To evaporate atoms, ideally “one at a time ”, a strong electric field is needed ( ∼1010Vm−1), which is achieved by the combina- tion of a positive high voltage source, between 2 and 12 kV, and aspecific shape of the sample under investigation: in APT, the sample usually has the shape of a sharp needle with tip radius of 50 to 150 nm. By measuring the (x, y)-coordinates for each atom ata given position, i.e., the position of the 2D detector (microchannelplate and delay-lines), the original position of the atom on thespecimen surface before evaporation can be deduced considering its trajectory in the applied electrostatic field. The third coordinate is then calculated based on the tip geometry and the ionic volumeresulting in a full 3D measurement of the atomic positions. To alsoobtain the time-of-fligh t, which is directly related to the mass-to-charge ratio of the detected atom/ion, laser pulses (or voltage pulses) are applied to the specimen tip, which controls the time of departure,making the APT method element specific. In modern APT instru- ments, up to 80% of the evaporated atoms are detected and for every single atom the mass-to-charge ratio can be determined.Studying materials in this atom-by-atom fashion results in out-standing chemical sensitivity which, combined with the sub-nanometer spatial resolution, puts APT in a unique position for 3D nanoscale investigations. 109,110 Originally relying on voltage pulsing, the application of APT used to be restricted to metals, shaped into needles by electropo-lishing. Because of this, the technique has been applied extensivelyto metals for studying the composition of precipitates, dislocations, and grain boundaries. 104,111Today, with the integration of the laser-based evaporation and sample preparation by dual-beam FIB,virtually any material can be investigated, opening the door forAPT studies of ferroelectric domain walls. However, due to thesmall analysis volumes (in the range of 10 7nm3), careful prepara- tion of specimens is key for a successful APT analysis of domain walls.112–115Here, SEM and its capability to image ferroelectric domain walls comes into play (see Sec. II), with FIB allowing extraction of individual walls with nanoscale spatial precision. Asan example of the potential of combined FIB-SEM and APT studies, we briefly discuss the recent work by Xu et al. 118on grain boundaries in oxides in Sec. IV B . B. Applications of APT to interfaces in oxides Analogous to ferroelectric domain walls, it is established that grain boundaries in ionic conductors can exhibit enhanced or reduced conductivity,116representing quasi-2D systems with specific properties different from the homogenous bulk. Theorigin of the anomalous transport behavior at the boundaries isoften attributed to impurity elements accumulating at the grain boundary. 117 Figure 11 presents an APT study of a grain boundary in Sm-doped CeO 2from the recent work by Xu et al.118InFig. 11(a) , a SEM image is shown of the needle specimen before and after the APT analysis. This specimen was extracted from the specific loca- tion of a grain boundary, which was identified with the SEM. Thecorresponding APT analysis is presented in Fig. 11(b) , which shows that Sm is homogeneously distributed in the bulk, with addi-tional impurity elements that are nominally only a few ppm accu- mulated at the boundary. As the APT technique has a sensitivity as low as a few ppm, it can readily detect such small concentrationswell below 0.1 at. %, in addition to their spatial distribution aroundthe grain boundary. The impurity elements where shown to cause a space charge potential at the grain boundary, leading to a depletion of oxygen vacancies. Similar findings have previously been made on dopedSrTiO 3,117,119where grain boundaries of doped samples show a negative potential at the grain boundary core, in contrast to pristinesamples that showed no sign of such electrostatic potential. This combination of SEM and APT led to a breakthrough in under- standing the origin of grain boundary transport properties, and thesame procedure could be applied to study the impact of pointdefects at domain walls, leading to novel insight regarding the nanoscale physics and defect chemistry at functional domain walls in ferroelectrics.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-13 ©A u t h o r ( s )2 0 2 0V. CONCLUSION For about half a century, scanning electron microscopy (SEM) has been used to visualize the domain and domain wall distribution in ferroelectrics, providing a unique opportunity for scale-bridging microscopy studies and continuously covering nano- to mesoscopiclength scales. In addition, the SEM measurement is contract-freeand fast, with typical data acquisition times in the order of a few seconds. On the one hand, it is an advantage of SEM that diverse mechanisms can be exploited to achieve domain and domain wallcontrast, making it applicable to a wide range of ferroelectric mate- rials. On the other hand, this diversity of contrast mechanismsoften makes it difficult to identify the dominant contribution,making the data analysis at ferroelectric domain walls highly non- trivial. At this point, we do not understand the contrast formation well enough, which is reflected by the patchwork-like selection ofmodels that have been proposed to explain observed SEM contrastat neutral and charged domain walls. The development of a comprehensive theory for the contrast formation at ferroelectric FIG. 11. Combining SEM and APT . (a) SEM image of a needle-shaped Sm-doped CeO 2sample, obtained before and after (inset) APT analysis. The SEM data shows how the needle is blunted during the APT analysis, resulting from successive field evaporation of the surface atoms. (b) 3D reconstruction of the evap orated area of the needle in (a). Here, only the impurity ions are shown. The Sm atoms are homogeneously distributed within the specimen. In contrast, at the grain bounda ry located in the center of the needle, an accumulation of various other elements is observed. (c) Cross-sectional data showing the variation in concentration for dif ferent elements across the grain boundary in (b). Adapted with permission from Xu et al., Nat. Mater. 19, 887 (2020). Copyright 2020 Springer Nature.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 128, 191102 (2020); doi: 10.1063/5.0029284 128, 191102-14 ©A u t h o r ( s )2 0 2 0domain walls is highly desirable in order to achieve quantitative insight and fully exploit the benefits offered by SEM. The perhaps biggest advantage of SEM is its outstanding flexi- bility when it comes to correlated studies beyond just imaging.Dual-beam FIB-SEM allows for combining imaging with nano-structuring, correlated microscopy measurements, and in situ switching experiments. In addition, in operando studies are feasible, allowing to study the performance of domain walls in devices anddevice-relevant geometries. Other opportunities that are yet to be explored fully are com- binations with advanced characterization methods such as atom probe tomography (APT), enabling 3D chemical structure analysis at domain walls with unprecedented precision. In addition, theoption to both image and cut in FIB-SEM instruments can be usedto resolve domain wall structures in 3D and with nanoscale spatialaccuracy; setups with micromanipulators are further capable of four-probe transport measurements, providing a pathway to deter- mine the intrinsic conductivity at domain walls. In summary, SEM has shown its value and still has a huge potential for studying ferroelectric domain walls that is yet to beunlocked, and with the field moving closer and closer to first device applications, it is likely that in situ/in operando studies by SEM will play an increasingly important role in the future. DATA AVAILABILITY The data that support the findings of this study are available within the article. ACKNOWLEDGMENTS D.M. acknowledges support by NTNU through the Onsager Fellowship Program and the Outstanding Academic FellowsProgram, and funding from the European Research Council (ERC)under the European Union ’s Horizon 2020 Research and Innovation Programme (Grant Agreement No. 863691). The Research Council of Norway is acknowledged for support to the Norwegian Micro-and Nano-Fabrication Facility, NorFab (Project No. 295864). REFERENCES 1J. Valasek, Phys. Rev. 17, 475 (1921). 2M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials (OUP, 2001). 3S.-W. Cheong and M. Mostovoy, Nat. Mater. 6, 13 (2007). 4T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Nature 426, 55 (2003). 5A. K. Yadav, C. T. Nelson, S. L. Hsu, Z. Hong, J. D. Clarkson, C. M. Schlepütz, A. R. Damodaran, P. Shafer, E. Arenholz, L. R. Dedon, D. Chen, A. Vishwanath, A. M. Minor, L. Q. Chen, J. F. Scott, L. W. Martin, and R. 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1.1868057.pdf	Precessional switching on exchange biased patterned magnetic media with perpendicular anisotropy M. Belmeguenai, T. Devolder, and C. Chappert Citation: Journal of Applied Physics 97, 083903 (2005); doi: 10.1063/1.1868057 View online: http://dx.doi.org/10.1063/1.1868057 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Recording performance and thermal stability in perpendicular media with enhancement of grain isolation as well as magnetic anisotropy field J. Appl. Phys. 111, 07B705 (2012); 10.1063/1.3677307 Critical fields and pulse durations for precessional switching of perpendicular media J. Appl. Phys. 97, 10E509 (2005); 10.1063/1.1849531 Magnetic and recording characteristics of perpendicular magnetic media with different anisotropy orientation dispersions J. Appl. Phys. 97, 10N503 (2005); 10.1063/1.1847911 Switching speed limitations in perpendicular magnetic recording media J. Appl. Phys. 93, 6199 (2003); 10.1063/1.1567801 Recording performance and magnetization switching of CoTb/CoCrPt composite perpendicular media J. Appl. Phys. 91, 8058 (2002); 10.1063/1.1452273 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.159.70.209 On: Sun, 14 Dec 2014 15:36:47Precessional switching on exchange biased patterned magnetic media with perpendicular anisotropy M. Belmeguenai,a!T. Devolder, and C. Chappert Institut d’Electronique Fondamentale, UMR CNRS 8622, Bâtiment 220, Université Paris-Sud, 91405 Orsay, France sReceived 23 July 2004; accepted 11 January 2005; published online 5 April 2005 d We propose to use an in-plane exchange bias ﬁeld to assist the applied ﬁeld to obtain precessional switching of the magnetization in a high perpendicular anisotropy media. Our calculation is madein the limit of a nondamped macrospin particle. From the energy conservation, we derive themagnetic trajectories for any applied ﬁeld magnitude and orientation. Precessional switching isshown to occur only for sufﬁciently large but not too high applied ﬁelds. The writing windowassociated with this switching ﬁeld interval enlarges when the exchange bias ﬁeld increases, ascalculated quantitatively in the case of perpendicular recording. The potential application of thisconcept to magnetic recording is discussed. Compared to conventional media, the switching ﬁelddistribution will be narrowed because the exchange bias provides a better immunity to texturedispersion.Sinceexchangebiasreducessigniﬁcantlythereversalﬁeld,wecanusehigheranisotropymaterials and enhance the areal density while preserving the thermal stability. © 2005 American Institute of Physics .fDOI: 10.1063/1.1868057 g I. INTRODUCTION Obtaining subnanosecond magnetization switching of submicron nanostructures is a key technological issue for theapplications of nanomagnetism, and especially for perpen-dicular magnetic recording. This technology is based on thehigh anisotropy media required for thermally stable ultrahighdensity storage. 1The main remaining difﬁculties of perpen- dicular recording are the low switching speed2and the poor signal-to-noise ratio sSNR dresulting from a large distribu- tion of switching ﬁeld inside the media.3Despite the need for faster data rates, little work has been dedicated to the fastdynamics of these media 4–6in contrast with the case of soft magnetic materials. In order to get better SNR and narrower switching ﬁeld distribution, the case of tilted anisotropy media wassuggested. 3,6–8This could bring a signiﬁcant increase in SNR, hence, in the areal density together with a faster mag-netization switching than perpendicular media. 6However, a critical issue is the manufacturability of such tilted aniso-tropy media. 3Another alternative to gain a better SNR is to redesign the recording head so that the write ﬁeld is angledwith respect to the perpendicular direction. 9The main draw- back of this proposed design is that shielding will be re-quired, in both the down and cross-track directions, to avoidthat the ﬁeld causes multipass erasure. 3Thus, none of the earlier approaches completely solves the mentioned issues. Patterned magnetic media is one proposed approaches for extending magnetic storage densities beyond the limit setby thermal decay for conventional granular media. Thispromising technology, where real densities as high as200 Gbits/in 2have been demonstrated, dramatically reduces jitter and improves SNR.10Therefore, in this article, we show that fast precessionnal magnetization reversal of highperpendicular anisotropy nanostructures can be achieved at reduced applied ﬁeld by polarizing the media using an in-plane exchange bias ﬁeld. The exchange bias ﬁeld lowers theswitching ﬁeld so that high anisotropy materials can be usedto enhance thermal stability, at the same cost in writing ﬁeld. Some aspects of this exchange biasing strategy have been partly presented elsewhere, notably in the case of pre-cessional dynamics triggered by in-plane applied ﬁeld. 11,12 We have underlined that a beneﬁt is obtained on the switch-ing ﬁeld and writing time. 11However, this case restricted to toggle switching and does not allow direct overwrite. The aim of this article is to study the general case of any applied ﬁeld orientation, and to determine the orientationsthat allow direct overwrite. The article is organized as fol-lows: we ﬁrst deﬁne our model and derive the analyticalexpressions of magnetization trajectory sSec. II d. These are obtained from the energy conservation in the case of negli-gible damping and for variable ﬁeld magnitude. Then wediscuss these trajectories and the effect of the exchange biason the switching ﬁeld for applied ﬁelds collinear to the per-pendicular easy axis sSec. III A d. We also describe the mini- mum reversal ﬁeld for various orientations of the appliedﬁeld sSec. III B d. The effect of the damping on the obtained results in the previous sections is also discussed sSec. IV d. Finally, we show that the storage density can be enhancedwhile maintaining the thermal stability.Applications to mag-netic recording are ﬁnally discussed sSec. V d. II. MAGNETIZATION DYNAMICS AND TRAJECTORY ALGEBRA Magnetization dynamics is described by the well-known Landau–Lifshitz–Gilbert equation13 adElectronic mail: mohamed.belmeguenai@ief.u-psud.frJOURNAL OF APPLIED PHYSICS 97, 083903 s2005 d 0021-8979/2005/97 ~8!/083903/6/$22.50 © 2005 American Institute of Physics 97, 083903-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.159.70.209 On: Sun, 14 Dec 2014 15:36:47dM dt=g0HeffˆM−a iMiSdM dt3MD, s1d where g0=0.221 MHz/ sAm−1dis the gyromagnetic factor, andais the damping parameter. The magnetization is sup- posed uniform and given by M=Msm, whereMsis the spon- taneous magnetization and mis a unit vector. Heffis the effective ﬁeld, comprising the applied ﬁeld H, the anisotropy ﬁeldHk, the demagnetizing ﬁeld HD, and an exchange bias ﬁeldHexcalong the sxdaxis. This exchange bias ﬁeld can for instance arises from a properly ﬁeld annealed antiferromag- netic layer adjacent to the magnetic layer of interest. We consider that the easy anisotropy axis is along szd. We write Hkeff=sHk−Msdthe amplitude of the effective an- isotropy ﬁeld and we consider a macrospin approximation sthe intrinsic contribution to the exchange ﬁeld is zero d.I n the following, lowercase letters are used to indicate magne-tization and ﬁelds normalized to M s. For instance, hkeff =Hkeff/Ms.A tt=0, the system is at equilibrium with mz.0. A ﬁeld pulse is applied in the sxzdplane at an angle uwith thes+zdorientation sFig. 1 d. Standard media for perpendicular magnetic recording typically exhibit Ms=300 kA/m8,14andHkhigher than 800 kA/m. In terms of exchange biasing, exchange bias ﬁeldof about 27 kA/m have been reached at room temperature,on a fPts20 Å d/Cos4Ådg 5/Pt/s2Åd/FeMn s130 Å d multilayer with perpendicular anisotropy.15The damping constant of CoPt systems is typically in the range of0.02–0.1. 4,16 Here, we restrict to the case hkeff.hexc. When the applied ﬁeld is fast rising and has a short duration, the magnetizationswitching is dominated by precession. Since our primarygoal is to present simple and indicative solution in that fast-rising limit, we shall neglect the damping in Eq. s1d, in con- trast with other studies. 17 The system energy per unity of volume is given by E m0Ms2=−mxshexc+hsinud−hkeff 2mz2−hmzcosu. s2d In the equilibrium state the total energy fEq.s2dgis mini- mum, which givesmx=hexc hkeff,my=0 andmz=±˛1−hexc2 hkeff2. s3d The initial state corresponds to a positive value of mz. Now let us derive the expression of the magnetization trajectories. Using the energy conservation and the initialconditions fEq.s3dg, the trajectory in the plane sxzdis di- rectly obtained from Eq. s2d. We get m x=−1 hexc+hsinuFhkeff 2mz2+hcossudmz−hkeff 2+hexc2 2hkeff −hcossud˛1−hexc2 hkeff2G+hexc hkeff. s4d Injecting the magnetization norm invariance smx2+my2 +mz2=1din Eq. s4d, we obtain the trajectory in the syzdplane my2=1 shexc+hsinud2H−hkeff2 4mz4−hhkeffcossudmz3 −Sh2cos2u−hkeff2 2+hexc2 2−hhkeff 3cossud˛1−hexc2 hkeff2+hshexc+hsinudsinsudDmz2 −2hcosuF−hkeff 2+hexc2 2hkeff−hcossud 3˛1−hexc2 hkeff2−hexc hkeffshexc+hsinudmzG+cJ,s5d where c=2hcosuF−hkeff 2+hexc2 2hkeff−hexc hkeffshexc+hsinudG 3˛1−hexc2 hkeff2−S1−hexc2 hkeff2DFh2cos2u+hkeff2 4−hexc2 4 −hshexc+hsinudsinuG. In the same manner the trajectories in sxydplane can be obtained. III. MAGNETIZATION TRAJECTORIES In magnetic recording systems, practical write heads generate ﬁelds either in the ﬁlm plane su=±90° dor out of the ﬁlm plane su=0° and u=180° d. The switching of ex- change biased perpendicular magnetic anisotropy ﬁlms wasstudied by means of in-plane ﬁelds elsewhere. 11We here focus ﬁrst on the case of perpendicular applied ﬁeld hsu =180° and u=0°d. The magnetization trajectory to any inter- mediate applied ﬁeld will be studied later sSec. III B din analogy with the representative case of u=180°. FIG. 1. Field geometry used in calculation. uis the angle between the applied ﬁeld Hand thezaxis.Hexc,Hk, andHDare, respectively, the exchange bias, the anisotropy, and the demagnetizing ﬁelds.083903-2 Belmeguenai, Devolder, and Chappert J. Appl. Phys. 97, 083903 ~2005 ! [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.159.70.209 On: Sun, 14 Dec 2014 15:36:47A. Field collinear with the perpendicular easy axis Some representative magnetization trajectories are plot- ted in Fig. 2 for hexc=0.5hkeffand for u=180°. For small values of h, the trajectory is a small elliptic oscillation starting from the initial state and precessingaround the in ﬁeld equilibrium position. Two energy degen-erate lobes are present fFigs. 2 sbdand 2 scd: curves 1 and 1 8g but only one scurve 1 dis gone through due to the initial conditions. The trajectory is anisotropy dominated and themagnetization stays in the positive s+zdhalf space fFig. 2scdg. Such a low ﬁeld does not permit switching. When increasing the ﬁeld above the so-called minimal switching ﬁeld sh swmindthe small and large lobes get connected by a bifurcation point fFig. 2 scdcurve 2 g, so that magnetiza- tion can pass from one lobe to the other and get effectivelyswitched from m z.0t omz,0. For an applied ﬁeld slightly above hswmin, the system un- dergoes a high amplitude precessional motion passingthrough the positive s+zdand the negative s−zdhalf spaces following a pear like trajectory fFig. 2 scd: curve 3 g. For larger ﬁeld, the excursion of the trajectory in the negative half space s−zddecreases with the applied ﬁeld strength.Above h=h swmax, the magnetization cannot switch be- cause the trajectory does not reach any longer the mz,0 half space fFig. 2 scdcurve 4 g. This is due to our choice of a=0, so the system cannot dissipate its high Zeeman energy andnever reaches the negative half space s−zd. Therefore, the magnetization precesses mostly around the applied ﬁeld, i.e., along szd. For an applied ﬁeld along s+zddirection, the magnetiza- tion always stays in positive s+zdhalf space and no switching occurs snot shown here d. Therefore, if h swmin,h,hswmaxthe ﬁ- nal state depends on the applied ﬁeld direction fhis+zdor his−zdgand on its duration and not on the initial magnetiza- tion state. Direct overwrite can be achieved systematically provided that hswmin,h,hswmaxandu=0° or 180°.We point out that having hswmin,h,hswmaxforu=180° is only necessary and not sufﬁcient to switch the magnetization. In this case themagnetization reversal depends also on the pulse ﬁeld dura-tion ssee Sec. IV d. The minimal reversal ﬁeld is determined by the lobe merging, while the maximal ﬁeld is determined by the con-ditionz 28=0fsee Fig. 2 scdg. These properties have been used for a numerical estimate of the reversal ﬁeld marginfh swmin,hswmaxg, plotted in Fig. 3. This margin vanishes shswmin =hswmaxdforhexcł0.11hkeffand enlarges monotonously for higher exchange coupling. For hexcless than 0.11 hkeff, switching is not possible with zero damping. The minimal reversal ﬁeld can also be derived based on the energy surface model.18An example illustrating this pro- cedure where analytic solution was possible is represented inRef. 11 for the case of u=90°. B. Arbitrary applied ﬁeld orientation Our aim now is to determine the minimal applied ﬁeld to switchmzfrommz.0t omz,0 for variable applied ﬁeld orientations. Two cases have to be distinguished when deter-mining the switching ﬁeld:1. 90° <u<270° For an applied ﬁeld in the negative s−zdhalf space s90°,u,270° d, the trajectories are similar in shape and in FIG. 2. Magnetization trajectories for exchange bias ﬁeld hexc=0.5hkeffand for an applied ﬁeld hcollinear with the zaxis su=180° d.sadMagnetization trajectories in sxzdplane. sbdTrajectories in sxydplane and scdinsyzdplane. Stars s*dand cross s1dindicate, respectively, the initial position and the degenerate initial position. The bold lines indicate the not visited part of thetrajectories.083903-3 Belmeguenai, Devolder, and Chappert J. Appl. Phys. 97, 083903 ~2005 ! [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.159.70.209 On: Sun, 14 Dec 2014 15:36:47characteristics to those shown in Fig. 2. In this case, magne- tization switching occurs when the small and large lobes getconnected by a bifurcation point fidem Fig. 2 scd2gfor a ﬁeld value equal to h swmin. The corresponding ﬁeld value hswminis obtained by solving Eq. s5dnumerically with respect to mz, for given ﬁeld and uvalues, to ﬁnd the mzextrema fz1,z2, z18, andz28in Fig. 2 scdg. This process is repeated while in- creasing the applied ﬁeld huntil we get z2=z18and 0 łmx2 ł1. 2. 0° <u<90° and 270° <u<360° For an applied ﬁeld in the positive s+zdhalf space s0° ,u,90° and 270° ,u,360° d, the switching depends on the applied ﬁeld orientation. Switching and no switching ar-eas, which are separated by critical orientations u1andu2, respectively, for 0° ,u,90° and 270° ,u,360°, can be differentiated. Foru2,u,u1, the magnetization always stays in posi- tives+zdhalf space and no switching occurs. Foru1,u,90°, the trajectory is a large elliptic preces- sion starting from the initial state and trying to join the otherlobe which can take place in the m z,0 half space. This lobe disappears for uvalues far from 90°. Here it is important to give the switching ﬁeld expression and not the condition ofthe existence of the lobe in the negative half space. Theconnection between the two lobes occurs in the m zł0 half space. Therefore, the hswminrequired is obtained from Eq. s4d by putting mz=0 andmx=1fEq.s6dg. Figure 4 shows a typi- cal magnetization trajectory for u=70°,hexc/hkeff=0.5 and for an applied ﬁeld equal to hswmingiven by Eq. s6d. For this ﬁeld value, magnetization passes by mz=0 andmx=1 and switch- ing is possible hswmin=1 2hkeff˛S1−hexc hkeffD3 ˛1−hexc hkeffsinu−˛1+hexc hkeffcosu. s6dFor −90° ,u,u2the trajectory is very similar to that of the case of u1,u,90°. The switching ﬁeld hswminis obtained fEq.s7dgby putting mz=0 andmx=−1 in Eq. s4d: hswmin=−1 2hkeff˛S1+hexc hkeffD3 ˛1+hexc hkeffsinu+˛1−hexc hkeffcosu. s7d Figure 5 shows the polar representation of hswmin, obtained numerically using the method described in case 1 and usingEq.s6d. These numerical results show that for all the ex- change ﬁeld values the minimal switching ﬁeld is obtainedfor uaround 135°. The plots in Fig. 5 are compared to the switching ﬁeld in Stoner–Wohlfarth model.19This clearly shows that the exchange ﬁeld signiﬁcantly lowers the switch-ing ﬁeld. For example, for u=180° and hexc=0.25hkeff, the reversal ﬁeld is equal to 42% of its value with zero exchangeﬁeld sh swmin=hkeffd. Moreover, our exchange bias strategy offers a signiﬁcantly better immunity to the texture dispersion of a typical media since u]hswmin/]uuu=180°hexcÞ0 FIG. 3. Minimal shswmindand maximal hswmaxswitching ﬁelds vs the normalized exchange bias ﬁeld hexc/hkefffor an applied ﬁeld collinear with the zaxis su=180° d. FIG. 4. Magnetization trajectory for exchange bias ﬁeld hexc=0.5hkeffand applied ﬁeld h=0.719876 hkeffwith an angle u=70° with the zaxis. FIG. 5. Polar representation of the minimal switching ﬁelds for two differ- ent values of the exchange bias ﬁeld hexc. The bold line represents the Stoner–Wohlfarth astroid. The dashed and dotted lines indicate the no switching limit su1dforhexc=0.5hkeffand 0.25 hkeff, respectively. The scale in the left indicates circles radius shswmin/hkeffmagnitudes d.083903-4 Belmeguenai, Devolder, and Chappert J. Appl. Phys. 97, 083903 ~2005 ! [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.159.70.209 On: Sun, 14 Dec 2014 15:36:47au]hswmin/]uuu=180°Stoner–WohlfarthsFig. 5 d. For example, the switch- ing ﬁeldhswminvaries only from 0.41 hkeffto 0.34hkeffsi.e., 17% d when uchanges from 180° to 170° for hexc=0.5hkeffand from hkeffto 0.673hkeffsi.e., 32% dfor the Stoner–Wohlfarth model. For practical reasons, we studied in more details the case of an applied ﬁeld along u=180°. According to Fig. 2 scd, hswminis greater than the ﬁeld given by Eq. s6dbecause the trajectories get connected in the positive half space s+zd. Therefore, we used Eq. s6dto obtain an analytical expression fEq.s8dg, which is an underestimate for minimal switching ﬁeld sFig. 6 d: hswmin hkeffø0.7˛s1−hexc/hkeffd3 1+hexc/hkeffforu=180° . s8d Comparing to the case of in-plane-applied ﬁeld, we observe that, forhexc.0.3hkeff, negative easy axis applied ﬁeld allows switching magnetization with less ﬁeld compared ssee Fig. 6 d to the in-plane applied case.11 IV. EFFECT OF THE DAMPING The aim of this section is to examine the effect of the damping on the earlier-obtained results. Figure 7 shows OOMMF simulation smacrospin case d for the magnetization trajectory in yzplane during the ﬁrst precession period with a=0.003. It is obtained for an applied ﬁeldHmaking an angle u=180° with the normal and for m0Hexc=0.15Hkeff=0.21 T. This magnetization trajectory is very similar to the corresponding one obtained with ourmodel in case of a=0. It is apparent that our conclusions are still valid despite the non-negligible damping. This ﬁgure shows that the maximal excursion of mzin the negative half space gets reduced as the applied ﬁeld in-creases and thus conﬁrms the existence of a ﬁeld intervalfh swmin,hswmaxgfor the switching. In this case the minimal shswmind and the maximal shswmaxdswitching ﬁelds are increased respec- tively by 6% and 4% while the ﬁeld margin fhswmin,hswmaxgis decreased by 10% compared to those obtained in case of a =0.As indicated earlier, the cases of u=90° or u=180° are the most practical for the magnetic recording technology. For u=90°, the two values of mzgiven by Eq. s3dare the in ﬁeld maxima of mz. Therefore, the ringing can be suppressed if the ﬁeld pulse is switched off once mzis near its maxima sprecessional ballistic switching d. This is obtained if we ap- ply a ﬁeld pulse of duration TPequal to the half of the pre- cession period stime that mzneeds to go from one maximum to other d.11 For u=180°,Eist=0+,mx=hexc/hkeff,mz=˛1−hexc2/hkeff2d ÞEst,mx=hexc/hkeff,mz=−˛1−hexc2/hkeff2dsEiis the initial en- ergy just after switching on the ﬁeld dwhatever tsa=0dand thusmznever reaches its equilibrium position in the negative half space smz=−˛1−hexc2/hkeff2d. Therefore, no precessional ballistic switching is possible and the ringing cannot be to- tally suppressed. In this case, the magnetization ﬁnal statedepends on the ﬁeld pulse duration and having m z,0 is not sufﬁcient for magnetization reversal after switching off theﬁeld pulse. The switching is obtained by precise choice ofthe ﬁeld pulse duration and larger is afew are the oscilla- tions sringing dafter switching off the ﬁeld and shorter is the reversal time. V. THERMAL STABILITY This section is devoted to discuss the consequences of our previous results on the potential use of the exchange biasfor magnetic recording. Indeed, the exchange bias ﬁeld de-creases the switching ﬁeld which may be detrimental to thethermal stability of the magnetization, a crucial criterion formagnetic storage applications. To estimate this effect, wecompare the energy barrier DEthat separates the two stable remnant states, with the 10 years stability requirements ie:DEV.40k BT,20whereT=300 K, kBis Boltzmann’s constant andVis the bit volume. This barrier energy per unit volume DEis given by FIG. 6. Switching ﬁeld as function of exchange ﬁeld for an applied ﬁeld in sxzdplane with u=180° sout-of-plane applied ﬁeld dand u=90° sin-plane applied ﬁeld dssee Ref. 11 d. For the out-of-plane applied ﬁeld, the numerical values are underestimated by Eq. s8d. The dashed line indicates the hexcfor which the reversal ﬁeld margin fhswmin,hswmaxgvanishes in the case of out-of- plane applied ﬁeld. FIG. 7. OOMMF simulation of mmagnetization trajectory in yzplane ob- tained for a=0.003, an exchange bias ﬁeld m0Hex=0.15Hkeff=0.21 T, Ms =298 kA/m and for an applied ﬁeld making an angle u=180° with the normal. The applied m0H=0.93 T is the maximal switching ﬁeld for a=0.083903-5 Belmeguenai, Devolder, and Chappert J. Appl. Phys. 97, 083903 ~2005 ! [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.159.70.209 On: Sun, 14 Dec 2014 15:36:47DE=m0Ms2 2hkeffF1−hexc hkeffS2−hexc hkeffDG. s9d It is the total energy difference, at zero applied ﬁeld, sDE =uEuh=0mx=1,mz=0−uEuh=0inidbetween the maximal sobtained for mx=1,my=0, andmz=0dand the minimal sinitial denergies uEuh=0inicorresponding to initial magnetization given by Eq. s2d. Only the case of u=pis analysed in this section. We shall illustrate this through a numerical example using a headﬁeldH head=1114 kA/m.21To gain in the areal density in perpendicular recording, we try to maximize DE, while keep- ing the writing possible sHswmin,Hheadd, withHexcandHkeffas free parameters. Assuming cubic bits with volume V=D3, this offers stable bits down to sizes D=s40kBT/DEd1/3. Using Eq. s8d andHswmin=Hhead, the bit sizes are reported in Table I for sev- eral exchange ﬁeld values and for Ms=298 kA/m.22The in- crease in stable bit size Dcorresponds to our increase of 24% in areal density for optimum hexc=0.3hkeff. This value of the exchange bias is, however, higher than the demonstrated val-ues at room temperature. Therefore, in practical cases thehigherH excshexcł0.3hkeffd, the higher will be the beneﬁt for perpendicular recording. VI. CONCLUSIONS In summary, our strategy can use the effect of exchange biasing on switching ﬁeld to increase the recording arealdensities, or decrease the switching ﬁeld. This can beachieved by going to higher anisotropy materials provided we can generate high exchange bias ﬁelds. The angular de-pendence of the switching ﬁeld showed that the minimalswitching ﬁeld is obtained for 135° between the applied ﬁeldands+ozd. Moreover, another interest of this strategy is the precessional mode that can lead to a very fast switching, as will be studied in a forthcoming paper. This may ﬁnd appli-cation to magnetic storage. ACKNOWLEDGMENT The work is supported in part by the European Commu- nities Human Potential programme under Contract No.HRPN-CT-2002-00318 ULTRASWITCH. 1D. A. Thompson, J. Magn. Soc. Jpn. 21,9s1997 d. 2Q. Peng and H. Bertram, J. Appl. Phys. 81, 4384 s1997 d. 3K. Z. Gao and H. N. Bertram, IEEE Trans. Magn. 38, 3675 s2002 d. 4C. H. Back and H. Siegmann, J. Magn. Magn. Mater. 200,7 7 4 s1999 d. 5A. Lyberatos, J. Appl. Phys. 93, 6199 s2003 d. 6Y. Y. Zou, J. P. Wang, C. H. Hee, and T. C. Chong, Appl. Phys. Lett. 82, 2473 s2003 d. 7C. H. Chee, Y. Y. Zou, and J. P. Wang, J. Appl. Phys. 91, 8004 s2002 d. 8K. Z. Gao and H. N. Bertram, IEEE Trans. Magn. 39,7 0 4 s2003 d. 9M. Mallary, A. Torabi, and M. Benakli, IEEETrans. Magn. 36,3 6s2000 d. 10M. Albrecht, C. T. Rettner, A. Moser, M. E. Best, and B. D. Terris, Appl. Phys. Lett. 81, 8275 s2002 d. 11T. Devolder and C. Chappert, J. Phys. D 36,3 1 1 5 s2003 d. 12T. Devolder, M. Belmeguenai, C. Chappert, H. Bernas, and Y. Suzuki, Mater. Res. Soc. Symp. Proc. 777, T6.4 s2003 d. 13L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8,1 5 3 s1953 d;T .L . Gilbert, Phys. Rev. 100, 1243 s1955 d. 14H. N. Bertram and M. Williams, IEEE Trans. Magn. 36,4s2000 d. 15F. Garcia, J. Sort, B. Rodmacq, S. Auffert, and B. Dieny, Appl. Phys. Lett. 83,3 5 3 7 s2003 d. 16A. Misra, P. B. Visscher, and D. M. Apalkov, J. Appl. Phys. 94, 6013 s2003 d. 17G. Bertotti, I. Mayergoyz, C. Serpico, and M. Dimian, J. Appl. Phys. 93, 6811 s2003 d. 18K. Z. Gao, E. Boerner, and H. N. Bertram, J.Appl. Phys. 93, 6549 s2003 d. 19E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. London 240,7 4 s1948 d. 20D. Weller and A. Moser, IEEE Trans. Magn. 35, 4423 s1999 d. 21F. Liuet al., IEEE Trans. Magn. 38, 1647 s2002 d. 22D. Weller et al., IEEE Trans. Magn. 36,1 0 s2000 d.TABLE I. Minimal stable grain diameter Dfor different magnitudes of exchange ﬁeld Hexcand anisotropy ﬁeld Hkeff. The media is writable with a head ﬁeld of Hswmin=1114 kA/m oriented at u=180°. HkeffskA/m d1114 2020 2343 2712 3142 4269 Hexc/Hkeff0 0.15 0.2 0.25 0.30 0.4 HswminskA/m d1114 1114 1114 1114 1114 1114 Dsnmd 9.26 8.46 8.38 8.34 8.31 8.32083903-6 Belmeguenai, Devolder, and Chappert J. Appl. Phys. 97, 083903 ~2005 ! [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.159.70.209 On: Sun, 14 Dec 2014 15:36:47
1.5066573.pdf	J. Chem. Phys. 150, 084701 (2019); https://doi.org/10.1063/1.5066573 150, 084701 © 2019 Author(s).Structural variation of anatase (101) under near infrared irradiations monitored by sum-frequency surface phonon spectroscopy Cite as: J. Chem. Phys. 150, 084701 (2019); https://doi.org/10.1063/1.5066573 Submitted: 15 October 2018 . Accepted: 18 January 2019 . Published Online: 27 February 2019 Xinyi Liu , Tao Zhou , and Wei-Tao Liu ARTICLES YOU MAY BE INTERESTED IN Development of ultrafast broadband electronic sum frequency generation for charge dynamics at surfaces and interfaces The Journal of Chemical Physics 150, 024708 (2019); https://doi.org/10.1063/1.5063458 Water structure at the interface of alcohol monolayers as determined by molecular dynamics simulations and computational vibrational sum-frequency generation spectroscopy The Journal of Chemical Physics 150, 034701 (2019); https://doi.org/10.1063/1.5072754 Surface van der Waals forces in a nutshell The Journal of Chemical Physics 150, 081101 (2019); https://doi.org/10.1063/1.5089019The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Structural variation of anatase (101) under near infrared irradiations monitored by sum-frequency surface phonon spectroscopy Cite as: J. Chem. Phys. 150, 084701 (2019); doi: 10.1063/1.5066573 Submitted: 15 October 2018 •Accepted: 18 January 2019 • Published Online: 27 February 2019 Xinyi Liu, Tao Zhou, and Wei-Tao Liua) AFFILIATIONS State Key Laboratory of Surface Physics, Key Laboratory of Micro and Nano Photonic Structures (MOE) and Department of Physics, Fudan University, Shanghai 200433, China Note: This article is part of the Special Topic “Nonlinear Spectroscopy and Interfacial Structure and Dynamics” in J. Chem. Phys. a)Author to whom correspondence should be addressed: wtliu@fudan.edu.cn ABSTRACT We probed the anatase (101) surface irradiated by near-infrared and infrared photons in different ambient gases by monitoring the surface lattice phonon mode using sum-frequency spectroscopy. We found that even under the irradiation of such low energy photons, the stability of surface oxygen vacancies, in comparison to sub-surface oxygen vacancies, can increase sensibly. The variation of this surface phonon mode is also in accordance with the photo-induced hydrophilicity of titanium oxide surfaces, which may provide the microscopic insight into this phenomenon. Published under license by AIP Publishing. https://doi.org/10.1063/1.5066573 I. INTRODUCTION Titanium dioxides (TiO 2) are among the most promising and most studied photocatalytic materials exhibiting a wide range of exciting properties ranging from the photocatalytic hydrogen production to dye-sensitized solar cells.1–6One important application of TiO 2is on the self-cleaning coating,7 which has been commercialized based on the photo-induced hydrophilicity discovered by Fujishima and co-workers in 1997.8Under the illumination of ultraviolet (uv) light, the TiO 2surface could turn from slightly hydrophobic to highly hydrophilic, and the reversal process happened by stock- ing the TiO 2surface in darkness for a few days. This phe- nomenon has aroused much interest and received intensive studies afterwards.9–11However, there still remain controver- sies on its microscopic mechanism, particularly on whether it is induced by the removal of organic adsorbates9or change in the TiO 2surface itself, for example, the creation of sur- face oxygen vacancies.12For the latter, it was usually consid- ered to only occur under the illumination of high energy uv photons; yet, self-cleaning coatings often function in ambientenvironment without much uv dosage,13which apparently contradict the vacancy scenario. One major difﬁculty in studying this phenomenon is that the reaction occurs under ambient conditions. Regarding that, the surface speciﬁc sum-frequency generation (SFG) is arguably the most viable technique to study surface reactions in ambient and yield molecular-level information. Most exist- ing SFG studies on TiO 2surfaces have focused on behaviors of adsorbates.14–19 Recently, we have further applied SFG to probe surface phonon mode of TiO 2, which can yield struc- tural information of surface lattices under reaction condi- tions.20In this study, by monitoring the surface phonon mode of the anatase (101) surface, we show that the relative sta- bility of surface oxygen vacancies can increase even under the illumination of low energy near infrared (NIR) or infrared photons. The sub-surface oxygen vacancies can thus migrate to the topmost surface lattice, increasing the density of sur- face oxygen vacancies and possibly the surface reactivity as well.20,21 This is most likely due to the photodoping of the sur- face conduction band through mid-gap defect absorption.21 Moreover, the evolution of the surface phonon mode, or J. Chem. Phys. 150, 084701 (2019); doi: 10.1063/1.5066573 150, 084701-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp density of the surface oxygen vacancies, is closely correlated with that of the photo-induced hydrophilicity, which may help us to gain insight into the phenomena at the microscopic level. II. THEORY AND EXPERIMENTAL DETAILS The basic theory of SFG is described elsewhere.22Brieﬂy, when the IR frequency ( !IR) is near the phonon resonance, the SF signal ( SSF) generated by two incident beams is proportional to$NR+$R2, where$NRis the non-resonant background, and $R=X q$ Aq !IR!q+iq(1) is the resonant contribution, with$ Aq,!q, and qbeing the amplitude, frequency, and damping coefﬁcient of the qth resonance mode, respectively. The experimental geometry is similar to that depicted elsewhere.20The laser system consisted of a Ti:sapphire oscil- lator (MaiTai SP, Spectra Physics), a regenerative Ti:sapphire ampliﬁer (Spitﬁre, Spectra Physics), and an optical parametric ampliﬁer (TOPAS-C, Spectra Physics) followed by a difference frequency generation stage. An ampliﬁer seeded by an oscil- lator was used to produce 4 W of 800 nm, 35 fs pulses at 1 kHz repetition rate. The beam was divided into two parts by a beamsplitter. About 2.6 W of the beam passed through a Bragg ﬁlter (N013-14-A2, OptiGrate), generating narrow- band pulses of 0.5 nm bandwidth. The rest was used to obtain broadband IR pulses ( 200 cm1) centered at about 880 cm1from the optical parametric ampliﬁer and the dif- ference frequency generation stage. The narrowband 800 nm beam of12J/pulse and the IR input beam, tunable from 780 to 980 cm1of1.4J/pulse, overlapped at the sam- ple surface with incident angles of 45and 57, respectively. The incident NIR intensity is 21 W/cm2. The generated SF signal was collected by a spectrograph (Acton SP2300) and recorded on a CCD camera (Princeton Instruments PyLoN 1340100).We bought anatase (101) sample (Fe-doped mineral crys- tal) of 2 mm thick from MaTeck. After re-polishing by Hefei Kejing Materials Technology Co., Ltd., the epi rough- ness reached0.4 nm. Before the measurement, it was cleaned by sonicating in acetone, ethanol, and deionized water (18.2 M cm) consecutively. We then placed it in a chamber purged with pure oxygen, followed by uv-ozone treatment for 10 min to remove organic contamination and redundant defects. During the measurement, the sample was mounted in a chamber with a base pressure <100 Pa. Pure nitrogen and oxygen gas could ﬁll the chamber to different pressures. The NIR and IR beams could be controlled on and off by mechanical shutters separately. All experiments were conducted at room temperature. III. RESULTS AND DISCUSSION Figure 1(a) illustrates the lattice structure of the stoi- chiometric anatase (101) surface. Figure 1(b) presents sum- frequency vibrational spectra from the surface in the spectral range of700-1000 cm1, with the beam polarization com- bination being PPP (referring to P-polarized SF output, P- polarized NIR input, and P-polarized IR input, respectively), and the beam incident plane being parallel to the [10 ¯1]-axis.20 A prominent resonant mode shows up at 850 cm1, which is a surface phonon mode due to the stretching vibration of the bond between the ﬁve-fold surface titanium ion [Ti(5c)] and the three-fold oxygen ion beneath it [O(3c)] [Fig. 1(a)].20 As time elapsed, the mode intensity gradually drops, with the lineshape and the central frequency remained nearly the same [Fig. 1(b)]. Figure 1(c) shows the time variation of the integrated mode intensity, with the initial intensity at time zero (the onset of NIR and IR beams) set as 100%. In our previous work,20we observed a sharp drop of the phonon mode upon uv irradiation that is known to gener- ate surface oxygen vacancies on anatase (101).23The surface oxygen vacancies locate preferentially on 2-fold surface oxy- gen sites [O(2c), Fig. 1(a)]. Since O(2c) are directly bonded to Ti(5c) sites, the generation of surface oxygen vacancies low- ers the coordination of the conjoint Ti(5c) to four-fold, thus FIG. 1. Phonon spectra of anatase (101) and intensity variation under photo illumination. (a) Lattice structure of the anatase (101) surface. Preferential positions for the surface and sub-surface oxygen vacancies are indicated by white circles. (b) Series of SF spectra from anatase (101) under the irradiation of NIR and IR pulses. All spectra are taken in a vacuum with the polarization combination being PPP. The beam polarization plane is parallel to the [10 ¯1]-axis. (c) The intensity variation of the surface phonon mode under continuous NIR and IR irradiations (solid squares and line). The intensity change upon uv irradiation is shown for comparison (dashed line). J. Chem. Phys. 150, 084701 (2019); doi: 10.1063/1.5066573 150, 084701-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp causing the corresponding phonon mode to diminish.20To see whether the same mechanism applies here, we compared the intensity variation in a vacuum to that in pure oxygen (O 2) ambient (6000 Pa) [Fig. 2(a)], and the latter clearly dropped at a much lower rate. We also checked the effect of different ambient gases [Figs. 2(b) and 2(c)]. We recorded the phonon intensity in a vacuum for 15 min, then turned off the inci- dent laser beams, purged the sample with different gases for another 15 min, and recorded the phonon intensity again. It turns out that the intensity can partly recover after the purge of O 2[Fig. 2(b)] but not after nitrogen (N2) [Fig. 2(c)]. Notably, the phonon intensity can com- pletely recover after being purged by ozone (O 3) to the ini- tial level [Fig. 2(c)]. Since both O 3and O 2are able to heal the surface oxygen vacancies,20and O 3has even stronger oxidation power, the measurements above strongly suggest that the phonon intensity drop under NIR and IR irradi- ations is also related to the increment of surface oxygen vacancies. We further examined the effect of incident photon ener- gies. Figure 3(a) shows that by turning off both beams for 15 min in a vacuum, the phonon intensity can recover by about 5%. This recovery is due to that on anatase (101), surface oxy- gen vacancies are not as stable as the sub-surface vacancies in a vacuum, and the former could migrate down to become the latter.21,24 We then turn off the NIR beam, but leave the IR beam on. After 15 min, the phonon intensity again recovered by almost 6% [Fig. 3(b)]. On the other hand, if we turn off the IR beam, but leave the NIR beam on, the intensity did not recover after 15 min [Fig. 3(c)]. The above measurements clearly indi- cate that it was the NIR beam at about 800 nm, or 1.5 eV, that mainly caused the gradual phonon intensity drop during the measurement. Meanwhile, the drop/recovery of the phononmode with/without irradiation has an analogous trend to that of the photo-induced hydrophilicity of titania surfaces, which can be turned on/off by illumination. Now we discuss the mechanism of this effect. Upon uv irradiation, the high energy uv photons can break surface Ti–O bonds and generate surface oxygen vacancies directly [Fig. 3(d), left panel]. However, since the bandgap of anatase is about 3.2 eV,25the 1.5 eV photons cannot directly excite from the oxygen 2p band via one-photon absorptions. To exam- ine whether two-photon effect could occur [Fig. 3(d), right panel], we varied the NIR power (relative to the full power) and recorded the percentage of photon intensity drop per unit time, as shown in Fig. 3(e). On the double logarithmic scale, the power dependence of the signal drop rate has a slope of about 1.2, indicating that the two-photon effect is minor in our case. On the other hand, it is known from previous studies on anatase (101) that there can exist a large amount of sub- surface vacancies underneath the sample surface.24These vacancies, together with other defects, form defect levels inside the anatase bandgap, the NIR irradiation can excite electrons from such defect levels to the bottom of the con- duction band [Fig. 3(d), right panel].13As we pointed out in our previous study,20,21 doping of the conduction band can increase the stability of the structural conﬁguration with sur- face oxygen vacancies. Therefore, though NIR cannot generate surface oxygen vacancies directly, it can dope the anatase and increase their relative stability, and effectively “attract” sub- surface vacancies to the top-most surface layer, and cause the slow drop of the phonon intensity. To further test that our observation is due to the migration of sub-surface vacancies, we stored the sample in an oxygen deﬁcient dark environment for about 14 days, during which period surface oxygen vacan- cies cannot be healed, but could migrate downward to form FIG. 2. (a) Variation of the surface phonon mode intensity in a vacuum (black squares and line) and in pure O 2 (red circles and line). [(b) and (c)] The variation of the surface phonon mode intensity with 15 min purge in darkness by (b) O 2, (c) N 2, and O 3, respectively. All measurements in (b) and (c) were done in a vacuum. J. Chem. Phys. 150, 084701 (2019); doi: 10.1063/1.5066573 150, 084701-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3. The variation of the surface phonon mode intensity with 15 min (a) in darkness, (b) IR irradiation only, and (c) NIR irradiation only. (d) Excitation schematics upon the uv or NIR irradiation. (e) Power dependence of intensity drop and ﬁtting. All measurements were done in a vacuum. sub-surface vacancies. As expected, under the same NIR irra- diance, the phonon intensity dropped more rapidly to about 70% in 30 min, in accordance with an increased amount of sub-surface vacancies. IV. CONCLUSION To conclude, we observed a gradual drop in the surface phonon intensity of the anatase (101) surface upon the con- tinuous irradiation of the NIR and IR photons. We found that the intensity drop was mainly due to the NIR beam, which dope the sample with electrons via defect band absorption and increases the relative stability of surface oxygen vacancies. This shows that even low energy photons can increase the sur- face vacancy densities of anatase surfaces, which may explain the reactivity of self-cleaning coatings (which are mainly of the anatase phase) under indoor/visible light illumination. More- over, the drop of the surface phonon intensity upon irradia- tion, and the recovery of the mode in darkness, has a similar trend with that of the light-induced hydrophilic/hydrophobic transitions of titanium dioxides. Further studies on this subject may reveal the microscopic mechanism of the phenomenon. ACKNOWLEDGMENTS This research was funded by the National Natural Science Foundation of China and the National Basic Research Program of China under Grant Agreement Nos. 2016YFA0300900 and 11622429. W.T.L. acknowledges the support from the National Program for Support of Top-Notch Young Professionals, and “Shuguang Program” supported by Shanghai Education Development Foundation and Shanghai Municipal Education Commission.REFERENCES 1M. Z. Jacobson and M. A. Delucchi, Energy Policy 39(3), 1154 (2011). 2A. Fujishima and K. Honda, Nature 238, 37 (1972). 3T. Kawai and T. Sakata, Nature 286, 474 (1980). 4L. Kavan, M. Grätzel, S. E. Gilbert, C. Klemenz, and H. J. Scheel, J. Am. Chem. Soc. 118(28), 6716 (1996). 5Q. Guo, C. Xu, Z. Ren, W. Yang, Z. Ma, D. Dai, H. Fan, T. K. Minton, and X. Yang, J. Am. Chem. Soc. 134(32), 13366 (2012). 6C. Xu, W. Yang, Q. Guo, D. Dai, M. Chen, and X. Yang, J. Am. Chem. Soc. 136(2), 602 (2014). 7K. Hashimoto, H. Irie, and A. Fujishima, Jpn. J. Appl. Phys. 44(12), 8269 (2005). 8R. Wang, K. Hashimoto, A. Fujishima, M. Chikuni, E. Kojima, A. Kitamura, M. Shimohigoshi, and T. Watanabe, Nature 388, 431 (1997). 9T. Zubkov, D. Stahl, T. L. Thompson, D. Panayotov, O. Diwald, and J. T. Yates, J. Phys. Chem. B 109(32), 15454 (2005). 10Q. F. Xu, Y. Liu, F. Lin, B. Mondal, and A. M. Lyons, ACS Appl. Mater. Interfaces 5(18), 8915 (2013). 11H. Kang, Y. Liu, H. Lai, X. Yu, Z. Cheng, and L. Jiang, ACS Nano 12(2), 1074 (2018). 12R. Wang, K. Hashimoto, A. Fujishima, M. Chikuni, E. Kojima, A. Kitamura, M. Shimohigoshi, and T. Watanabe, Adv. Mater. 10(2), 135 (1999). 13S. Banerjee, D. D. Dionysiou, and S. C. Pillai, Appl. Catal. B: Environ. 176, 396 (2015). 14C. Wang, H. Groenzin, and M. J. Shultz, J. Phys. Chem. B 108(1), 265 (2004). 15C. Wang, H. Groenzin, and M. J. Shultz, J. Am. Chem. Soc. 126(26), 8094 (2004). 16A. Liu, S. Liu, R. Zhang, and Z. Ren, J. Phys. Chem. C 119(41), 23486 (2015). 17S. Liu, A. Liu, B. Wen, R. Zhang, C. Zhou, L. Liu, and Z. Ren, J. Phys. Chem. Lett. 6(16), 3327 (2015). 18J. H. Jang, F. Lydiatt, R. Lindsay, and S. Baldelli, J. Phys. Chem. A 117(29), 6288 (2013). 19S. Kataoka, M. C. Gurau, F. Albertorio, M. A. Holden, S. M. Lim, R. D. Yang, and P. S. Cremer, Langmuir 20(5), 1662 (2004). J. Chem. Phys. 150, 084701 (2019); doi: 10.1063/1.5066573 150, 084701-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 20Y. Cao, S. Chen, Y. Li, Y. Gao, D. Yang, Y. R. Shen, and W. T. Liu, Sci. Adv. 2(9), e1601162 (2016). 21Y. Li and Y. Gao, Phys. Rev. Lett. 112(20), 206101 (2014). 22Y. R. Shen, J. Opt. Soc. Am. B 28(12), A56 (2011). 23S. Mezhenny, P. Maksymovych, T. L. Thompson, O. Diwald, D. Stahl, S. D. Walck, and J. T. Yates, Chem. Phys. Lett. 369(1-2), 152 (2003).24P. Scheiber, M. Fidler, O. Dulub, M. Schmid, U. Diebold, W. Hou, U. Aschauer, and A. Selloni, Phys. Rev. Lett. 109(13), 136103 (2012). 25D. O. Scanlon, C. W. Dunnill, J. Buckeridge, S. A. Shevlin, A. J. Logsdail, S. M. Woodley, C. R. A. Catlow, M. J. Powell, R. G. Palgrave, I. P. Parkin, G. W. Watson, T. W. Keal, P. Sherwood, A. Walsh, and A. A. Sokol, Nat. Mater. 12, 798 (2013). J. Chem. Phys. 150, 084701 (2019); doi: 10.1063/1.5066573 150, 084701-5 Published under license by AIP Publishing
1.5130458.pdf	AIP Advances 9, 125322 (2019); https://doi.org/10.1063/1.5130458 9, 125322 © 2019 Author(s).Precession damping in [Co60Fe40/Pt]5 multilayers with varying magnetic homogeneity investigated with femtosecond laser pulses Cite as: AIP Advances 9, 125322 (2019); https://doi.org/10.1063/1.5130458 Submitted: 03 October 2019 . Accepted: 04 November 2019 . Published Online: 23 December 2019 M. A. B. Tavares , L. H. F. Andrade , M. D. Martins , G. F. M. Gomes , L. E. Fernandez-Outon , and F. M. Matinaga COLLECTIONS Paper published as part of the special topic on 64th Annual Conference on Magnetism and Magnetic Materials Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials. ARTICLES YOU MAY BE INTERESTED IN Stabilization and tuning of perpendicular magnetic anisotropy in room-temperature ferromagnetic transparent CeO 2 films Journal of Applied Physics 126, 183903 (2019); https://doi.org/10.1063/1.5125321 Spin Hall effect investigated by spin Hall magnetoresistance in Pt 100−x Aux/CoFeB systems AIP Advances 9, 125312 (2019); https://doi.org/10.1063/1.5129889 Magnetization dynamics and damping behavior of Co/Ni multilayers with a graded Ta capping layer Journal of Applied Physics 121, 163903 (2017); https://doi.org/10.1063/1.4982163AIP Advances ARTICLE scitation.org/journal/adv Precession damping in [Co 60Fe40/Pt] 5multilayers with varying magnetic homogeneity investigated with femtosecond laser pulses Cite as: AIP Advances 9, 125322 (2019); doi: 10.1063/1.5130458 Presented: 7 November 2019 •Submitted: 3 October 2019 • Accepted: 4 November 2019 •Published Online: 23 December 2019 M. A. B. Tavares,1 L. H. F. Andrade,1,a) M. D. Martins,1G. F. M. Gomes,2,3L. E. Fernandez-Outon,1,3 and F. M. Matinaga3 AFFILIATIONS 1Centro de Desenvolvimento da Tecnologia Nuclear, CDTN, 31270-901 Belo Horizonte, M.G., Brazil 2PPGMCS, UNIMONTES, 39401-089 Montes Claros, M.G., Brazil 3Departamento de Fisica, UFMG, 31270-901 Belo Horizonte, M.G., Brazil Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials. a)Corresponding author. E-mail address: lhfa@cdtn.br (L. H. F. Andrade) ABSTRACT We report on the ultrafast magnetization dynamics of [Co 60Fe40/Pt] 5multilayers studied with femtosecond laser pulses. The samples were grown at room temperature by DC magnetron sputtering with Ta capping and Pt buffer layers and present the same thickness and per- pendicular magnetic anisotropy as determined by vibrating sample magnetometry. Controlled growth rate of the Pt buffer layer modified the anisotropy fields and magnetic domain sizes as measured by magnetic force microscopy (MFM). An estimation of the average magnetic domain sizes was obtained from the profile of the self-correlation transform of the MFM images. For multilayers having an average magnetic domain size of 490 nm, we report a damped precession of the magnetization which decays with a time constant of ∼100 ps and which has a frequency which varies from 8.4 GHz to 17.0 GHz as the external field increases from 192 mT to 398 mT. Fitting the precession dynamics with the Landau-Lifshitz-Gilbert equation we evaluated the damping α, which decreases from 0.18 to 0.05 with increasing magnetic domain sizes (127 nm to 490 nm). These αvalues are higher than for single layers suggesting an enhanced scattering and spin pumping effects from the Pt adjacent layers. In addition, the precession frequency increases from 2.04 GHz to 11.50 GHz as the anisotropy field of the multilayers increases from 6.5 kOe to 13.0 kOe. Finally, a comparative analysis between micromagnetic simulations and MFM images allowed us to determine the exchange stiffness (A ex) in the [Co 60Fe40/Pt] 5multilayers. ©2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5130458 .,s Using femtosecond light pulses to study ultrafast magnetiza- tion dynamics has been a fruitful approach for investigating mag- netic materials and their applications. It has been allowing a bet- ter understanding of magnetization dynamics at sub-picosecond timescales which is a pre-requisite for improving the speed of current devices.1–3 Initially, it was shown that magnetic materials could be demagnetized locally by femtosecond pulses at sub-picosecond time scales.4In the following, that the ultrafast photoexcitation of electrons and spins could lead to changes of the effective field launching a magnetization precession that may be followed inreal space in the time domain.5,6Now, it is well known that the analysis of this precession may be used, analogously to fer- romagnetic resonance, to evaluate important material parameters, like the dynamic anisotropy, the magnetic anisotropy field and the damping in magnetic films and magnetic nanostructures.3,7,8 Thereafter, this field of research has flourished with the investiga- tions of all-optical switching in ferrimagnetics,9the inverse Fara- day effect in dielectrics and antiferromagnets1and more recently, it has been extended to the attosecond time scale with element speci- ficity and new probing wavelengths.10Nowadays, much attention has been devoted to the investigation and generation of ultrafast AIP Advances 9, 125322 (2019); doi: 10.1063/1.5130458 9, 125322-1 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv superdiffusive spin currents and terahertz radiation following ultra- fast photoexcitation.11–13 Here we report on the ultrafast magnetization dynamics of [Co 60Fe40/Pt] 5multilayers which were synthesized with varying magnetic textures. Magnetic multilayers with Co xFeyand Pt or Pd interlayers have attracted much interest because they have been proposed as reference layers in STT-MRAM and magneto-resistive sensors.14,15The investigated samples were grown at room tem- perature by DC magnetron sputtering with Ta capping and Pt buffer layers and present perpendicular magnetic anisotropy (PMA) as determined by vibrating sample magnetometry (VSM). Con- trolled growth rate of the Pt buffer layer modified the anisotropy fields and magnetic domain sizes as measured by magnetic force microscopy (MFM) and VSM. In order to have a set of samples with varying magnetic homogeneity, a set of [Co 60Fe40/Pt] 5multi- layers (S1-S4) with the same thickness and which distinguish them- selves only by the growth rate of the buffer layers were synthe- sized. Details of sample growth and characterization are presented in Ref. 16. The time-resolved measurements were carried out with 100 fs laser pulses from a Ti:Saphire oscillator. The experiments were done at low fluency ( ∼0.1 mJ.cm-2) in a degenerate pump and probe setup in the Kerr configuration in a standard time resolved fem- tosecond magneto-optical setup.17After photoexcitation by 100 fs pump pulses, we report an ultrafast demagnetization of the mul- tilayers in hundreds of fs followed by a partial remagnetization, which occurs typically with a time constant of ∼1.5 ps (not shown). Thereafter, as displayed in Figure 1a, a damped precession of the magnetization is observed to decay with a time constant of ∼100 ps for the multilayers with larger domain sizes (S4). In Figure 1b we present the frequency dependence of the precession which in this case varies from 8.4 GHz to 17.0 GHz as the external field increases from 192 mT to 398 mT for the sample S4. Expect- edly, the decay time, η, increases for higher fields reaching a con- stant value, in this case of ∼100 ps for fields greater than 200 mT (Figure 1c). From the decay time it is possible to determine a value for the damping, α, in Co 60Fe40/Pt multilayers fitting the experimental data with the Landau-Lifshitz-Gilbert (LLG) equation (Figure 1b).3 Assuming a sample uniformly saturated to magnetization, perpen- dicular magnetic anisotropy, small deviations of saturation magne- tization, Ms, from the equilibrium direction, an external DC field ⃗Happlied in the plane of incidence under an angle θHand an effec- tive field comprising the applied field plus and effective anisotropy field defined as Heff K=HK−4πMs(HKbeing the sample’s intrinsic anisotropy field) it is possible to solve the LLG equation to obtain the frequencies of precession and the decay time. The solution gives the following expression for the frequency of precession and the decay time ω=γ⋅√ H1⋅H2 (1) and 1 η=1 2⋅α⋅γ⋅(H1+H2) (2) where ⎧⎪⎪⎨⎪⎪⎩H1=H⋅cos(θH−θ0)+Heff K⋅cos(2⋅θ0) H2=H⋅cos(θH−θ0)+Heff K⋅(cos2θ0)(3) andγ=gLμB/̵his the gyromagnetic factor, gLis the Landé fac- tor,μBis the Bohr magneton and̵h=h/2πwhere his the Planck constant. For fitting the experimental data in Figure 1b we used the saturation magnetization and anisotropy field measured from VSM ( ±10%) and a value of g=5.18 for [Pt/Co 60Fe40/Pt] 5 multilayers. The anisotropy field was derived from M-H loops as Hk=Hs+ 4πMs.18We note that this surprisingly large value of g was necessary for fitting the experimental results and may be related to additional orbital moment contribution from the Co/Pt inter- face.17,19,20Clearly, it would be interesting to check this hypothesis by using other techniques like ferromagnetic resonance and x-ray mag- netic circular dichroism (XMCD) but it goes beyond the scope of the present work. As shown in Figure 2, we observed a decreasing damping with increasing the magnetic domain sizes observed by MFM in the multilayers (inset Figure 2). Let’s stress that all the investi- gated Co 60Fe40/Pt multilayers have the same thickness and vary- ing magnetic domain sizes due to the fact that they were grown over Pt buffer layer which had distinct crystalline textures and magnetic homogeneity. Fitting the experimental data with the LLG equation, we may evaluate αin each sample which decreases from 0.18 to 0.05 for multilayers with increasing magnetic domain sizes. Note that these values are higher than for single lay- ers and may be correlated, as reported before in multilayers, FIG. 1 . (a) TR-MOKE, (b) precession frequency and (c) decay time as a function of the external magnetic field for S4. AIP Advances 9, 125322 (2019); doi: 10.1063/1.5130458 9, 125322-2 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv FIG. 2 . Precession damping for multilayers with increasing magnetic textures: S1 (Fig. 2a), S2 (Fig. 2b), S3 (Fig. 2c), S4 (Fig. 2d). The insets show the corresponding MFM images. with an enhanced scattering and spin pumping effects from the Pt adjacent layers.21,22We note that the precession frequency (extracted fitting the data to LLG) increases from 2.04 GHz to 11.50 GHz as the anisotropy field of the multilayers increases from 6.5 kOe to 13.0kOe. For characterizing the magnetic homogeneity of the multilay- ers, we did MFM measurements (inset of Figure 2) and micro- magnetic simulations (Figure 3). The simulations were made by using MuMax3 software, developed at the DyNaMat group by Prof. Van Waeyenberge at Ghent University.23MFM measurements were obtained using a NTEGRA Aura (NT-MDT Co). The MFM measurements were done at demagnetized state. Let’s stress that the domain in the demagnetized states also mirror local inho- mogeneities.22An estimation of the average magnetic domain sizes of the samples was obtained from the profile of the self- correlation transform of the MFM images.24The results show that we have an average domain size of 127.5nm for the sample S2, 176.5nm for the sample S3 and 490.0nm for the sample S4. For the sample S1 it was not possible to infer the average domain size since the domains size of this sample is smaller than the tip resolution. In order to estimate the exchange stiffness, Aex, value for each multilayer we used the adjustment of micromagnetic simulations to MFM measurements (Figure 3). Usually Aexa key parameter con- trolling magnetization reversal in magnetic materials. The simula- tions are carried varying the Aexin order to most closely match the MFM images. The obtained values of Aexfor each sample are pre- sented in Table I. Pre-requisite for simulating the domain pattern with MuMax3 and determinating Aex, is the knowledge of the Ms,Ku andα, which here were determined respectively from VSM measure- ments and from analysis of the magnetization dynamics measure- ments as discussed previously. The static magnetic properties weredetermined for each sample by measuring the magnetic hysteresis loops at parallel and perpendicular directions to the film plane and thereafter calculating the anisotropy constant, Ku=HkMs/2.18The simulations that best adjusted to the MFM images are displayed in FIG. 3 . Micromagnetic simulation results of sample (a) S1, (b) S2, (c) S3 and (d) S4. All images correspond to areas of 5 μm x 5μm. AIP Advances 9, 125322 (2019); doi: 10.1063/1.5130458 9, 125322-3 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv TABLE I . Saturation magnetization (M s), uniaxial anisotropy (K u), damping ( α), exchange stiffness (A ex), magnetostatic exchange length (l ex1) and magnetocrys- talline exchange length (l ex2) values for the samples. Msat Ku Aex lex1 lex2 Sample (kA/m) (MJ/m3) α (pJ/m) (nm) (nm) S1 472 1.91 0.180 15.5 10.5 2.85 S2 770 4.95 0.179 42.8 10.7 2.94 S3 551 2.75 0.076 24.3 11.3 2.97 S4 560 2.75 0.053 26.3 11.5 3.09 Figure 3 and the obtained values of Aexfor each sample are presented in Table I. The exchange stiffness is defined by Aex=nJS2/a, where n is the number of nearest neighbours, S2the square of spin, Jthe exchange integral and athe lattice constant and therefore the dis- tance between the spins. The exchange length, which is defined as lex1=√ 2Aex/μ0M2scorresponds to magnetostatic exchange length and lex2=√ Aex/Kucorresponds to magnetocrystalline exchange length25and both increase with the increase of the average domain size. We note that the obtained values of Aexare in the same order of the values reported before in Co-based films ∼10pJ/m18,26and present some variation in the set of investigated samples. It has been reported that Aexmay vary with fabrication conditions, such as Ar gas pressure, substrates, seed layers, compositions, and annealing conditions because it is sensitive to the distance between magnetic atoms and number of nearest neighborhoods.27In our case, since the Pt buffer layer crystallinity and magnetic homogeneity vary with the growth processes, we could expect that Aexcould be different for each sample. In conclusion, we report on the ultrafast magnetization dynam- ics of a set of [Co 60Fe40/Pt] 5multilayers with varying magnetic homogeneity and same thickness which present a decreasing damp- ing (0.18 to 0.05) as we increase the magnetic domain sizes (127 nm to 490 nm) and which frequency of precession increases from 2.04 GHz to 11.5 GHz as the anisotropy field of the multilay- ers increases from 6.5 kOe to 13.0kOe. Fitting the magnetization dynamics with LLG equation we extracted from the decay time of the precession αvalues which are higher than for single layerssuggesting an enhanced scattering and spin pumping effects from the Pt adjacent layers. Finally, a comparative analysis between micromagnetic simulations and MFM images allowed us to deter- mine the exchange stiffness, Aex, which is a key parameter control- ling magnetization reversal in magnetic materials. As magnetic mul- tilayers with Co xFeyand Pt or Pd interlayers have been proposed as reference layers in STT-MRAM and magneto-resistive sensors, we hope that the investigation of its dynamic properties may be useful for the design of new fast magnetic devices. The authors thank the financial support from FAPEMIG, CNPq, CDTN/CNEN and CAPES. REFERENCES 1A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010). 2A. Barman and J. Sinha, Spin Dynamics and Damping in Ferromagnetic Thin Films and Nanostructures (Springer, 2018). 3C. Papusoi et al. , J. Phys. D: Appl. Phys. 51, 325002 (2018). 4E. Beaurepaire et al. , Phys. Rev. Lett. 76, 4250 (1996). 5M. van Kampen et al. , Phys. Rev. Lett. 88, 227201 (2002). 6M. Vomir et al. , Phys. Rev. Lett. 94, 237601 (2005). 7J.-Y. Bigot et al. , Chemical Physics 318, 137 (2005). 8S. Mizukami et al. , Appl. Phys. Lett. 96, 152502 (2010). 9C. D. Stanciu et al. , Phys. Rev. Lett. 98, 207401 (2007). 10C. La-O-Vorakiat et al. , Phys. Rev. Lett. 103, 257402 (2009). 11V. Shokeen et al. , Phys. Rev. Lett. 119, 107203 (2017). 12D. Rudolf et al. , Nature Communications 3, 1037 (2012). 13T. Sant et al. , Scientific Reports 7, 15064 (2017). 14J. Qiu et al. , AIP Advances 6, 056123 (2016). 15H. Meng and J.-P. Wang, Appl. Phys. Lett. 88, 172506 (2006). 16L. E. Fernandez-Outon et al. , JMMM 467, 139 (2018). 17P. Neilinger et al. , Appl. Surface Science 461, 202 (2018). 18D.-T. Ngo et al. , Journal of Magnetism and Magnetic Materials 350, 42 (2014). 19N. Nakajima et al. , Phys. Rev. Lett. 81, 5229 (1998). 20D. Weller et al. , Phys. Rev. B 49, 12888 (1994). 21E. Barati et al. , Phys. Rev. B 90, 014420 (2014). 22J. Walowski et al. , J. Phys. D: Appl. Phys. 41, 164016 (2008). 23A. Vansteenkiste et al. , AIP Advances 4, 107133 (2014). 24D. Navas et al. , Phys. Rev. B 81, 224439 (2010). 25G. S. Abo et al. , IEEE Trans. Magn. 49, 4937 (2013). 26R. Moreno et al. , Phys. Rev. B 94, 104433 (2016). 27C.-Y. You, Appl. Phys. Express 5, 103001 (2012). AIP Advances 9, 125322 (2019); doi: 10.1063/1.5130458 9, 125322-4 © Author(s) 2019
1.1452685.pdf	Thermal magnetization fluctuations in CoFe spin-valve devices (invited) Neil Smith, Valeri Synogatch, Danielle Mauri, J. A. Katine, and Marie-Claire Cyrille Citation: Journal of Applied Physics 91, 7454 (2002); doi: 10.1063/1.1452685 View online: http://dx.doi.org/10.1063/1.1452685 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/91/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Characteristics of microwave oscillations induced by spin transfer torque in a ferromagnetic nanocontact magnetoresistive element J. Appl. Phys. 105, 07D124 (2009); 10.1063/1.3076047 Anomalous magnetoresistance behavior of CoFe nano-oxide spin valves at low temperatures J. Appl. Phys. 93, 7690 (2003); 10.1063/1.1540149 Comparative study of magnetoresistance and magnetization in nano-oxide specular and nonspecular MnIr/CoFe/Cu/CoFe spin valves from 10 to 300 K J. Appl. Phys. 91, 5321 (2002); 10.1063/1.1459617 Spin-filter spin-valve films with an ultrathin CoFe free layer J. Appl. Phys. 89, 5581 (2001); 10.1063/1.1359169 Layer selective determination of magnetization vector configurations in an epitaxial double spin valve structure: Si(001)/Cu/Co/Cu/FeNi/Cu/Co/Cu Appl. Phys. Lett. 77, 892 (2000); 10.1063/1.1306395 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 15:32:59Thermal Magnetic Stability of Nano-Sized Magnetic Devices Mike Mallary, Chairman Thermal magnetization ﬂuctuations in CoFe spin-valve devices invited Neil Smith,a)Valeri Synogatch, Danielle Mauri, J. A. Katine, and Marie-Claire Cyrille IBM Almaden Research Center, San Jose, California 95120 Thermally induced magnetization ﬂuctuations in the Co 86Fe14free~sense !layer of micron-sized, photolithographically deﬁned giant magetoresistive spin-valve devices are measured electrically, bypassing a dc current through the devices and measuring the current-dependent part of the voltagenoise power spectrum. Using ﬂuctuation–dissipation relations, the effective Gilbert dampingparameter afor 1.2, 1.8, and 2.4 nm thick free layers is estimated from either the low-frequency white-noise tail, or independently from the observed thermally excited ferromagnetic resonancepeaksinthenoisepowerspectrum,asafunctionofappliedﬁeld.Thegeometry,ﬁeld,andfrequencydependence of the measured noise are found to be reasonably consistent with ﬂuctuation–dissipation predictions based on a quasianalytical eigenmode model to describe the spatialdependence for the magnetization ﬂuctuations. The extracted effective damping constant a’0.06 found for the 1.2 nm free layer was close to 3 3larger than that measured in either the 1.8 or 2.4 ﬁlms, which has potentially serious implications for the future scaling down of spin-valve readheads. © 2002 American Institute of Physics. @DOI: 10.1063/1.1452685 # I. INTRODUCTION It was recently demonstrated1that broadband ~white !re- sistance noise generated from thermally induced magnetiza-tion ﬂuctuations, or ‘‘mag-noise,’’ in the soft, ferromagneticfree layer of the giant magnetoresistive ~GMR !spin-valve read heads used in hard disk drives can contribute to a sub-stantial, if not dominant portion of the head’s intrinsic outputnoise power. Hence, mag-noise will pose a fundamental limitto the signal/noise ratio of any form of magnetoresistive~MR!sensor employing thin-ﬁlm ferromagnetic sensing lay- ers. Due to the potentially important implications for presentand particularly future read heads with ever decreasing de-vice sizes, this topic has gained considerable furtherattention 2–4in the past few months. In addition to its practical implications, mag-noise in MR sensors can be used to provide a simple electrical mea- surement technique to quantitatively study the basic dampingproperties and loss mechanisms of the constituent soft ferro-magnetic thin ﬁlms. This includes, in particular, any geo-metrical dependencies of these properties or mechanisms insmall submicron ‘‘nanostructures’’ not as easily probed bymore traditional ferromagnetic resonance ~FMR !methods. The relations between the measured noise and the dampingproperties of the thin ﬁlms can be described via theﬂuctuation–dissipation theorem ~FDT!. 5 This article will brieﬂy report on mag-noise measure- ments in GMR spin valves with ultrathin ~1.2–2.4 nm ! Co86Fe14free-layers, which were more fully presented at the46th MMM conference in Seattle.6Unfortunately, due to the very recent nature of the experimental results, publicationconstraints for these proceedings did not permit a more thor-ough description of the complete set of measurements, nordetails of the noise modeling techniques which are describedin general terms elsewhere. 7A considerably more complete description of this work is planned to be submitted for pub-lication in the near future. II. EXPERIMENT The experimental device structure used in the measure- ments is described pictorially in Fig. 1. The 2 mm thick a!Author to whom correspondence should be addressed; electronic mail: neils@almaden.ibm.com FIG. 1. ~Top!Image of planar view of test structure; contact leads/pads are approximately to scale. ~Bottom !Image of cross section view of test struc- tures; contact leads/pads not included;gap g550nm.JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 10 15 MAY 2002 7454 0021-8979/2002/91(10)/7454/4/$19.00 © 2002 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 15:32:59Ni80Fe20plated ‘‘keeper’’layer ~easy axis along 6zˆ!was left unpatterned over the entire wafer, and chemically–mechanically polished smooth. The sputtered spin-valve ma-terial had the following basic layer structure: seedlayers/PtMn/CoFe/Ru/CoFe/Cu/CoFe( t free)/cap layers, with free layer saturation induction Bs54pMs>17 kG and thick- ness,tfree>1.2, 1.8, and 2.4 nm ~fromB–Hloops against calibrated Ni 80Fe20standards !. All of the magnetic layers were sputtered from a Co 86Fe14target. The spin-valve ﬁlms were photolithographically pat- terned into 30- mm-long, narrow stripes of width L~Fig. 1 !, with 0.5 mm<L<1mm. Au/Ta leads of much lower sheet resistance were deposited by liftoff, and electrically deﬁnedan ‘‘active trackwidth’’ W a>3L, such that device resistance was sensitive magnetoresistively to only the magnetizationinside this active region. The pad geometry was tailored tothe probe tips of a high-frequency voltage probe, with mini-mized contact area to reduce the series pad-shield capaci-tance to ,1p F . Strong exchange pinning between PtMn and the bottom CoFe ‘‘pinned’’ layer, combined with considerably strongerantiparallel ~AP!coupling 8between pinned and ‘‘reference’’ CoFe layers across the ’1 nm Ru spacer, keeps the magne- tization in this combined ‘‘synthetic ferrimagnet’’ stifﬂyaligned along 6xˆeven in the presence of applied ﬁelds on the order of 1 kOe. From the well-known GMR cosine law, 9 the device resistance should then vary as R5R01DRm¯x, wherem¯xis the transverse ~xaxis!component of the free- layer unit magnetization mˆ(x,z), averaged over the ‘‘active volume’’ Va[LWatfree. Depending on tfree, the present de- vices had R0’73–80 VandDR/R0’6%–8%, including lead/parasitic resistance. When applying a nonzero dc biascurrentI b, ﬂuctuations IbdRin spin-valve output voltage provide a direct electrical measure of the mean free-layermagnetization ﬂuctuations dm¯xinsideVa. For the present work, the mag-noise contribution SVmag(f;Ib)5SV(f;Ib)2SV(f;I50) to the total device volt- age noise power spectral density, SV(f;Ib) has been mea- sured in spin-valve devices as a function of both frequency f and quasistatic longitudinal ~z-axis!ﬁeldHzusing commer- cially available instrumentation.10In all the present cases Ib 55 mA. A static x-axis bias ﬁeld was applied to cancel out ﬁelds from both interlayer coupling as well as the ‘‘image’’currents in the lower NiFe keeper, in order to maintain ap-proximate longitudinal alignment mˆ(x,z)’zˆof the quiescent free layer magnetization inside the active volume. III. RESULTS Using arguments similar to those described previously,1,2 the FDT can be used to derive the following relationship: 4kTa gMsVa>SVmag~f!0;Ib,Hz! ~IbdR/dHx!2, ~1! where ais the Gilbert form of the phenomenological damp- ing parameter. The right half of Eq. ~1!contains only mea- surable quantities. @Here,SVmag(f!0) was estimated by SVmag(f>170 MHz; Hz), anddR/dHxversusHzwas mea- sured using a lock-in ampliﬁer and a small, transverse ac‘‘tickle ﬁeld’’ dHx>0.1 Oerms. #Hence, Eq. ~1!may be used to extract the effective damping parameter a, as all other physical parameters are known ~T>300 K, and gyromag- netic ratio g>19 Mrad/Oes !. The result of this procedure, for spin-valve devices with tfree>1.8 nm and L>0.5, 0.7, and 1.0 mm is shown in Fig. 2. Restricting attention to the Hz.25 Oe data, one can estimate a ‘‘mean’’ value of a¯’0.023 to within about 610% peak– peak variation. As was typical of this type of measurement,no discernible systematic dependence of the extracted ais seen with either HzorL. The latter conﬁrms the predicted 1/Vadependence from Eq. ~1!over a 4 3variation. The rem- nant, oscillatory variations with Hzof the extracted a, par- ticularly for small Hz,25 Oe, is believed due to signiﬁcant anisotropy dispersion in the free layer, which typically re-sulted in nonmonotonic behavior of udR/dHxuwithHz. The damping properties and/or avalues for the free layer may be alternatively measured from the broadband fre-quency dependence of the normalized mag-noise power spectral density S V8mag(f)[SVmag(f)/SVmag(f!0), which FIG. 2. Extracted values of damping parameter aby the method of Eq. ~1!, using test devices with tfree51.8 nm and stripe widths L50.5, 0.7, and 1.0 mm. Dashed line represents ‘‘eyeball’’-estimated mean value of a¯’0.023. FIG.3. ~Solidlines !Measurednormalizedmag-noisepowerspectraldensity vs frequency for test device with tfree51.2 nm and stripe width L 50.7 mm; discrete values of dc bias ﬁeld as indicated. ~Dashed lines !Theo- retically predicted with an ‘‘eyeball’’ ﬁtted value of a¯(tfree)50.060, using methods described in Ref. 7 and brieﬂy herein.7455 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Smithet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 15:32:59scales out explicitdependencies on kTandMsVa, as well as ampliﬁer gain and ﬁeld source calibration factors. Figure 3 showsSV8mag(f;Hz) for anL>0.7mm,tfree>1.2 nm device for a series of discrete values of dc longitudinal bias ﬁeldH z. The upper bound of 2.9 GHz for the measured spectra was limited by our presently available spectrum analyzer.10 Since the measurement is itself completely passive, the peaksin the spectra clearly demonstrate the thermal excitation oflow order ferromagnetic resonance mode ~s!, which give non- zero contribution to the volume-averaged magnetizationﬂuctuation dm¯x. However, unlike Eq. ~1!, quantitative analysis is here considerably more modeling intensive, the details of which,for reasons mentioned above, will be postponed until a laterpublication. In brief, mag-noise spectra were computed usingthe general formalism described previously, 7by expanding thex,ymagnetization components m(x,z) in terms of the static eigenmodes n(x,z) of the ‘‘stiffness-ﬁeld’’ tensor7HI for a patterned stripe in the presence of a ‘‘keeper’’ layer. Then(x,z) were analytically approximated with trigonomet- ric functions ~standing spin waves !, corrected for ﬁnite stripe widthLin ways similar to that described elsewhere11to im- prove eigenvalue accuracy. The modeled mag-noise power spectral density for the L>0.7mm,tfree>1.2 nm device geometry is also shown in Fig. 3. The extracted value of a¯’0.06 was determined by an ‘‘eyeball-ﬁt’’ to the FMR peak heights which, scaling ap-proximately as 1/ a2, are the most sensitive parametric to a. However, the calculated results here also provide a fairlygood representation of the FMR resonance frequencies andlinewidths, along with their H zdependence, despite the rela- tively ideal, simplistic nature of the model. Extracted values a¯(tfree) using the method of Eq. ~1!, and by modeling SV8mag(f;Hz), are summarized in Fig. 4. @Being less dependent on modeling approximations, a¯from Eq.~1!is considered here to be more reliable. #For the thin- nest free layer tfree>1.2 nm, the value of a¯’0.06 was con- sistent between both methods, and there is no doubt thatthese ultrathin Co 86Fe14free layers show anomalously large damping. The mechanism for this appears conﬁned to ex-tremely thin Co 86Fe14ﬁlms, since a¯extracted from both tfree>1.8 and 2.4 nm spin valves was roughly the same within experimental error, and close to three times smallerthan that for the case of t free>1.2 nm. Vibrating sample magnetometer ~VSM !measurements with perpendicular to plane ﬁelds on 1 in. witness couponsshown in Fig. 5 unambiguously indicate that there exists alarge thickness-dependent perpendicular uniaxial anisotropy H k’>9.8 and 4.5 kOe in free layers with tfree>1.2 and 1.8 nm, respectively. The linearity of the in-plane ﬁeld B–H loops with deposition time conﬁrms that value of Bs (>17 kG) is virtually independent of free layer thickness. The VSM measurements with perpendicular ﬁelds were mo- tivated by initial failure, assuming Hk’50, to account for the observed resonance frequencies in the data of Fig. 3, atwhich point in time all witness coupons for t free52.4 and 3.0 nm spin-valve ﬁlms had been used up during ion-millingcalibration for lithography purposes. The two available datapoints suggest a somewhat stronger than 1/ t freedependence forHk’, the latter being expected for effects of surface/ interface-induced anisotropy, e.g., as believed observed inAu/Co/Au and Au/Cu/Co/Cu/Au sandwiches 12with similar Co ﬁlm thickness. IV. DISCUSSION AND CONCLUSIONS The observed sharp increase in the effective damping parameter awith decreasing free-layer thickness in ultrathin (tfree<1.5 nm) CoFe ﬁlms, is here at least suggestive of surface/interface inhomogeneity acting as a mechanism foradditional mode coupling between quasidegenerate eigen- FIG. 4. Summary of present experimental results for extracted ~ﬁtted!val- ues of mean, effective Gilbert damping parameter a, using either Eq. ~1! ~solid circles !or normalized power spectral density ~open circles !. FIG. 5. Coupon VSM measurements of magnetic moment mvs perpendicu- lar to plane ﬁeld H’. The moment is expressed in units of equivalent NiFe magnetic thickness t5tNiFe(m/msNiFe), where msNiFe5BsNiFetNiFeis the satu- rated moment of a similar size NiFe coupon of known thickness tNiFeand BsNiFe510kG. Onset of saturation ﬁeld ~indicated by dashed vertical lines ! provides a measure of BsCoFe2Hk’.~The ﬁelds applied here are much less than required to overcome the very strong antiparallel coupling between pinned and reference CoFe layers, so that tis virtually that of only the free layer. !Inset shows low-ﬁeld ~,100 Oe !in-plane saturated measurements of tfree*5tfree(BsCoFe/BsNiFe) vs free-layer deposition time ~}physical thickness tfree!.7456 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Smithet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 15:32:59modes of high ( kzWa>2p) and low ( kzWa<p) wave num- berskz.~The present data further suggest a possible link between this hypothesized surface/interface effect, and oneresponsible for the strongly thickness-dependent increase in H k’.!It is only the latter, ‘‘low’’- kzmodes that contribute signiﬁcantly to the volume-averaged ﬂuctuations dm¯x, which are presently observed magnetoresistively. Such modecoupling ~two-magnon process !could simulate excess damp- ing by channeling thermal energy out of low- k zinto high- kz degenerate modes, and if interfacial in nature, would be expected13to have a strong ;1/tfree2dependence. Interest- ingly, such a mechanism would notbe expected to be scale invariant once the active device ‘‘trackwidth’’ Wawas sufﬁ- ciently small ~!1mm!such that exchange stiffness would signiﬁcantly break the degeneracy between modes of highand lowk zvalues. Use of the present technique, accompa- nied perhaps by structures alternatively fabricated viae-beam lithography, would be a relatively straightforward,direct way to further investigate this. In addition to the explicit 1/ V ascaling from Eq. ~1!, the issue of the ﬁlm thickness ~and other geometrical !dependen- cies of mag-noise in future MR read head gains in impor-tance as device size will approach and eventually drop belowthe 0.1 mm range. Previous scaling analysis2suggests that the magnetic stiffness of read sensors will tend to increasewith smaller device size, such that the lowest order FMRfrequencies will remain above the read channel bandwidth. Ifso, signal/noise limitations due to mag noise will be deter- mined by the low frequency tail S Vmag(f!0), which scales proportionally with the damping parameter a@Eq.~1!#. Assuming a tfree-independent a, the same scaling analy- sis has already argued that mag-noise will likely tend to sub-stantially reduce the otherwise expected beneﬁt of increasing read head magnetic sensitivity by thinning the free layer.Having a simultaneously increasing damping constant withdecreasing t freewould only further aggravate this situation. ACKNOWLEDGMENTS The authors wish to thank Linda Lane for mask design of the test structures measured here, and Tsann Lin and TyChen for assistance in spin-valve ﬁlm deposition. 1N. Smith and P. Arnett, Appl. Phys. Lett. 78, 1448 ~2001!. 2N. Smith, IEEE Trans. Magn. ~to be published !. 3N. H. Bertram, Z. Jin, and V. L. Safonov, IEEE Trans. Magn. ~to be published !. 4J. Zhu, J. Appl. Phys. 91, 7273 ~2002!; Y. Zhou, A. Roesler, and J. Zhu, ibid.91, 7276 ~2002!; S. B. Shueh and L. Liu, paper CB-08 presented at the 46th Annual Conference on Magnetism and Magnetic Materials, Se-attle, WA, November 12–16, 2001; V. L. Safonov and H. N. Bertram, J.Appl. Phys. 91, 7279 ~2002!;ibid.91, 7285 ~2002!. 5R. Kubo, Rep. Prog. Phys. 29, 255 ~1966!; L. D. Landau and E. M. Lifshitz, in Statistical Physics ~Pergamon, Oxford, 1980 !, Chap. 12, pp. 384–96. 6N. Smith et al., J. Appl. Phys. 91, 7454 ~2002!. 7N. Smith, J. Appl. Phys. 90, 5768 ~2001!. 8S. S. P. Parkin and D. Mauri, Phys. Rev. B 44, 7131 ~1991!. 9B. Dieny, V. S. Speriosu, S. S. P. Parkin, B.A. Gurney, D. R. Wilhoit, and D. Mauri, Phys. Rev. B 43, 1297 ~1991!. 10Picoprobe 20 GHz probe, MITEQ 0.1–8 GHz preamp ~43 dB gain, 1.3 dB noise ﬁgure !, Agilent 8560-EC spectrum analyzer ~2.9 GHz !. 11P. H. Bryant, J. F. Smyth, S. Shultz, and D. R. Fredkin, Phys. Rev. B 47, 1255 ~1993!. 12C. Chappert and P. Bruno, J. Appl. Phys. 64, 5736 ~1988!. 13R. D. McMichael, M. D. Stiles, P. J. Chen, and W. Engelhoff, Jr., J.Appl. Phys.83, 7037 ~1998!; S. M. Rezende,A.Azevedo, M.A. Lucena, and F. M. deAguiar, Phys. Rev. B 63, 214418-1 ~2001!.7457 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Smithet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 15:32:59
1.3635782.pdf	Spin-torque-driven ballistic precessional switching with 50 ps impulses O. J. Lee, D. C. Ralph, and R. A. Buhrman Citation: Applied Physics Letters 99, 102507 (2011); doi: 10.1063/1.3635782 View online: http://dx.doi.org/10.1063/1.3635782 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/99/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Microscopic theory of spin torque induced by spin dynamics in magnetic tunnel junctions J. Appl. Phys. 109, 07C909 (2011); 10.1063/1.3540411 Micromagnetic study of switching boundary of a spin torque nanodevice Appl. Phys. Lett. 98, 102501 (2011); 10.1063/1.3561753 Perpendicular spin-torque switching with a synthetic antiferromagnetic reference layer Appl. Phys. Lett. 96, 212504 (2010); 10.1063/1.3441402 Spin-torque-driven vortex dynamics in a spin-valve pillar with a perpendicular polarizer Appl. Phys. Lett. 91, 242501 (2007); 10.1063/1.2822436 Self-consistent simulation of quantum transport and magnetization dynamics in spin-torque based devices Appl. Phys. Lett. 89, 153504 (2006); 10.1063/1.2359292 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.251.14.35 On: Tue, 23 Dec 2014 16:59:34Spin-torque-driven ballistic precessional switching with 50 ps impulses O. J. Lee,1,a)D. C. Ralph,1,2and R. A. Buhrman1 1Cornell University, Ithaca, New York 14853, USA 2Kavli Institute at Cornell, Ithaca, New York 14853, USA (Received 29 May 2011; accepted 16 August 2011; published online 7 September 2011) We demonstrate reliable spin-torque-driven bal listic precessional switching using 50 ps current impulses in a spin-valve device that includes both in- plane and out-of-plane spin polarizers. Different threshold currents as the function of switching direc tion and current polarity enable the ﬁnal orientation of the magnetic free layer to be steered, in accord wi th a macrospin analysis, by the sign of the pulse, eliminating the need for read-before-write toggl e operation. The pulse amplitude windows for this deterministic operation are wider and more symme tric as a function of current polarity for shorter impulses, while inhomogeneous fringe ﬁelds from t he polarizers lead to asymmetries as a function of current direction. VC2011 American Institute of Physics . [doi: 10.1063/1.3635782 ] The fast reversal of a nanomagnet is of active interest because its study can enhance understanding of fundamental magnetic dynamics and because of the technological advan-tages that a successful high-speed non-volatile magnetic memory could provide. Several schemes 1–7have been explored for fast nanomagnet switching, with the perhapsmost scalable approach being demonstrated by recent experiments 8–11which achieved reliable high-speed reversal of a thin ﬁlm nanomagnet by using the spin torque (ST) froma spin-polarized current pulse as short as 100 ps. These experiments utilized devices in which a thin ﬁlm free layer (FL) is located between an out-of-plane (OP) spin polarizer(OPP) and an in-plane (IP) analyzer/polarizer (IPP). In this conﬁguration, the ST ( s OP) generated by a strong OP- polarized current pulse incident upon the FL forces theFL moment out of plane, inducing a demagnetization ﬁeld (H demag ) about which the FL begins to precess.6,7If the pulse width and amplitude are properly controlled, the result canbe a rapid rotation of the moment by 180 /C14to the reversed equilibrium position. The simplest form of this OP-precessional reversal scheme has the potential disadvantage of being a toggle opera- tion, in which both parallel (P) to anti-parallel (AP) and AP-to-P switching occur for either sign of current. This is incontrast to a deterministic operation in which the ﬁnal state is controlled by the current polarity, as is the case for ST devices only utilizing IP polarized currents. However, previousOP-ST experiments 8–10in which the IPP also exerted a strong ST,sIP, on the FL obtained differences in the threshold cur- rents for switching as a function of current polarity andswitching direction. This indicated that the ﬁnal state in OP-ST devices may be determinable by pulse-current polarity, although with pulse widths /C21100 ps, only one current polarity showed a sufﬁciently wide window between the switching currents for P-to-AP and AP-to-P to yield reliable writes. 8 Here, we report the achievement of reliable and deter- ministic spin torque ballistic precessional switching (STBPS) by using 50 ps current impulses, demonstrating that shorter and stronger pulses can enhance the inﬂuence of sIP, provid-ing wider current windows for deterministic switching. Based on micromagnetic simulations, we also conclude that inhomogeneous stray magnetic ﬁelds from the two polarizersinduce asymmetries in the deterministic switching windows for the two current polarities. We fabricated nanopillar spin valve devices from thin- ﬁlm multilayers with the structure: bottom lead/OPP/Cu(6)/ Py(5)/Cu(12)/Py(20)/top lead (thicknesses in nm), where Py is Ni 80Fe20. The OPP was Pt(10)/[Co(0.44)/Pt(0.68)] 4/ Co(0.66)/Cu(0.3)/Co(0.66). The 5 nm Py layer served as the magnetic FL and the 20 nm Py layer was the IPP. The devices were fabricated into approximately elliptical cross-sectionswith dimensions 50 /C2170 nm 2, with the etch producing slightly tapered side walls (20-30/C14from vertical).12The thick- ness of the IPP (20 nm) was chosen to be much greater thanthe spin-diffusion length ( /C245 nm) to ensure a strong s IP. For pulses longer than 200 ps, these devices exhibited preces- sional switching characteristics similar to previous meausure-ments 8. Here, we focus on results obtained with 50 and 100 ps impulses. We generated 50 ps current impulses (Fig. 1(b)) by dif- ferentiating a sharply falling step pulse, while pulses with 100 ps widths were generated with a commercial pulse gen- erator. The current through the device was calculated takinginto account the impedance mismatch between the load re- sistance and the 50 Xtransmission line. 13All measurements were performed at room temperature under an applied mag-netic ﬁeld canceling the average in-plane dipole ﬁeld from the IPP. Ten devices were studied in detail and similar behavior was obtained in all cases. We deﬁne positive cur-rent to correspond to electron ﬂow from the OPP to the FL (and to the IPP). Figures 1(c) and1(d) show switching probabilities ( P s) obtained from one device as a function of current amplitude, polarity, and switching direction (P-to-AP or AP-to-P) using both 50 ps and 100 ps current impulses. For the 50 ps case,reliable P-to-AP switching ( P s/C2195%) was achieved for cur- rent pulse amplitudes beyond Iþ r;P/C0AP/C2411 mA, but AP-to-P switching was not observed up to the highest pulse levelemployed, from which we conclude that the threshold current ( P s/C215%) to initiate switching is Iþ th;AP/C0P>17 mA. This yields a deterministic window at positive pulsea)Author to whom correspondence should be addressed. Electronic mail: ol29@cornell.edu. 0003-6951/2011/99(10)/102507/3/$30.00 VC2011 American Institute of Physics 99, 102507-1APPLIED PHYSICS LETTERS 99, 102507 (2011) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.251.14.35 On: Tue, 23 Dec 2014 16:59:34amplitudes for ST switching to the AP state of Dþð50 ps Þ/C17Iþ th;AP/C0P/C0Iþ r;P/C0AP>6 mA (Fig. 1(c)), while Dþð100 ps Þ/C244 mA (Fig. 1(c)). For negative 50 ps impulses, the pulse amplitude required for reliable AP-to-P switching was larger in magnitude, I/C0 r;AP/C0P/C24/C015 mA, resulting in a window D/C0ð50 ps Þ/C17/C0 ð I/C0 th;P/C0AP/C0I/C0 r;AP/C0PÞ>3 mA (Fig. 1(d)), while D/C0ð100 ps Þwas negligible. The increase in the switching windows DþandD/C0with the reduction of pulse width to 50 ps demonstrates the possibility of implementing ultra-fast deterministic STBPS. Certain aspects of the switching behavior, including the origin of the deterministic switching windows, can be under- stood with a simple zero-temperature (T ¼0) macrospin model that utilizes the Landau-Lifshitz-Gilbert ( LLG) equa- tion including the effects of ST14–16from the two spin polarizers dbm dt¼/C0 cbm/C2H! effþabm/C2dbm dtþca1ðh1Þbm/C2bp1/C2bm /C0ca2ðh2Þbm/C2bp2/C2bm; (1) where aiðhÞ¼/C22h 2eIpðtÞ loMoVPigiðhÞand H! eff¼ðHkmxþHdx þHaÞ^xþðHdz/C04pMomZÞ^z. Here, cis the gyromagnetic ra- tio,^mis the unit vector of the FL, Hkis the anisotropy ﬁeld of the FL, HdxandHdzare the IP and OP components of the effective dipole ﬁeld Hdacting on the FL, Hais the external applied ﬁeld along the FL easy axis, ^p1is the spin polariza- tion axis of the OPP ( ^z) and ^p2is of the IPP ( /C0^x), gðhÞ¼2K2=½ðK2þ1ÞþðK2/C01Þcosh/C138,h1ð2Þis the angle between the FL and OPP (IPP), Kis a torque asymmetry pa- rameter due to spin accumulation effects (we assume sym- metric electrodes), Mois the saturation magnetization of the FL, and Piis the spin polarization.17 In the STBPS, the reversals are mostly governed by the initial out-of-plane rotation angle generated by the currentpulse because this angle determines the strength of H demag . From Eq. (1), we have the initial equation of motion for the out-of-plane rotation1þa2 cde dt¼ða1þa2eoÞþð a2/C0a/C14pMoÞe; (2) where eis the out-of-plane offset angle of the FL moment relative to the equilibrium angle ( eo¼Hdz=4pMo). The rota- tion angle e(sp)when the pulse is terminated is a1ðeða2/C0a/C14pMoÞsp/C01Þ=ða2/C0a/C14pMoÞ, assuming that a square pulse with the width sp¼ctp=ð1þa2Þ/C25ctpis applied and eo¼0. In the absence of an IPP ( a2¼0), the cur- rent required to achieve a given out-of-plane rotation angleis independent of both current polarity and switching direc- tion. However, in the presence of an IPP, as s OPforces the FL out of plane, the IPP causes an additional non-zero torqueperpendicular to the sample plane, s IP/ð^m/C2^p2/C2^mÞZ(see Fig. 1(a)), that, assuming the a1>0 case, either accelerates (a2>0) or retards ( a2<0) the out-of-plane rotation of the FL moment driven by sOP. This additional torque ðsIPÞZ causes a difference between the currents required for AP-to- P and P-to-AP switching for a given pulse polarity (compareFig.1). From Eq. (2)and using parameters appropriate to our devices (see below), the macrospin calculation yields D þ¼D/C0/C241.5 mA, 2.5 mA, and 4.5 mA for tp¼200 ps, 100 ps, and 50 ps, respectively. Our devices have the interesti ng features that the P-to-AP switching current at positive bias is always less than the magni-tude of the AP-to-P switching current at negative bias ( s IPis favorable to the switching direct ion in both cases) and that the deterministic window for the posit ive current pulses is invaria- bly larger than for the negative. To understand these features and the details of the STBPS, we performed T ¼0m i c r o m a g - netic simulations that utilized Eq. (1)and employed the follow- ing magnetic parameters: Mo(IPP)¼850 emu/cm3, Mo(FL)¼650 emu/cm3,Mo(OPP) ¼870 emu/cm3, exchange constants A(IPP)¼A(FL)¼13/C210/C06erg, A(OPP) ¼26 /C210/C06erg, OPP anisotropy K?ðOPPÞ¼8/C2106erg/cm3,a n d FL damping a¼0:03.18The simulated nano-pillar had an el- liptical cross-section of 50 /C2170 nm2and the mesh size was 5/C25/C22.5 nm3. The static magnetic conﬁgurations were ﬁrst calculated by the energy minimization method19for an external magnetic ﬁeld located at the center of the minor loop for the Pand AP states. Then a current impulse ( I p(t)) was applied at time t¼0 taking into account nonzero rise and fall times, with the calculated ST exerted on the interface cells of each mag-netic layer. The micromagnetic ST simulations reveal that the STBPS in this device structure is initiated by reversal at oneend of the FL ellipse, with the remainder of the FL follow- ing. 20Fig.2shows the simulated time-traces of hmxi(Figs. 2(a) and2(b)) and hmzi(Figs. 2(c) and2(d)) for the left 60 nm and the right 60 nm of the FL for positive (Figs. 2(a)and 2(c)) and negative (Figs. 2(b) and2(d)) 50 ps impulses. In this simulation, K¼1.5,P1¼0.20, and P2¼0.37. For both current polarities, and even for P2¼0.0, the FL reversal is an inhomogeneous process in which the right (left) side rotates faster for P-to-AP (AP-to-P) reversal, rather than theuniform rotation of a macrospin. 7 This nonuniform reversal occurs because the reference layers’ dipole ﬁelds are inhomogeneous at the position of theFL (H dxis plotted in Fig. 2(a)inset) and the local critical cur- rent density Jcfor OP ST excited precession depends on this FIG. 1. (Color online) (a) Scheme of STBPS. In the case shown, the current polarity and orientations of the ﬁxed layers are such that the OP torque pro- motes upward (positive z) displacement of bmand the IP torque retards it. (b) Measured waveform of the /C2450 ps (FWHM) current impulse injected into the device. Measured switching probabilities ( Ps) using (c) positive and (d) negative 50 ps or 100 ps current impulses.102507-2 Lee, Ralph, and Buhrman Appl. Phys. Lett. 99, 102507 (2011) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.251.14.35 On: Tue, 23 Dec 2014 16:59:34ﬁeld, Jc/Heff k;x¼j6Hk=2þHaþHdxj(Ref. 21;þcorre- sponds to AP-to-P, /C0to P-to-AP). For the simulated device in the AP conﬁguration, Heff k;L/C25230 Oe when averaged over the leftmost 60 nm of the FL, while Heff k;R/C25460 Oe for the rightmost 60 nm. For the P conﬁguration, the variation in Heff kis even greater, with Heff k;L/C25490 Oe and Heff k;R/C25140 Oe. Furthermore, HdZcauses the magnetization of the FL to tilt out of plane, with this effect being stronger (weaker) on the right (left) end of the FL due to the additive (subtractive) combination of the IPP and OPP ﬁelds. The effect of HdZfor þIis to increase the inﬂuence of the sIPdue to the signiﬁcant equilibrium hmzi(eo>0) on the right side of the FL (Fig. 2(c) and Eq. (2)) for P-to-AP ( sIPis favorable to this direc- tion) but eo/C250 on the left side for AP-to-P. For /C0I, the effect ofHdZprovides a eothat is opposite to the displacement driven by the OP-ST, resulting in a smaller maximum valueofhm zifor a given pulse amplitude and hence a reduction in the inﬂuence of sIP. The consequences of the inﬂuence of the sIPwithin the micromagnetic simulations can be seen in Fig. 3which com- pares the P2¼0.37 and P2¼0.0 cases, and the consequences of the nonuniform dipole ﬁelds can be seen by comparingFigs. 3(a)–3(d) (w/H d) to Figs. 3(e)and3(f)(w/o Hd). With-outHd, we obtain, as in the original macrospin model, more uniform FL reversals and large and symmetric values for D/C0 andDþ, with only a small difference between jIþ th;P/C0APjand jI/C0 th;AP/C0Pjarising from our use of K¼1.5 (Figs. 3(e) and 3(f)). With Hd, we ﬁnd that for 50 ps pulses jIþ th;P/C0APjis reduced while jI/C0 th;AP/C0Pjis increased and Dþ>D/C0. For 100 ps pulses (Fig. 3(c)and3(d)), the simulations show that both D/C0andDþare reduced further relative to the 50 ps case (the reduction in Dþis not visible in Fig. 3because Iþ th;AP/C0P ð100 ps Þ>18 mA). This is consistent with the experiment and Eq. (2). In summary, we have demonstrated reliable and deter- ministic STBPS with a 50 ps spin polarized impulse current where the shorter current impulse enhances the deterministic write operation. If the fringe ﬁelds can be reduced close tozero, then nearly symmetric deterministic windows should be achievable for both pulse polarities, enabling very fast, energy efﬁcient STBPS. This research was supported by the Ofﬁce of Naval Research and by the NSF/NSEC program through the Cornell Center for Nanoscale Systems. We also acknowledge NSF support through use of the Corne ll Nanofabrication Facility/ NNIN and the Cornell Center for Materials Research facilities. 1J. A. Katine and E. E. Fullerton, J. Magn. Magn. Mater. 320, 1217 (2008). 2H. W. Schumacher, C. Chappert, P. Crozat, R. C. Sousa, P. P. Freitas, J. Miltat, J. Fassbender, and B. Hillebrands, Phys. Rev. Lett. 90, 017201 (2003). 3K. Vahaplar, A. M. Kalashnikova, A. V. Kimel, D. Hinzke, U. Nowak, R.Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, and Th. Rasing, Phys. Rev. Lett. 103, 117201 (2009). 4S. Garzon, L. Ye, R. A. Webb, T. M. Crawford, M. Covington, and S. Kaka, Phys. Rev. B 78, 180401(R) (2008). 5R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett. 92, 088302 (2004). 6A. D. Kent, B. O ¨zyilmaz, and E. del Barco, Appl. Phys. Lett. 84, 3897 (2004). 7K. J. Lee, O. Redon, and B. Dieny, Appl. Phys. Lett. 86, 022505 (2005). 8O. J. Lee, V. S. Pribiag, P. M. Braganca, P. G. Gowtham, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 95, 012506 (2009). 9C. Papusoi, B. Delat, B. Rodmacq, D. Houssameddine, J.-P. Michel, U. Ebels, R. C. Sousa, L. Buda-Prejbeanu, and B. Dieny, Appl. Phys. Lett. 95, 072506 (2009). 10H. Liu, D. Bedau, D. Backes, J. A. Katine, J. Langer, and A. D. Kent, Appl. Phys. Lett. 97, 242510 (2010). 11G. E. Rowlands, T. Rahman, J. A. Katine, J. Langer, A. Lyle, H. Zhao, J. G. Alzate, A. A. Kovalev, Y. Tserkovnyak, Z. M. Zeng, H. W. Jiang, K.Galatsis, Y. M. Huai, P. Khalili Amiri, K. L. Wang, I. N. Krivorotov, and J.-P. Wang, Appl. Phys. Lett. 98, 102509 (2011). 12P. M. Braganca, O. Ozatay, A. G. F. Garcia, O. J. Lee, D. C. Ralph, and R. A. Buhrman, Phys. Rev. B 77, 144423 (2008). 13S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, J. A. Katine, and M. Carey, J. Magn. Magn. Mater. 286, 375 (2005). 14J. C. Slonczewski, J. Magn. Magn. Mater. 247, 324 (2002). 15D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008). 16J. Xiao, A. Zangwill, and M. D. Stiles. Phys. Rev. B 72, 014446 (2005). 17The ﬁeld-like spin torque (J. C. Slonszewski, Phys. Rev. B 71, 024411 (2005)) is not included here as it is negligible in spin valve devices. 18I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, D. C. Ralph,and R. A. Buhrman, Science 307, 228 (2005). 19M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, National Institute of Standards and Technology Technical Report No. NISTIR 6376 1999. 20See supplementary material at http://dx.doi.org/10.1063/1.3635782 for movies of the micromagnetic simulation results. 21D. Houssameddine, U. Ebels, B. Delae ¨t, B. Rodmacq, I. Firastrau, F. Pon- thenier, M. Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda, M.-C. Cyrille, O. Redon, and B. Dieny, Nature Mater. 6, 447 (2007). FIG. 2. (Color online) (a) Micromagnetic simulation of the time trace of hmxiat the two ends (blue, as averaged over the left 60 nm of the FL, and red, averaged over right 60 nm (see inset)) of the FL for AP ( hmxi/C251) and P (hmxi/C25/C0 1) initial conﬁgurations using a positive impulse Ip¼þ10 mA, tp¼50 ps (FWHM). Dashed lines: initial P conﬁguration, solid lines: ini- tially AP. Inset: the calculated easy-axis component of the inhomogeneous dipole ﬁelds acting on the FL. (b) Simulated time trace of hmxifor AP and P initial conﬁgurations using a negative impulse Ip¼/C013.6 mA and tp¼50 ps. (c) and (d) Simulated time trace of hmzifor the same conditions as (a) and (b), respectively; note different time scale for (c) and (d). FIG. 3. (Color online) Simulated switching probabilities Psat zero tempera- ture. Top ﬁgures: P1¼0.2 and P2¼0.37, bottom ﬁgures: P1¼0.2 and P2¼0.0, i.e., no spin-torque from the in-plane polarizer/analyzer. Rectan- gles: P-to-AP switching, circles: AP-to-P. (a) and (b) Psfor 50 ps Ipimpulses for (a) positive and (b) negative pulses. (c) and (d) Psfor 100 ps impulses. (e) and (f) Psfor 50 ps impulses assuming zero dipole ﬁeld from the polar- izer layers acting on the FL.102507-3 Lee, Ralph, and Buhrman Appl. Phys. Lett. 99, 102507 (2011) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.251.14.35 On: Tue, 23 Dec 2014 16:59:34
5.0056995.pdf	Reversible strain-induced spin–orbit torque on flexible substrate Cite as: Appl. Phys. Lett. 119, 042402 (2021); doi: 10.1063/5.0056995 Submitted: 17 May 2021 .Accepted: 19 July 2021 . Published Online: 29 July 2021 Grayson Dao Hwee Wong,1,2 Calvin Ching Ian Ang,1 Weiliang Gan,1 Wai Cheung Law,1,2 Zhan Xu,1,3 Feng Xu,3 Chim Seng Seet,2and Wen Siang Lew1,a) AFFILIATIONS 1School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371 2GLOBALFOUNDRIES Singapore Pte. Ltd., 60 Woodlands Industrial Park D St 2, Singapore 738406 3MIIT Key Laboratory of Advanced Metallic and Intermetallic Materials Technology, School of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China a)Author to whom correspondence should be addressed: wensiang@ntu.edu.sg ABSTRACT We propose the use of mechanical strain and mild annealing to achieve reversible modulation of spin–orbit torque (SOT) and Gilbert damping parameter. X-ray diffraction results show that the residual spin–orbit torque enhancement and Gilbert damping reduction, due to the post-mechanical strain treatment, can be reset using mild annealing to alleviate the internal strain. The spin Hall efﬁciency of the heat-and strain-treated Pt/Co bilayer was characterized through spin-torque ferromagnetic resonance, and it was found that the device couldswitch between the strain enhanced SOT and the pristine state. The Gilbert damping parameter behaves inversely with the spin Hall efﬁ- ciency, and therefore, strain can be used to easily tune the device switching current density by a factor of /C242 from its pristine state. Furthermore, the resonance frequency of the Pt/Co bilayer could be tuned using purely mechanical strain, and from the endurance test, thePt/Co device can be reversibly manipulated over 10 4cycles demonstrating its robustness as a ﬂexible device. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0056995 The ability to manipulate magnetization has helped the current- induced spin–orbit torque (SOT) gather a considerable amount of interest in recent decades.1–6SOT is induced by a pure spin current that is generated as the result of a spin–orbit interaction when a charge current passes through a non-magnetic metal.6,7In heavy-metal/ ferromagnetic (HM/FM) heterostructures, the SOT is contributed by two established phenomena: the spin Hall effect (SHE) in the HMand/or the Rashba–Edelstein effect at the HM/FM interface. 8–11The spin Hall efﬁciency heffis commonly used to quantify the performance of this charge-to-spin conversion, and it is ideal to have a large hefffor better energy-efﬁcient memory. To date, most studies are concentrated around HM with a strong spin–orbit coupling (SOC), such as Pt, b-Ta, and b-W, topological insulators, and even antiferromagnetic materials.12–18 To further push the boundaries of the heff, many efforts have been devoted to manipulating the extrinsic contribution of the spin Hall effect (SHE). Such works include alloying of the HM with lighter conductive metals, usage of insertion layers within the HM and the varying deposition condition of the HM, and many others.19–24The extrinsic SHE mechanism capitalizes on electron scattering caused byimpurities within the HM, and the two most prominent scattering processes are skew scattering and side-jump scattering.25,26Although theheffcan be easily enhanced through tuning the resistivity of the HM, its manipulation after the device fabrication is irreversible.22–24 Among them, the use of mechanical strain is a promising candidatenot only for enhancing the h effbut also for tuning it reversibly.27,28 Previous works have demonstrated SOT enhancement with the use of strain;29however, the ability to revert the enhancement has yet to be demonstrated and research is required to further develop the use of mechanical strain into a feasible option for the manipulation of the SOT. In this work, we demonstrate the ability to manipulate the strain- mediated SOT enhancement reversibly in Pt/Co using a combination of mechanical strain and mild annealing. By annealing Pt/Co at mild temperatures, the internal strain induced by mechanical tensile strain is alleviated, and this has been conﬁrmed using x-ray diffraction (XRD). When the internal strain is removed, the device behaves simi- larly to its pristine state making further manipulation of the device possible. The generated spin current was characterized using the spin- torque ferromagnetic resonance technique (ST-FMR), and the Gilbert Appl. Phys. Lett. 119, 042402 (2021); doi: 10.1063/5.0056995 119, 042402-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldamping parameter of Pt/Co was found to behave inversely with the manipulated SOT. Furthermore, using the mechanical tensile strain, the resonance ﬁeld of Pt/Co devices can be tuned allowing for micro- wave detection applications. These ﬁndings establish a unique tech-nique to inﬂuence the strain-mediated SOT and have a considerablecontribution to the development of ﬂexible spintronics devices. The effects of tensile strain and mild annealing on the spin cur- rent generation of Pt/Co bilayers are characterized using the spin-torque ferromagnetic resonance (ST-FMR) measurement. The bilayerPt(5 nm)/Co(5 nm) ﬁlms used in this study were deposited using mag-netron sputtering onto unstrained ﬂexible Kapton at room tempera- ture using an Ar pressure of 2 mTorr and a base pressure lower than 5/C210 /C08Torr. A Ti(5 nm) seed and a cap layer were used for ﬁlm adhesion and oxidation prevention, and from the previous study, itwas shown that Ti does not contribute to spin current generation. 29 Within the bilayer, Pt takes up the role of the SOT generation via SHE due to its strong SOC. ST-FMR devices and coplanar waveguides (CPWs) were patterned using optical lithography. Figure 1(a)illustrates the ST-FMR measurement setup and device. During the ST- FMR measurement, a microwave radio frequency (RF) charge current ðJCÞis injected into the CPW and along the longitudinal direction of the microstrip device (10 /C250lm2). Simultaneously, an in-plane external magnetic ﬁeld ðHextÞis applied at a 45/C14angle with respect to the longitudinal direction of the device. The RF current passing through the Pt layer generates an oscillating transverse spin current by SHE, which will then enter the adjacent Co layer. The magnetizationof the Co layer experiences an in-plane and out-of-plane torque fromthe RF current. 16,17When the RF spin current frequency matches the precessional frequency of the magnetization, the FMR is established, and the oscillating torques will result in the oscillation of the device resistance due to anisotropic magnetoresistance in the Co layer. Byusing a bias tee, the mixing of the RF current and the oscillating resis-tance is measured as a rectiﬁed DC voltage signal ðV mixÞ. The magnitude of the strain ewas approximated using e¼T=2R,w h e r e Tand Rare the total thickness of the substrate (120lm) and the bilayer structure and the curvature radius of the FIG. 1. (a) Schematic illustration of the Pt/Co bilayer device for the ST-FMR measurement. The green and navy blue arrows represent the precessing magnetiza tion in the Co layer and the applied external ﬁeld, respectively. An RF current was applied along the longitudinal direction ( x-axis) of the device generating two orthogonal torques as it passes through the heavy metal. Photo of strained ST-FMR devices on the ﬂexible Kapton substrate and the optical image of the device are as shown in the i nset. (b) X-ray reﬂectivity proﬁles for Pt(5 nm)/Co(10 nm) ﬁlms at different steps of the process: step ‹is the pristine ﬁlm, step ›is the pristine ﬁlm annealed at 150/C14C for 1 h, step ﬁis the tensile strain treatment of epost¼1:5% for 1 h, and step ﬂis the annealing process at 150/C14C for 1 h. (c) X-ray diffraction spectra of Pt(25 nm)/Co(25 nm) ﬁlms demonstrating a right shift in the Pt(111) peak shift when strained and back when treated with mild annealing.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 042402 (2021); doi: 10.1063/5.0056995 119, 042402-2 Published under an exclusive license by AIP Publishingmold, respectively.30All strains used in this work are mechanical ten- sile strain, and the direction of the strain is along the longitudinaldirection of the microstrip. Two methods of strain measurement wereused. The ﬁrst is the strain treatment, where the sample is strained at a speciﬁc e postfor 1 h and measured at its relaxed state, while the second is an in situ strain measurement where the sample is strained at ein during measurement. X-ray reﬂectivity (XRR) spectroscopy was performed on Pt(5 nm)/Co(10 nm) at different steps of the characterization processto determine the effects of strain and mild annealing on the interfacialroughness between Pt and Co. The sample used throughout the differ-ent step processes is the same, and the spectra are shown in Fig. 1(b) . The different steps of the process are: step ‹is where the ﬁlm is pris- tine, and this is used as a reference; step ›is the pristine ﬁlm after being vacuum annealed at 150 /C14C for 1 h; step ﬁis the annealed ﬁlm treated with a tensile strain of epost¼1:5% for 1 h; and step ﬂis the strain-treated ﬁlm after annealing at 150/C14Cf o r1 h .F r o mt h eX R R measurements (see the supplementary material ), no signiﬁcant change in interfacial roughness was observed, and this concurs with previous study that the strain-mediated SOT enhancement is a bulk effect dueto the extrinsic SHE. 29,31X-ray diffraction measurement was also per- formed on Pt(25 nm)/Co(25 nm) ﬁlms for steps ‹,›,ﬁ,a n dﬂ. Similar to the XRR measurement, the XRD sample used for all foursteps is the same. From Fig. 1(c) , the Pt(111) peak shifts right after the strain treatment indicating that the internal strain persists within theﬁlm even after the strain has been removed. However, upon treatingthe ﬁlm with mild annealing, the Pt(111) peak shifted back. This dem-onstrates the use of annealing as a means to relieve the residual inter-nal strain-induced and suggests that the strain-mediated SOTenhancement can be reversed. Unlike the Pt(111) peak, the Co(002)peak remains stationary, and this difference in response found in theCo and Pt layers is due to their different Poison’s ratios. 32This implies that the Co layer is unaffected by both the strain and mild annealing. Figure 2(a) shows the ST-FMR spectra for bilayer Pt/Co mea- sured at a microwave power of 12 dBm with a frequency range of8–17 GHz in steps of 1 GHz. The measured V mixconsists of a symmet- ric and anti-symmetric Lorentzian function, which can be expressed as Vmix¼V0SFSHextðÞ þAFAHextðÞ ½/C138 ; (1) where V0¼/C01 4dR ducl0IRFcosu 2pDHd f =dHext ðÞ jHext¼Hres; FSHextðÞ ¼DH2 Hext/C0Hres ðÞ2þDH2; and FAHextðÞ ¼DHH ext/C0Hres ðÞ Hext/C0Hres ðÞ2þDH2: Here, V0,DH,Hext,S,a n d Aare the scaling factors, linewidth, the applied external ﬁeld, the magnitude of the symmetric and anti-symmetric components of the V mix, respectively. The symmetric com- ponent is proportional to the damping-like torque, and theanti-symmetric component is the result of the sum of the Oersted ﬁeld and the ﬁeld-like torque. 16,33The peak-to-peak voltage VP/C0Pof theST-FMR spectra decreases with increasing einas shown in Fig. 2(b) . From Eq. (1), there are several contributing factors such as DH,a n d ðdf=dHextÞjHext¼Hrescan lead to a change in VP/C0P. However, the magni- tude of VP/C0Pis primarily inﬂuenced by the resistivity of Pt as an increased resistivity would decrease the current density through the Pt layer. When the tensile strain is employed along the longitudinal direc- tion of the microstrip, the strip elongates and narrows along the direc- tion of strain resulting in an enhancement in resistivity. To determine the change in Pt resistivity, a separate set of single layer Pt(5 nm) microstrips were fabricated and characterized using a semiconductor analyzer at different steps as shown in Fig. 3(a) . At step ‹, the ﬁlm is in its pristine state after fabrication. To set the device, the sample is annealed at 150/C14Cf o r1 hi ns t e p ›. The resistivity slightly decreased as a result of the improvement in ﬁlm quality from the mild annealing. Thereafter in step ﬁ, a tensile strain of epost¼1:5% was applied for 1 h. During the strain treatment, the resistivity increases as the microstrips are stretched along the longitudinal direction, resulting in a narrower cross-sectional area. Relaxing the ﬁlm after the treat- ment for measurement, the residual strain within the Pt retains the enhanced resistivity as shown in the plot. For step ﬂ, the sample was annealed at 150/C14C for 1 h before characterization, and upon mild annealing, the resistivity of Pt decreases as the internal strain caused by the strain treatment is relieved. Finally, steps /C176and–are repeated steps that are the same as steps ﬁandﬂ, respectively, that show the repeatability of the process. The strain response of Co resistivity was measured and found to be negligible in contrast to Pt (refer to the sup- plementary material ). Since the Pt layer thickness is much larger than its spin diffusion length, the ﬁeld-like torque in bilayer Pt/Co can be assumed to be neg- ligibly small as shown in previous work.16,29,34Using this approxima- tion, the spin Hall efﬁciency for the Pt/Co bilayer is calculated by the following expression: FIG. 2. (a) Measured ST-FMR spectra of the Pt/Co bilayer while applying ein¼1:5% for frequencies between 8 and 17 GHz using a microwave power of 12 dBm. (b) In situ strain dependence of VP/C0Pmeasured at 12 GHz. (c) Leftward shift of ST-FMR spectra due to the tensile strain at varying ein. (d) In situ strain dependence of HRes.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 042402 (2021); doi: 10.1063/5.0056995 119, 042402-3 Published under an exclusive license by AIP Publishingheff¼S Ael0MStCotPt /C22hﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ4pMeff Hresr ; (2) where tCoandtPtare the thicknesses of Co and Pt layers, respectively, MSis the magnetization saturation, and Meffis the effective magneti- zation. The MSof bilayer Pt/Co measured at epost¼0% and 1.5% was obtained to be 1220 630 and 1130 640 emu/cc3,r e s p e c t i v e l y ,w h i c hi s within a range consistent with other works, and therefore, the mag- netic proximity effect is assumed to be negligible in this study.35–38To obtain the required Meff, the in-plane magnetization Kittel equation f¼c=2pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ðHresþHKÞð4pMeffþHresþHKÞp was used, where cis the gyromagnetic ratio and HKis the total magnetic anisotropy ﬁeld. Having similar behavior as the qPt,t h e heffat various measurement s t e p si sa ss u m m a r i z e di n Fig. 3(b) . From the proportionality between both the qPtandheff, the strain-mediated SOT enhancement is a result of the extrinsic SHE in the Pt layer. Predominantly, the DHof the ST-FMR spectra is broadened by extrinsic contribution such as the ﬁlm inhomogeneous broadening term ðDHÞand two magnon scattering. Two magnon scattering results in a nonlinear frequency dependence of the DH, which is not observed in the measured samples.39The effective Gilbert damping parameter ðaeffÞwas calculated from the DHdependences of the fre- quency expressed as DH¼DH0þ4pfaeff=c;where DHis a result of sample imperfections that are assumed to be frequency independent, and the data are as shown in Fig. 3(c) . The two main aeffcontributors of bilayer Pt/Co are the intrinsic Gilbert damping ðaintÞfrom Co and the damping introduced by the spin pumping effect ðaSPÞdue to the adjacent Pt.40,41aintremains unchanged as it is independent of the strain-induced magnetic anisotropy.42–44TheaSP, however, is highly dependent on the spin pumping effect at the interface between the Ptand Co layers. An enhancement in the extrinsic SHE will result in agreater spin pumping effect and, hence, larger a SPcontribution. Therefore, aeffh a sa ni n v e r s et r e n da sc o m p a r e dt ot h e qPtandheff.The effects of strain and mild annealing on the critical switching current density JC0of an in-plane magnetization SOT device can be evaluated using the following equation: JC0/C252e /C22haeff heff4pMeff 2/C18/C19 MStFM: (3) From this equation, JC0is proportional to the ratio aeff=heff, and a decrease in this ratio will denote a lower JC0.35,45The inset in Fig. 3(c) shows how the JC0can be controlled using a combination of mechani- cal strain and mild annealing. This method allows the JC0to alternate between /C2490% and /C2450% of the pristine JC0, allowing for an addi- tional degree of freedom in inducing magnetization reversal of the SOT device. Aside from SOT manipulation, the mechanical strain can also be used to tune the resonance frequency by shifting the FMR spectrum, and from Fig. 2(c) , a left shift motion of the ST-FMR spectra is observed as the in situ tensile strain applied increases.46The shift in HResis attributed by the magnetoelastic anisotropy induced by the mechanical tensile strain. This additional anisotropy has an easy axis perpendicular to the uniaxial anisotropy generated by the external magnetic ﬁeld, which will result in a shift in the magnetic easy axis of the Pt/Co bilayer.30Figure 2(d) shows the HResdependence of ein.T h e Pt/Co device has a tunable HReswith a magnitude of /C012366O ep e r unit ein. Using this tuning capability, the detectable HRescan be adjusted based on the applied strain and then reversed by relaxing thedevice. Figure 4(a) demonstrates how the Pt/Co device can switch between two states of H Resby applying strain and relaxing it. The ﬁrst cycle begins with the device in the pristine state measured at the relaxed position. Subsequently, the even cycles refer to the in situ strain d e v i c ew h i l et h eo d dc y c l e sa r em e a s u r e dw h e nt h ed e v i c ei sr e l a x e d . With every cycle, a distinct shift in HResis observed. This cycle of FIG. 3. (a) Resistivity of a single layer Pt(5 nm) microstrip measured at different steps of Nwith the inset illustrating the individual steps: step ‹is the pristine ﬁlm, step ›is the pristine ﬁlm annealed at 150/C14C for 1 h, step ﬁis the annealed ﬁlm treated with a tensile strain of epost¼1:5% for 1 h, step ﬂis the strain-treated ﬁlm annealed at 150/C14C for 1 h, and steps /C176and–are repeated treatment procedures that are the same as steps ﬁandﬂ, respectively. (b) From the ST-FMR measurements of the Pt/Co bilayer, heffand (c) Gilbert damping parameter as a function of Nare presented. The normalized switching current density of the Pt/Co bilayer is shown in the inset.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 042402 (2021); doi: 10.1063/5.0056995 119, 042402-4 Published under an exclusive license by AIP Publishingstraining and relaxing is performed continuously for a repeatability test as shown in Fig. 4(b) . For consistency of the cycles, the straining and relaxing are performed using a linear actuator attached with a stepper motor. The HResat both 0% and 1% strain are highly stable for 104cycles demonstrating the device robustness against the mechanical strain. In summary, we investigated the use of mild annealing and mechanical strain for reversible manipulation of the SOT. In the Pt/Cobilayer, XRD spectra show that the tensile strain induces a residual strain within the Pt layer that can be alleviated by treating the ﬁlm with mild annealing. Using a combination of these two treatmentmethods, the spin Hall efﬁciency and Gilbert damping parameter become versatile and can be tuned with ease even after fabrication. Apart from SOT manipulation, strain can be used to tune the reso-nance ﬁeld of the Pt/Co bilayer, and in the endurance test performed, the tunability of the device remains highly stable even after 10 4cycles. These results pave an alternative avenue for manipulating the SOTreversibly that can also be used as a tunable microwave detector. See the supplementary material for the interfacial roughness of the Pt and Co layers measured using XRR and the Co resistivity change at different Nsteps. AUTHORS’ CONTRIBUTIONS G.D.H.W. conceived the idea, designed this work, drafted the manuscript, and fabricated the devices for measurement. W.C.L. assisted in the development of the experimental setup. W.L.G., C.C.I.A., and Z.X. made scientiﬁc comments on the result. W.S.L.,C.S.S., and F.X. coordinated and supervised the entire work. All authors contributed to the discussion and the revision of the ﬁnal manuscript.This work was supported by an Industry-IHL Partnership Program (No. NRF2015-IIP001-001) and an EDB-IPP (Grant No.RCA-17/284). This work was also supported by the RIE2020 ASTAR AME IAF-ICP Grant No. I1801E0030. The authors declare that they have no competing interest. DATA AVAILABILITY The data that support the ﬁndings of this study are available within the article and its supplementary material .T h ed a t at h a ts u p - port the ﬁndings of this study are available from the corresponding author upon reasonable request. REFERENCES 1Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910–1913 (2004). 2J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005). 3G.-Y. Guo, S. Murakami, T.-W. Chen, and N. Nagaosa, Phys. Rev. Lett. 100, 096401 (2008). 4E. Saitoh, M. Ueda, H. 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Mater. 28, 1705928 (2018).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 042402 (2021); doi: 10.1063/5.0056995 119, 042402-6 Published under an exclusive license by AIP Publishing
1.5079313.pdf	Appl. Phys. Lett. 114, 042401 (2019); https://doi.org/10.1063/1.5079313 114, 042401 © 2019 Author(s).Spin-orbit-torque-driven multilevel switching in Ta/CoFeB/MgO structures without initialization Cite as: Appl. Phys. Lett. 114, 042401 (2019); https://doi.org/10.1063/1.5079313 Submitted: 30 October 2018 . Accepted: 13 January 2019 . Published Online: 29 January 2019 S. Zhang , Y. Su , X. Li , R. Li , W. Tian , J. Hong , and L. You Spin-orbit-torque-driven multilevel switching in Ta/CoFeB/MgO structures without initialization Cite as: Appl. Phys. Lett. 114, 042401 (2019); doi: 10.1063/1.5079313 Submitted: 30 October 2018 .Accepted: 13 January 2019 .Published Online: 29 January 2019 S.Zhang,a)Y.Su,a)X.Li,R.Li,W.Tian, J.Hong, and L. Youb) AFFILIATIONS School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan 430074, China a)Contributions: S. Zhang and Y. Su contributed equally to this work. b)Author to whom correspondence should be addressed: lyou@hust.edu.cn ABSTRACT Spin-orbit torque (SOT) has been proposed as an alternative writing mechanism for the next-generation magnetic random access memory (MRAM), due to its energy efﬁciency and high endurance in perpendicular magnetic anisotropic materials. However, the three-terminal structure of SOT-MRAM increases the cell size and consequently limits the feasibility of implementing high den- sity memory. Multilevel storage is a key factor in the competitiveness of SOT-MRAM technology in the nonvolatile memory mar-ket. This paper presents an experimental characterization of a multilevel SOT-MRAM cell based on a perpendicularly magnetizedTa/CoFeB/MgO heterostructure and addresses the initialization-free issue of multilevel storage schemes. Magneto-optical Kerreffect microscopy and micromagnetic simulation studies conﬁrm that the multilevel magnetization states are created by chang- ing a longitudinal domain wall pinning site in the magnet. The realization of robust intermediate switching levels in the commonly used perpendicularly magnetized Ta/CoFeB/MgO heterostructure provides an efﬁcient way to switch magnets for low-power,high-endurance, and high-density memory applications. Published under license by AIP Publishing. https://doi.org/10.1063/1.5079313 The conventional spin-transfer torque magnetic random access memory (STT-MRAM) offers non-volatility, high den-sities, and complementary metal-oxide semiconductor(CMOS) process compatibility. 1–4However, the key drawback of STT-MRAM is that reading and writing current share thesame path through the junction. 3–6Hence, the aging of the tunnel barrier is accelerated on injecting high-write-currentdensities, especially for switching on the nanosecond timescale. Moreover, the read disturb challenge grows with thescaling technology as the read-to-write current ratiodecreases. Fortunately, the concept of a three-terminal mag-netic memory device based on a spin-orbit torque (SOT)effect has been recently proposed in heavy metal (HM)/fer-romagnetic metal (FM)/oxide heterostructures, where themagnetic bit is written by a current pulse injected throughthe bottom HM, and a magnetic tunnel junction (MTJ) can beemployed to read the state of the magnetic bit, namely, theSOT-MRAM. 3,4,7–11The decoupled write and read paths of SOT devices can naturally resolve the problems related to the endurance and reliability of conventional two-terminalSTT-MRAMs. Moreover, separate optimization is available fortuning the two independent read and write channels, relaxing the high magnetoresistance ratio and low resistance-areaproduct simultaneously required for MRAM. H o w e v e r ,t h el o ws t o r a g ed e n s i t yp r o b l e mi naS O T - M R A M becomes serious owing to its three-terminal architecture. Onekey solution is the multilevel cell (MLC) conﬁguration, a maturetechnique already utilized in ﬂash memories, which can providean enhanced integration density. Until now, the demonstration of MLC in SOT-MRAM has been rarely reported. Four-bit SOT- MLC was demonstrated by connecting two MTJs in one storageelement. 12However, the fabrication process of such a design is complicated. Recently, SOT-MLC design was reported in a Co/Ptmultilayer ferromagnet by controlling the multidomain forma-tion through the current pulse and also in Pt/Co/Ta/Co multi-layers by tilting the magnetization through an external magneticﬁeld or antiferromagnetic interlayer coupling. 13–16By contrast, from the material point of view, materials such as CoFeB whichcombine high spin polarization and a low Gilbert damping con-stant are highly desirable for spintronic devices. In addition, Ta/ CoFeB/MgO heterostructures with perpendicular magnetic anisotropy offer high scalability as the shape anisotropy ﬁeld Appl. Phys. Lett. 114, 042401 (2019); doi: 10.1063/1.5079313 114, 042401-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apleffect is negligible and have been commonly used in perpendicu- lar MTJ with high tunnel magnetoresistance.2,17,18 Here, we report that multilevel magnetization states can be controlled in perpendicularly magnetized Ta/CoFeB/MgO het- erostructures by modulating the in-plane ﬁeld and are indepen- dent of the initial state. The ﬁlm stack consists of Ta (10 nm)/Co 40Fe40B20(CoFeB, 1.2 nm)/MgO (1.6 nm)/Ta (20 nm) (from the bottom), which is sputtered on a thermally oxidized Si substrateat room temperature. This heterostructure closely resemblesthe one described in our previous work. 19As depicted in Fig. 1(a) , when an in-plane charge current ﬂows through the Ta layer, thespin current is generated due to the spin Hall effect (SHE) in Taand is transmitted across the Ta/CoFeB interface. This results inthe application of a torque on the CoFeB layer. The devicesbased on such a heterostructure are patterned into a Hall barwith dimensions of 400 (length) /C250 (width) lm 2[Fig. 1(b) ]. The anomalous Hall resistance RH, which is proportional to the aver- age out-of-plane magnetization component of CoFeB, is mea-sured under a small current of 0.1 mA. Measurements as afunction of a vertical magnetic ﬁeld exhibit sharp switching,indicating a good perpendicular magnetic anisotropy (PMA) inthe cells [ Fig. 1(c) ]. Current-induced switching is measured by applying a current pulse (0.5 s in duration) under the assistanceof an in-plane magnetic ﬁeld H xcollinear to the current. With a ﬁxed magnetic ﬁeld in the þx direction, sweeping a quasistaticin-plane current generates hysteretic magnetic switching between the M z>0 and M z<0 states, with the positive current favoring M z>0[Fig. 1(d) ]. The switching direction changes when the in-plane ﬁeld is inversed. This is consistent with nega- tive spin Hall angle hSHin Ta, which was reported to be –0.09 in our previous work.19 According to SHE, when the charge current ﬂows along x, the spin moments within the generated spin current point in the ydirection. The effective magnetic ﬁeld produced by SOT can be expressed as ~HSOT/~M/C2y. As a result, the z-component of the SOT effective ﬁeld, HSOT z,i se x p r e s s e da s8 HSOT z¼/C22h 2eM sthSHJm x; (1) where /C22his Planck’s constant, eis the electron charge, Msis the saturation magnetization, tis the thickness of the CoFeB layer, hSHi st h es p i nH a l la n g l eo fT a ,a n d Jis the current density. It is reported that the function of the in-plane magnetic ﬁeld Hxis to orient the magnetic moments within the domain wall (DW) to have a signiﬁcant in-plane component mx.20Therefore, a tunable effective ﬁeld HSOT zc a nb ee x e r t e do nt h em a g n e t i cﬁ l mb yv a r y - ing the magnitude and the direction of Hx. Hence, the domain conﬁguration and the resultant Hall resistances of the Hall bar can be controlled by varying the external magnetic ﬁeld Hx. This concept is conﬁrmed by our measurement of RH–J loops under various Hx, as depicted in Fig. 2 . Under the in-plane magnetic ﬁeld of abs( Hx)<20Oe, the magnetization switching is incomplete and results in an RHvalue smaller than the value at Hx¼620 Oe. In addition, the magnetization switching even starts to occur at the very small in-plane ﬁeld abs( Hx)/C202O e , which indicates that DW could move under a small SOT effective ﬁeld HSOT zin our ﬁlm structure. This magnetic ﬁeld range (2–20 Oe) along with the switching current density ( <3.5/C2106A cm/C02) is lower than the reported value in similar Ta/CoFeB/ MgO structures.21,22With an Hxlarger than 20 Oe, for instance, when Hx¼50Oe, 100Oe, and 200 Oe, there is no signiﬁcant change in the RHvalue (see Sec. S1 of the supplementary mate- rial). By analyzing the dependence of the SOT efﬁciency on the in-plane ﬁeld, a Dzyaloshinskii–Moriya interaction (DMI) effec- tive ﬁeld ( HDMI) is obtained to be around 100 Oe (referring to Sec. S2 of the supplementary material ), stabilizing a right handed N/C19eel wall of the ferromagnet. The DMI constant was estimated to be around 0.13 mJ m/C02, which is close to the reported values in similar structures.21,23 FIG. 1. SHE-driven magnetization switching. (a) Sample geometry for SHE switch- ing measurements. In-plane current ﬂowing through tantalum along the x direction(electrons along the –x direction) causes spin separation across the thickness of the tantalum (z-direction). This results in the accumulation of electrons with y- polarized spins at the Ta/CoFeB interface, which offers SHE-induced spin torque tothe magnetization and thereby moves the domain wall with the assistance of the in-plane magnetic ﬁeld H x.Irepresents the applied direct current, and the green arrow denotes the current direction. The blue and pink arrows represent the orientations of accumulated spins. The red and orange colors in the CoFeB ﬁlm indicate themagnetic domains with opposite directions of magnetic moments, which is either“upwards” or “downwards”. (b) Optical micrograph of the fabricated Hall bar struc- ture. The width of the Hall bar ( W)i s5 0 lm, and the length ( L) is 400 lm. (c) Anomalous Hall resistance R Has a function of the perpendicular magnetic ﬁeld Hz (RH-Hzloop). (d) Current-induced magnetization reversal when Hxis parallel (red) or antiparallel (green) to the current direction. FIG. 2. RH-Jloops measured under different values of (a) positive and (b) negative in-plane magnetic ﬁelds Hx.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 114, 042401 (2019); doi: 10.1063/1.5079313 114, 042401-2 Published under license by AIP PublishingIn most multilevel storage mechanisms, extra initialization steps are required to reset the memory cell for the next writing event.24–26This results in a prolonged writing time and increased power consumption. MLC design using an SOT mech- anism in Co/Pt multilayers resolves this issue.13Here, we also emphasize that the initialization is no longer necessary in our MLC scheme. As shown in Fig. 3 , by applying different values of the magnetic ﬁeld Hxsuch as 620,612,67, and62O e ,t h eM L C can be set to eight resistance states at a ﬁxed current density of J¼4.3/C2106Ac m/C02. The Hall resistance solely relies on the Hx value and does not rely on the previous magnetic states. In sucha scheme, the state can be written to a target level from any of the eight levels, and it can also reach all other levels from oneparticular level. This means that any of the multistates can be achieved by setting the corresponding in-plane magnetic ﬁeld without any initialization step. Here, the demonstrated eightHall resistance states can represent an MLC with 8 bits. Themultilevel states hint that the evolution of domain structures must play an important role. The domain conﬁgurations can be visualized by using Magneto-optical Kerr effect (MOKE) spectroscopy, as shown in Fig. 4 .W es a t u r a t e dt h em a g n e t i z a t i o n M(pointing downwards) in the Hall bar by using a sufﬁciently high external ﬁeld and thenapplied a current pulse ( J¼3.5/C210 6Ac m/C02in current density and 0.5 s in duration) along the þx direction in the presence of different magnitudes of the in-plane magnetic ﬁeld Hx.U n d e r Hx¼20 Oe, most of the magnetic moments were reversed to point upwards [ Fig. 4(a) ]. Then, Hxswept to /C020 Oe step by step, and the MOKE images were captured at every step, as shown in Figs. 4(b)–4(i) . It can be seen clearly that the pinning site of the longitudinal domain wall depends on the magnitude of Hx.T h e magnetization orientation in the reversed domain was deter- mined by the zcomponent of the Oersted ﬁeld generated by the current pulse (a detailed analysis can be found in Sec. S3 in thesupplementary material ). By reversing the direction of the cur- rent, the position of the m z¼1a n d mz¼/C01 polarized domains was reversed (referring to Sec. S4 of the supplementary material ). As a complementary conﬁrmation, we also performed Object-Oriented Micro-Magnetic Framework (OOMMF) micro-magnetic simulations (details can be found in Sec. S5 in thesupplementary material ) based on the generalized Landau– Lifschitz–Gilbert (LLG) theory in order to obtain the relationship between H xand the DW pinning position.27Through the micro- magnetic simulations, the controlling of DW displacement bythe in-plane ﬁeld and an initialization-free process are demon- strated. As shown in Fig. 5(a) , with out-of-plane Oersted ﬁelds acting on the top and bottom edges and an in-plane currentﬂowing through the whole magnet simultaneously, 7 differentmagnitudes of in-plane ﬁelds induce 7 stable magnetizationstates of the magnet. Then, an identical in-plane ﬁeld of /C0200 Oe is applied on the magnet with the aforementioned 7 initial states. We observe that whatever the initial state be, thesame ﬁnal state is obtained under an identical in-plane ﬁeld [ Fig. 5(b)]. That is to say, in our multilevel cell device, ﬁnal states are only controlled by the in-plane ﬁeld, and no initialization is needed in the working process. The simulations show qualitativeagreement with our experimental results except the larger in-plane ﬁelds used in simulations. The main reason for such FIG. 3. Multilevel states in the Ta/CoFeB/MgO heterostructure controlled by eight different in-plane magnetic ﬁelds Hxunder a current with a current density of J¼4.3/C2106Ac m/C02. (a) Sequence of applied magnetic ﬁelds Hx. (b) Corresponding RHresponse to the Hxsequence shown in (a). FIG. 4. MOKE microscopy images of a Hall bar device after applying a series of external magnetic ﬁelds Hx: from (a) 20 Oe to (i) /C020 Oe in the presence of a cur- rent pulse with an amplitude of J¼3.5/C2106Ac m/C02and a duration of 0.5 s along theþx direction. Bright and dark regions in the channel correspond to magnetiza- tion pointing upward ( þmz) and downward ( /C0mz), respectively. FIG. 5. Micromagnetic simulations of initialization-free switching with the Oersted ﬁeld and SOT acting on the mag- net. (a) The various initial states achievedunder different in-plane ﬁelds. (b) The cor-responding ﬁnal states when the in-plane ﬁeld switches to –200 Oe.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 114, 042401 (2019); doi: 10.1063/1.5079313 114, 042401-3 Published under license by AIP Publishingdiscrepancy is the imperfection of real material structures, and the temperature has not been taken into account in simulation. InFig. 5(a) , the arrows in the DW represent the direction of magnetization moments. It is clearly shown that in the presenceo ft h ei n - p l a n em a g n e t i cﬁ e l d ,t h em a g n e t i cm o m e n ti n s i d et h eDW prefers to align with the external ﬁeld. The magnitude of H x determines the tilting angle of the magnetic moments within the DW. As a result, according to Eq. (1), perpendicular SOT effective ﬁelds are different under various in-plane ﬁelds. For our CoFeBﬁlm considered here, the longitudinal DW is likely to be of N /C19eel- type due to the DMI effective ﬁeld, as previously mentioned. Inthe case of H x¼0,HSOT zis zero as mx¼0 within the DW. The DW ﬁnally displaces to the center of the Hall bar under the com- bined effect of the Oersted ﬁeld, DMI ﬁeld, exchange ﬁeld, mag-netostatic ﬁeld, etc., in accordance with R H¼0X. However, for Hx6¼0,HSOT zdrives the DW, moving it to a new pinning position and resulting in an intermediate magnetization between full sat- uration and demagnetization. The DW pinning effect is preferredto lower the total energy. The ﬁnal steady position of the DWdepends on the orientation and magnitude of H SOT z. In our demonstration, in principle, the pinned position of the longitudinal DW, and consequently anomalous Hall effect resistance, can be tuned in an analogue manner by magneticﬁelds, which indicates that our devices may exhibit a memristivebehavior. On the other hand, to conﬁrm the feasibility of scalingdown, we performed OOMMF micromagnetic simulations with t h ec e l ls i z ed o w nt o8 0 /C280 nm 2, which still resulted in dual magnetic domains. It is possible to approach MLCs below100 nm by carefully adjusting the material properties.Consequently, our devices can be used as artiﬁcial synapses inartiﬁcial neural networks for neuromorphic computing. 28 Besides, due to the large memory capacity, MLC may become agood candidate for processing-in-memory. 29 In summary, we investigated that multilevel resistance states can be achieved in the perpendicularly magnetized Ta/ CoFeB/MgO heterostructure by controlling the magnetization conﬁgurations of the magnet through an in-plane magneticﬁeld. The DW moves orthogonally to the current ﬂow driven bySOT, and the ﬁnal pinning position of the DW is independent ofits initial state. These results are crucial in the realization of high-density and low-power-consumption SOT-MRAM and fer- romagnetic memristor applications. 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1.1453336.pdf	Microwave, structural, and magnetic properties of Cu/Fe/CoFe/Cu P. Lubitz, S. F. Cheng, F. J. Rachford, M. M. Miller, and V. G. Harris Citation: Journal of Applied Physics 91, 7783 (2002); doi: 10.1063/1.1453336 View online: http://dx.doi.org/10.1063/1.1453336 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/91/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Microwave, magnetic, and structural properties of nanocrystalline exchange-coupled Ni 11 Co 11 Fe 66 Zr 7 B 4 Cu 1 films for high frequency applications J. Appl. Phys. 103, 063917 (2008); 10.1063/1.2899094 Nanostructured magnetic Fe–Ni–Co/Teflon multilayers for high-frequency applications in the gigahertz range Appl. Phys. Lett. 89, 242501 (2006); 10.1063/1.2402877 Magnetic and microwave properties of CoFe PtMn CoFe multilayer films J. Appl. Phys. 99, 08C901 (2006); 10.1063/1.2163843 Evolution of magnetic coupling in ferromagnetic tunnel junctions by annealing J. Appl. Phys. 91, 7478 (2002); 10.1063/1.1447874 Effects of grain cluster size on coercivity and giant magnetoresistance of NiFe/Cu/CoFe/Cu/NiFe pseudo spin valves Appl. Phys. Lett. 77, 3435 (2000); 10.1063/1.1328053 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.239.20.174 On: Fri, 22 Aug 2014 09:54:34Microwave, structural, and magnetic properties of Cu ÕFeÕCoFeÕCu P. Lubitz,a)S. F. Cheng, F. J. Rachford, M. M. Miller, and V. G. Harris Naval Research Laboratory, Washington, DC 20375 The structure and the static and dynamic magnetic properties of pure Fe ﬁlms with a surface overlayer of Co 9Fe1were studied. These structures are potential components of spin-valve or tunneling devices in which small magnetic damping, large moment, low anisotropy and high spinpolarization may be advantageous.The ﬁlms are polycrystalline and have Cu under and over layers.The Fe layers studied are from 3 to 20 nm thick and the CoFe layer was usually 1 nm. With a CoFeoverlayer we found a range of Fe thicknesses from below 4 to near 6 nm in which low coercivityandnarrowferromagneticresonance ~FMR !linewidthresulted.Bothbelowandabovethisrangethe properties degraded, apparently because the 2 nm Cu top layer was inadequate to protect the Feagainst oxidation. Using extended x-ray absorption ﬁne structure, we found only bcc Fe; atomicforce microscopy shows a systematic decrease in roughness with increasing thickness of Fe, whichmay explain the magnetic hardness for the thinnest ﬁlms. Fe ~5n m !/CoFe ~1n m !, with a Gilbert a;0.004, has FMR linewidths about 2/3 those of Permalloy ﬁlms of comparable thickness. For some applications investigated, distinct advantages can be obtained using the high Qof the ferromagnetic system. © 2002 American Institute of Physics. @DOI: 10.1063/1.1453336 # I. INTRODUCTION Polycrystalline iron with grain size less than about 20 nm can be a nearly ideal soft magnetic material. Embodi-ments such as nano-crystalline alloys, 1multilayer structures with negligible magnetostriction2and single layer polycrys- talline thin ﬁlms3have been shown to display, respectively, very small coercivity, very large rotational permeability~;4000!and the minimum reported ferromagnetic linewidth for a metal, ~;15 Oe at 10 GHz !. To date, however, pure Fe has not been extensively explored as a ‘‘sense’’layer in spin-valve-type structures, although it was the soft component ofthe ﬁrst reported such material, Co/Cu/Fe. 4For some appli- cations its relatively small giant magnetoresistance ~GMR ! effect, its high conductivity, and its large moment are unat-tractive, but these liabilities can be overcome, or possiblyused advantageously, for other applications such as GMRbased magnetic sensors. The Gilbert relaxation parameter athat describes intrin- sic relaxation effects is usually obtained from ferromagneticresonance ~FMR !linewidths. Small values of ahave been reported for Fe.5,6However, the actual widths observed to date are somewhat larger than those attributable only to in-trinsic damping. For instance, Ref. 5 ﬁnds damping equiva-lent to aof about 0.002, corresponding to a linewidth in- crease of 0.7 Oe/GHz. However, the measured linewidthextrapolated to 9.5 GHz is about 45, roughly a factor of 3more than reported here. Linewidth additional to intrinsicmay arise from a variety of inhomogeneities, although thestrong exchange coupling of the moments in Fe averagesregions with size smaller than ;20 nm 1,7so that ﬁne grains and surface irregularities do not add signiﬁcantly to line-width. Many features of the Cu/Fe/Cu multilayer system have been studied extensively, both using single crystal or poly-crystalline Cu starting or ‘‘seed’’layers. The presence of Cu at both interfaces in our work is motivated by the need toprotect the Fe from oxidation and to simulate the interfacerequired in actual spin valves. Characteristics of this systeminclude initial growth of low moment fcc Fe on Cu up toabout 1 nm, then conversion of the total Fe layer to bccaccompanied by roughening of the surface and developmentof essentially the full moment for the entire layer. Kicuchielectron diffraction of Fe grown on Cu ~111!, the dominant texture of our Cu layers, indicates that a sixfold symmetrypersists at least to 11 nm of Fe, 8although this probably rep- resents a mosaic of ~110!twinned facets. In this study, we address the question of whether addi- tion ofa1n mC o 9Fe1at the top surface of a polycrystalline Fe layer, as is customarily used with permalloy ~Py!when it is the soft or sense layer in spin valves, adversely affects thesoft magnetic properties of the Fe. We used the alloy with90% Co, which we found earlier 7has essentially no magne- tostriction under our deposition conditions, as opposed to the95% alloy sometimes used; we also found linewidths as lowas 30 Oe for this alloy, i.e., somewhat wider than Fe or Py.We observed the ferromagnetic resonance of the system aswell as its magnetic switching properties. FMR, in additionto probing linewidths, allows us to determine the magneticmoment and anisotropies of the system. We ﬁnd that under some conditions, a thin CoFe layer may be added to the Fe surface with negligible degradationof the desirable magnetic switching properties, or of theFMR linewidth. II. EXPERIMENT Polycrystalline Cu/Fe/CoFe/Cu structures were made by magnetron sputtering at ambient temperature. Polished Siwafers with native oxide surfaces were used as substrates.Some ﬁlms of Cu/Fe/Cu were made by electron beam depo-sition using substrates held at 250 °C. These had marginally a!Electronic mail: lubitz@anvil.nrl.navy.milJOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 10 15 MAY 2002 7783 0021-8979/2002/91(10)/7783/3/$19.00 © 2002 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.239.20.174 On: Fri, 22 Aug 2014 09:54:34better linewidths and were thus selected for study of the fre- quency dependence of the linewidth from 1 to 40 GHz. Our other FMR studies were conducted at about 9.5 GHz and at 300 K using a commercial microwave spectrometer.Because of the high moment of our materials, only in-planeresonance could be observed with our equipment. Switchingbehavior, e.g., H c, was determined using vibrating sample magnetometry. The surface roughness was obtained usingatomic force microscopy ~AFM !. RMS roughness values and lateral size of representative features were obtained using thesystem software. Extended x-ray absorption ﬁne structure~EXAFS !was employed to determine if any of the Fe re- mained in the bcc phase. III. RESULTS Figure 1 shows the frequency dependence of the deriva- tive peak-to-peak linewidth for a Cu/Fe 20 nm/Cu structuremade by electron beam deposition, as a function of fre-quency. Clearly the nonintrinsic contribution is only a fewOe, while the slope is 1.24 Oe/GHz. The result correspondsto aof 0.004. When we used magnetron sputtering and reduced the Fe thickness, the widths were nearly as low, e.g., 17 Oe for 4–5nm of Fe. Furthermore, for Fe of thicknesses from 4 to about6 nm, the linewidths were not adversely affected by the pres-ence of a surface layer of 1 nm of CoFe. Figure 2 shows thedependence of the FMR linewidth on the Fe thickness forthis structure. For thinner or thicker Fe, the width increasedsigniﬁcantly. Changing the thickness of the CoFe layer af-fected the region of soft properties, with thicker CoFe in-creasing the ‘‘window.’’ For this reason, and because theseﬁlms were found to degrade further with time, we believe theharder properties are a result of surface oxidation. The coercivities show a pattern similar to the linewidth, indicating that oxidation increases both coercivity and line-width. In the range 4–6 nm of Fe the easy axis loop is nearlysquare with H cof about 6 Oe while in the hard direction Hc is about 2 Oe with a well deﬁned knee at about 4 Oe, con- sistent with the in-plane anisotropy as determined usingFMR. For structures with 7–10 nm Fe, the loops are stillmoderately square, but H cis about 16 Oe and there is only a slight different between easy and hard directions, althoughthe net in-plane uniaxial anisotropy ﬁeld is still only on the order of 5 Oe. Some Hcvalues are also included in Fig. 2. The in-plane FMR ﬁelds, Fig. 3, vary strongly with Fe thickness, and also correlate strongly with increased line-width and coercivity. The FMR ﬁelds can be related to aneffective moment, 4 pMeff, assuming the widely accepted value of g52.1. 4 pMeffis 19 kOe when the Fe layer is about 5 nm, only slightly less than that expected for bulk Fe.This falls considerably for the ﬁlms with wide lines. Thesereductions are not seen in direct measurements of Msuch as with a vibrating sample magnetometer and integrated FMR.This implies that a uniaxial ﬁeld with magnitude up to ap-proximately 6.5 kOe is acting on the magnetic layers. EXAFS did not reveal any Fe in bcc form, consistent with previous work on the Cu/Fe/Cu system in this thicknessrange. However, theAFM scans did show a systematic varia-tion in roughness with Fe thickness, indicating a rms rough-ness for 3, 5, 7 and 8 nm Fe layers, respectively, of 0.8, 0.4,0.34 and 0.23 nm. The lateral scale of observed surface fea-tures was much larger, being typically 50 nm with a slighttendency to be larger and more dispersed for the thickest ofthese. We also investigated some other candidates as protective or seed layers for Fe. Of these Ni was more effective than Cuin protecting the surfaces and also maintained soft magneticproperties. Al and Ag as protective layers produced verypoor quality Fe, often with linewidths of hundreds of Oe andcoercivity nearly as large, with nearly isotropic M-Hloops, and low effective moments. FIG. 1. Linewidth vs frequency. Some of these data were obtained by ﬁtting a Lorenztian line shape to the absorption and were converted to derivativepeak-to-peak widths by multiplying by A3/2. FIG. 2. Linewidth and selected coercivities vs frequency. Solid circles show linewidths as a function of Fe thickness for 1 nm of CoFe overlayer, opencircles for 0.5 nm Co on 4 nm Fe and 2 nm CoFe on 7 nm Fe. Trianglesindicate coercivity for 1 nm CoFe on 5- and 10-nm-thick Fe. FIG. 3. Resonant ﬁelds as a function of Fe thickness for Cu/Fe ~x!/CoFe 1 nm/Cu. Solid squares denote CoFe of 1nm thickness. Open squares are for0.5 and 2 nm of CoFe and 4 and 7 nm of Fe, respectively.7784 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Lubitz et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.239.20.174 On: Fri, 22 Aug 2014 09:54:34IV. DISCUSSION There is a moderate discrepancy between our work and some others as to the relaxation rate in pure Fe. The slope ofour data with frequency in Fig. 1 is clearly almost a factor of2 higher see Ref. 5, for example.The data of Schreiber et al. 5 seem to be consistent with a constant slope, whereas ours may even curve upward slightly, but our actual linewidthsare less than theirs at all common frequencies. Therefore it ispossible that the true frequency dependence of their line-width is masked by an inhomogeneous contribution which isreduced by the large applied ﬁelds needed for high frequencyFMR. A very recent article 9found the linewidths of single crystal Fe ﬁlms of thicknesses comparable to those of thisstudy to have nearly the same widths as ours at comparablefrequencies, leading to a relaxation rate of 1.26 310 8/s, i.e., essentially the same as we ﬁnd. The superior soft magnetic properties of these Fe layers, as indicated in Fig. 2, either without any CoFe, or for Fe/CoFe, at least when the Fe thickness is near 5 nm, are con-sistent with known soft magnetic properties 3,7of nanocrys- talline Fe or CoFe separately. The puzzle here is why, forsome thicknesses of CoFe deposited on top of Fe, the usuallyeffective exchange narrowing that commonly occurs is inef-fective and huge perpendicular anisotropy results. The EX-AFS data show that almost all of the Fe is in bcc form andmagnetometry indicates that it has essentially the bulk mo-ment. Therefore it is unlikely that any magnetic abnormali-ties are occurring in the Fe. However, surface or inter-granular oxidation is a likely problem for the thinner Felayers because the roughness was found to be comparable tothickness of the surface protecting Cu layer. Since the starting substrates were found to be ﬂat to about 0.2 nm, it is likely that the greater roughness seen isinduced during the martensitic transformation of initial fcciron to bcc, as reported in studies of the growth of Fe on Cu. 8 The persistence of large effects, particularly effective out-of-plane ﬁelds of many kOe, to thicknesses past 8 nm of Fe, iscurrently under investigation. Magnetostriction, even in thepresence of moderate strain at the Cu/Fe or CoFe/Cu inter-face, is not likely to be signiﬁcant since it is small in both ofthese magnetic materials. However, it is known that for Co, a low symmetry envi- ronment may confer uniaxial anisotropy ﬁelds of order 10kOe, e.g., hcp Co. In fact, the value of anisotropy in hcp Cois determined by the c/a axes ratio, 10among other factors. Anisotropy in hcp Co is so sensitive to c/a ratio that itsvariation with temperature causes changes in this quantity of;14 kOe, even over a moderate temperature range well be- lowT c.6Also note, in a low symmetry environment like Co/P~111!multilayers, perpendicular anisotropy is sufﬁcient to overcome demagnetization, resulting in easy out-of-planedirection. 11Hence, even though the CoFe is only ;20% of the magnetic structure, it is conceivable that it could inducethe up to 6.5 kOe of out-of-plane anisotropy seen. Further- more, slight variations of this anisotropy would be effectivein inducing coercivity and linewidth. V. CONCLUSIONS The system Cu/Fe/CoFe/Cu, which may be a good pros- pect as a soft, or sense layer component of spin-valve GMRstructures, has been shown to have nearly ideally soft mag-netic properties over a range of Fe thicknesses near 5 nm.However, for Fe thicknesses more than about 1 nm above orbelow this value, coercivity, linewidth and an out-of-planeanisotropy all increase sharply. Surface oxidation is impli-cated, especially for the thinner layers, since roughness wasfound to increase rapidly with decreasing Fe thickness in thisrange, reducing the effectiveness of the protective Cu layer.The effects for Fe thicker than 6 nm may arise from CoFehaving large out-of-plane anisotropy, as in the hcp Co, orCo/Pt multilayers, induced by the reduced symmetry andstrain arising from growth on textured Fe. Preliminary evaluations of GMR in spin-valve structures based on a sense layer of Fe/CoFe and a pinned layer ofCoFe/IrMn, indicate only moderate MR values, less thanpure CoFe. The linewidths, while narrow for parallel senseand pinned layers, broaden rapidly near the antiparallel, orhigh resistance conﬁguration, as is often seen when someNe´el coupling is present between the magnetic layers. 12 Possible adverse effects of high conductivity of pure Fe may be avoided by the addition of a few percent of suchsoluble elements as Ni or Si, either of which rapidly in-creases resistivity while Si also reduces magnetostriction. 13 The narrow linewidths of the Cu/Fe/CoFe/Cu structures studied indicate that the FMR has a very high Qat the lowest frequencies measured. While this would be a liability inrapid switching applications, it is advantageous for a class ofsensors we are considering as sensors of small microwavemagnetic ﬁelds in the 1–10 GHz range. 1G. Hertzer, IEEE Trans. Magn. 25,3 3 2 7 ~1989!. 2M. Senda and Y. Nagai, J. Appl. Phys. 65, 1238 ~1989!. 3P. Lubitz, M. Rubinstein, D. B. Chrisey, J. S. Horwitz, and P. R. Brous- sard, J. Appl. Phys. 75, 5595 ~1994!. 4A. Chaiken, P. Lubitz, J. J. Krebs, G. A. Prinz, and M. Z. Harford, Appl. Phys. Lett. 59, 240 ~1991!. 5F. Schreiber, J. Pﬂaum, Z. Frait, Th. Mu ¨hge, and J. Pelzl, Solid State Commun. 93, 965 ~1995!. 6S. M. Bhagat and P. Lubitz, Phys. Rev. B 10,1 7 9 ~1974!. 7P. Lubitz, S.-F. Cheng, K. Bussmann, G. A. Prinz, J. J. Krebs, J. M. Daughton, and D. Wang, J. Appl. Phys. 85, 5027 ~1999!. 8G. Gubbioti, L. Albini, S. Tacchi, G. Carlotti, R. Gunella, and M. De Crescenze, Phys. Rev. B 60, 17150 ~1999!. 9R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 ~2001!. 10W. J. Carr, Phys. Rev. 109, 1971 ~1958!. 11I. B. Chung, Y. M. Koo, and J. M. Lee, J. Appl. Phys. 87, 4205 ~2000!. 12W. Stoecklein, S. S. P. Parkin, and J. C. Scott, Phys. Rev. B 38, 6847 ~1988!. 13R. Bozorth Ferromagnetism ~Van Nostrand, New York, 1951 !.7785 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Lubitzet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.239.20.174 On: Fri, 22 Aug 2014 09:54:34
9.0000151.pdf	AIP Advances 11, 035316 (2021); https://doi.org/10.1063/9.0000151 11, 035316 © 2021 Author(s).Voltage-controlled spin-wave intermodal coupling in lateral ensembles of magnetic stripes with patterned piezoelectric layer Cite as: AIP Advances 11, 035316 (2021); https://doi.org/10.1063/9.0000151 Submitted: 15 October 2020 . Accepted: 15 February 2021 . Published Online: 08 March 2021 A. A. Grachev , E. N. Beginin , S. E. Sheshukova , and A. V. Sadovnikov COLLECTIONS Paper published as part of the special topic on 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials and 65th Annual Conference on Magnetism and Magnetic Materials ARTICLES YOU MAY BE INTERESTED IN Spin waves transport in 3D magnonic waveguides AIP Advances 11, 035024 (2021); https://doi.org/10.1063/9.0000185 The design and verification of MuMax3 AIP Advances 4, 107133 (2014); https://doi.org/10.1063/1.4899186 Spin–orbit-torque magnonics Journal of Applied Physics 127, 170901 (2020); https://doi.org/10.1063/5.0007095AIP Advances ARTICLE scitation.org/journal/adv Voltage-controlled spin-wave intermodal coupling in lateral ensembles of magnetic stripes with patterned piezoelectric layer Cite as: AIP Advances 11, 035316 (2021); doi: 10.1063/9.0000151 Presented: 6 November 2020 •Submitted: 15 October 2020 • Accepted: 15 February 2021 •Published Online: 8 March 2021 A. A. Grachev,a) E. N. Beginin, S. E. Sheshukova, and A. V. Sadovnikov AFFILIATIONS Laboratory “Magnetic Metamaterials,” Saratov State University, Saratov 410012, Russia Note: This paper was presented at the 65th Annual Conference on Magnetism and Magnetic Materials. a)Author to whom correspondence should be addressed: Andrew.A.Grachev@gmail.com. Laboratory “Magnetic metamaterials,” Saratov State University, Saratov 410012, Russia. ABSTRACT Here we report about the strain-tuned dipolar spin-wave coupling in the adjacent system of yttrium iron garnet stripes, which were strain- coupled with the patterned piezoelectric layer. Spatially-resolved laser ablation technique was used for structuring the surface of the piezoelec- tric layer and electrodes on top of it. Using a phenomenological model based on coupled modes equation, we demonstrate a voltage-controlled intermodal coupling in lateral magnonic stripes. The features of the tunable spin-wave coupling by changing the geometric parameters and the type of magnetization is demonstrated. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/9.0000151 I. INTRODUCTION The using elementary quanta of magnetic excitations (magnons or spin waves (SW)) as carriers of information signals has attracted more interest in recent years due to the possibility of transferring the magnetic moment (spin) of an electron without transferring an electric charge and without generating Joule heat inherent in semi- conductor technologies.1–5The properties of SWs are determined by the dipolar and exchange interactions in magnetic medias6–8and can change significantly during structuring of magnetic films. SWs are used for generation,9transmission10and the processing11,12of the information signals in micro- and nanoscale structures. A thin ferrites films is used for these purposes, for example, yttrium iron garnet (YIG) which demonstrates record low values of the spin-wave damping parameter.6 The voltage-controlled tunability of the spin-wave spectra in the thin magnonic films carried out due to the transformation of the effective internal magnetic field. The latter changes due to inverse magnetostriction (Villari effect) as a result of local deforma- tion of the magnetic film. It has been experimentally demonstrated that electrical field tunability of spin-wave coupling can be effec- tively used to control the magnon transport,13,14which led to thecreation of a class of spin-wave devices, such as two-channel directional couplers,15spin-wave splitters.16,17This demonstra- tion of spin-wave coupling phenomena opens the possibility of study the nonlinear dynamics of SW18and the mechanisms for the spin-wave coupling tunability. The using of YIG films opens a possibility to create functional elements of magnonic net- works based on the study the properties of spin waves propa- gating along irregular magnetic stripes with broken translational symmetry.19 Here, we use a numerical and experimental techniques to demonstrate the effects of voltage-controlled the spin-wave coupling in a system of three lateral magnetic stripes with a patterned piezo- electric layer. Spatially-resolved laser ablation technique was used for structuring the surface of the piezoelectric layer and electrodes on top of it. We show an effective tuning of the spin-wave charac- teristics using an electric field due to local deformation of the piezo- electric layer and the inverse magnetostriction effect in YIG stripes. Using a phenomenological model based on coupled modes equation, we demonstrate a voltage-controlled intermodal coupling in lateral magnonic stripes. The features of the tunable spin-wave coupling by changing the geometric parameters and the type of magnetization is demonstrated. AIP Advances 11, 035316 (2021); doi: 10.1063/9.0000151 11, 035316-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv A lateral structure consisted of three parallel-oriented magnonic stripes S1,S2andS3(Fig. 1(a)). Using the laser ablation technique based on fiber YAG:Nd laser with the high precision 2D galvanometric scanning module (Cambridge Technology 6240H) magnonic structure was fabricated from t=10μm-thick a yttrium iron garnet (YIG) film [(YIG) Y3Fe5O12(111)], grown on a 500 μm-thick gallium-gadolinium garnet [(GGG) Gd 3Ga5O12 (111)] substrate. A system of w=500μm-width with a distance of d=40μmbetween each other forms three spin-wave channels. The length of the magnonic stripes was 6 mm for S1,3and 8 mm for S2. The SW was excited using a microstrip antenna with 1 μm-thick and 30μm-width. The structure is placed in external static magnetic field, H0=1100 Oe. Field is oriented along the xaxis. This configuration is allows to excite the magnetostatic surface wave (MSSW) in stripe S2. A 200 μm-thick lead zirconate titanate (PZT) is used as a piezoelectric material. A 1 μm-thick copper electrode is placed (“GND” in Fig. 1(a)) on a top side of PZT, which does not have a significant effect on the propagation of SW in magnetic stripes. On the bottom side of PZT a 100 nm-thick titanium electrodes G1andG2were deposited above S1andS3, respectively. For more efficient magneto-electric coupling, we use a spatial resolution laser ablation technique to patterned a piezoelectric layer on 25 μm-thick. In the upper inset in Fig. 1(a) shows a SEM image of the edge of a piezoelectric layer in direct contact with a YIG stripes. A voltage Vg1,2was separately applied to each of the electrodes in the experiment. We use a two-component epoxy strain gauge adhesive (labeled “EA” - epoxy adhesive in bottom inset in Fig. 1(a)) to connect the magnonic stripes and the PZT layer. FIG. 1. (a) Scheme of the considered structure. The inset at the bottom shows the cross section of the x−zof the lateral structure. The inset at the top shows a SEM image of the edge of the piezoelectric layer. Distribution of the component of the tensor of mechanical stresses Syyin the case of an unpatterned (b) and patterned (c) piezoelectric layer.To demonstrate a processes of piezo-magnetic coupling, we are developed a numerical program based on a finite element method (FEM). First, we calculate elastic deformations caused by an external electric field in the piezoelectric layer. Next, we obtain the profiles of the internal magnetic field in lateral magnetic stripes. Then, the obtained profiles of the internal magnetic field were used in the micromagnetic simulation.20 The relative transformation in the size of the PZT layer is shown in Fig. 1(b), where the colour gradient is shows the distribution of the mechanical stress tensor component Syyin the case of Vg1=250 V. It means that the deformation of the piezoelectric layer occurs in the local region of the PZT layer under the electrode G1, which leads to a change in the value of the internal magnetic field Hintin the stripe S1due to the inverse magnetostrictive effect. In addition, we estimate the effective deformations in case of patterned piezo- electric layer (see Fig. 1(c)) and of unpatterned piezoelectric layer (see Fig. 1(b)). It should be noted that in the case of a patterned piezoelectric layer we obtain the amplification of local deforma- tions in the region of contact of the piezoelectric layer with the YIG stripe. We use the phenomenological model based on the idea of the coupling of the co-propagating spin wave. In the case of multimode spin-wave transport the intermodal coupling coefficients between the magnetic stripes can be obtained from experimentally observed beating of spin waves propagating in adjacent channels:21 −id dy⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪A1 1 A1 2 A2 1 A2 2 A3 1 A3 2⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ⎝β1 0 κ11 κ12 0 0 0 β2κ21 κ22 0 0 κ11 κ12 β1 0 κ11 κ12 κ21 κ22 0 β2κ21 κ22 0 0 κ11 κ12 β1 0 0 0 κ21 κ22 0 β2⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟ ⎠⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪A1 1 A1 2 A2 1 A2 2 A3 1 A3 2⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪(1) where coupling coefficients κ11,κ12,κ21,κ22represents the inter- modal interaction between 1st and 2nd transverse modes,6,22Aj i- is the dimensionless SW amplitude along xcoordinate, βiis the wave number of SW propagating in the single magnonic waveguide of the same width as the widht of one of the magnonic channel in the lat- eral structure, κis the coupling coefficient between the SW in the adjacent stripes. Figure 2(a) shows mode profiles of each separated magnetic stripe S2,3atE1=E3=0 kV/cm for 1st transverse mode in S2(blue solid curve), 1st transverse mode in S3(red solid curve) and 2nd transverse mode in S3(red dashed curve). The shaded area in this case is called the overlap integral C(f,E). In terms of the phe- nomenological theory of coupled waves, we can introduce the value C(f,E), the numerical value of which is equal to the integral of the overlap of the eigenmodes of two separate YIG stripes: C(f,Ei)=∫Φ1(x,f,E1)Φ2(x,f,E3)dx√ ∫Φ2 1(x,f,E1)dx∫Φ2 2(x,f,E3)dx, (2) whereΦi(x,f,Ei)is the distribution of the field of the eigenmodes of the SW propagating in the ith stripe. An exact calculation of the coupling coefficient C, including for a system of three stripes, can be performed using the FEM23and is beyond the scope of this work. AIP Advances 11, 035316 (2021); doi: 10.1063/9.0000151 11, 035316-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 2. (a) Mode profiles of each separated magnetic stripe S2,3atE1=E3 =0 kV/cm for 1st transverse mode in S2(blue solid curve), 1st transverse mode in S3(red solid curve) and 2nd transverse mode in S3(red dashed curve). 2D spatial maps of intensity of MSSW using system (1) for E1=E3=0 kV/cm (b) and E1 =10 kV/cm, E3=0 kV/cm (c). Figures 2(b), (c) shows the solution of the system (1), it should be noted that the SW intensity in the stripes is redistributed peri- odically along the stripes S1,2,3. In case of E1=10 kV/cm and E3=0 kV/cm (see Fig. 2(C)) spin-wave energy ceases to be transmitted in the stripe S1due to decrease internal magnetic field and in terms of this model due to changing β1. To demonstrate the voltage-control distribution of the spin- wave signal a numerical simulation was performed based on the solution of the Landau-Lifshitz-Hilbert equation (LLG):24–26 ∂M ∂t=γ[Heff×M]+α Ms[M×∂M ∂t], (3) where Mis the magnetization vector, Ms=139 G is the satu- ration magnetization of the YIG film, α=10−5is the damping parameter of SW, Heff=H0+Hdemag+Hex+Ha(E)is the effec- tive magnetic field, H0is the external magnetic field, Hdemag - demagnetization field, Hex- exchange field, Ha(E)is the anisotropy field, which includes taking into account the external electric field, γ=2.8 MHz/Oe is the gyromagnetic ratio in the YIG film. In order to reduce signal reflections from the boundaries of the computational domain, regions (0 <y<0.3 mm and 3.7 <y <4.0 mm) were introduced in numerical simulation with the damping parameter αwith exponential decreasing. Figure 3 shows the intensity distribution I(x,y)=√ m2y+m2zin the case of the MSSW propagation (see Figs. 3(a, b, e, f)) and in the case of propagation backward volume magnetostatic waves (BVMSW) (see Figs. 3(c, d, g, h)). If voltage is applied to the electrode G1, the spin-wave intensity distribution is transformed. So for E1=10 kV/cm and E3=0 kV/cm, a spin-wave power is transfer between S2andS3(see Fig. 3(e-h)). In this case, Lnumerically coincides with the coupling length in two identical laterally magnetic stripes.21,27It should be noted that changing of geometric parameters affects to the internal magnetic field distribution. Herewith the effective tuning of the coupling length via local strains changes in such a way that when the distance between the stripes changes from 20 μm(see Fig. 3(a, c)) to 60 μm(see Fig. 3(b, d)) the Lincreases by 1.25 times. FIG. 3. Results of calculating the spatial distribution of the SW intensity I(x)for E1=E3=0 kV/cm ((a-d) or left column) and for E1=10 kV/cm, E3=0 kV/cm ((e-h) or right column) in case of MSSW at f=4.9 GHz (a, b, e, f) and BVMSW at f=4.85 GHz (c, d, g, h). To fully describe the dipolar SWs in the lateral structure, the dipolar coupling efficiency was calculated for forward volume mag- netostatic waves (FVMSW) propagating in a laterally stripes in the case of equilibrium magnetization direction normal to the surface of the structure (along the zaxis). In this case, the internal magnetic field in the YIG stripes Hint≈H0−4πMs.6In case of propagating of the FVMSW in two lateral magnetic stripes,28it was found that the propagation regime of the FVMSW in the lateral geometry is inef- fective due to the pronounced increase in the value of the coupling length Lin the long-wavelength part of the spectrum. In a system of three lateral stripes, this leads to the fact that in the case of excitation of the FVMSW in S2, there is no directional coupling of SW into the lateral stripes S1andS3. To understand the influence of local elastic strains on the sta- tionary distribution of the MSSW, using a Brillouin light spec- troscopy (BLS) technique of magnetic materials.22,29A probe laser beam with a wavelength of 532 nm was focused on the transparent side of the GGG composite structure, as shown by the arrow in the inset in Fig. 1(a). We obtain the frequency dependence of IBLSin the section along the axis xaty=5.0 mm, in the case of E1=10 kV/cm andE3=0 kV/cm (see Fig. 4(a)), which demonstrate the transfor- mation of spin-wave intensity, when an electric field is applied. In the frequencies from f1=4.925 GHz to f2=4.985 GHz, we observe a damping of SW in S2, which corresponds the regimes when the spin-wave energy is localised in S3. It should be noted, in this case, the we observe the edge mode30propagation in the stripe S2. We see a coupling between the edge modes propagating along S1andS2. To AIP Advances 11, 035316 (2021); doi: 10.1063/9.0000151 11, 035316-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 4. The frequency dependence of the BLS signal (a) and SW intensity (b) in the section y=3 mm at E1=10 kV/cm and E3=0 kV/cm. (c, d) 2D spatial maps of the normal component of the dynamic magnetization of SW by variation of the electric field E1atE3=0 kV/cm, E3=−5 kV/cm (the value E3is shown in the figure). prove a strain-tuned SW switching we use the micromagnetic simu- lations and obtain a frequency dependence of the SW intensity (see Fig. 4(b)). We see a good agreement with the experimental data. Let us consider the effects of gradual variation of the elec- tric field polarity on the dynamic magnetization component mz, which can determine the SW phase in a section along the xaxis at y=3.0 mm. With a gradual variation of E1(see Fig. 4(c)), a change in the magnitude and sign is observed (phase change exceeds the value π)mzin the stripes S1andS2. When the field E3=−5 kV/cm is applied to the electrode G2, the internal fields in the S2and S3 become equal and the E1changes in the range −10...+10 kV/cm leads to a change in the sign of mzin all three stripes of the YIG. Thus, we observe the strain-tuned dipolar spin-wave coupling in the adjacent system of ferromagnetic stripes. As an experimental demonstration of the investigated physical processes, a configura- tion of the magnonic structure with a piezoelectric layer and struc- tured electrodes on its surface is proposed. We use the spatial resolu- tion laser ablation technique for structuring the piezoelectric layer. The latter is thus created by structuring the surface of the magnetic film and creating irregular waveguiding channels on it. As a demon- stration of this physical effect, using numerical and experimental methods we show a voltage-controlled spin-wave transport along a three-channel lateral structure. We demonstrate that the varia- tion in the geometric parameters of adjacent magnonic stripes leads to a change in the internal field and the effectiveness of the influ- ence of elastic strains on the properties of propagating coupled SWs and characteristic features are revealed that manifest themselves in a change in the modes of spin-wave transport. Using a phenomena- logical model based on coupled modes equation, we demonstrate a voltage-controlled intermodal coupling in lateral magnonic stripes. ACKNOWLEDGMENTS This work is supported by grant of the RFBR (19-37-90145). 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1.2838019.pdf	Toggle switching of weakly coupled synthetic antiferromagnet for high-density magnetoresistive random access memory Y. Fukuma, H. Fujiwara, and P. B. Visscher Citation: Journal of Applied Physics 103, 07A716 (2008); doi: 10.1063/1.2838019 View online: http://dx.doi.org/10.1063/1.2838019 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/103/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Low writing field with large writing margin in toggle magnetic random access memories using synthetic antiferromagnet ferromagnetically coupled with soft magnetic layers Appl. Phys. Lett. 89, 061909 (2006); 10.1063/1.2335810 Enhancement of writing margin for low switching toggle magnetic random access memories using multilayer synthetic antiferromagnetic structures J. Appl. Phys. 99, 08N905 (2006); 10.1063/1.2173962 Reduction of switching field distributions by edge oxidization of submicron magnetoresistive tunneling junction cells for high-density magnetoresistive random access memories J. Appl. Phys. 99, 08R702 (2006); 10.1063/1.2165138 Critical-field curves for switching toggle mode magnetoresistance random access memory devices (invited) J. Appl. Phys. 97, 10P507 (2005); 10.1063/1.1857753 Micromagnetic studies of domain structures and switching properties in a magnetoresistive random access memory cell J. Appl. Phys. 97, 10E310 (2005); 10.1063/1.1852193 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.193.242.161 On: Mon, 08 Dec 2014 22:41:02Toggle switching of weakly coupled synthetic antiferromagnet for high-density magnetoresistive random access memory Y . Fukuma,a/H20850H. Fujiwara, and P . B. Visscher MINT Center, University of Alabama, Tuscaloosa, Alabama 35487, USA /H20849Presented on 7 November 2007; received 24 September 2007; accepted 20 November 2007; published online 27 February 2008 /H20850 The toggle-switching behavior for circular and elliptic cylinder shaped memory cells of weakly coupled synthetic antiferromagnet with a diameter and thickness /H20849ferromagnetic layer /H20850ranging from 200 to 400 and from 2.5 to 5.0 nm, respectively, has been studied by micromagnetic simulation. Thecritical ﬁelds for a circular cylinder are much larger than those predicted by a single domain model.This is due to the formation of edge domains resulting in a so-called Sstate. The elliptic cylinder with an aspect ratio of /H110221.2 is found to be necessary to prevent the increase of the start ﬁeld by the edge domains. © 2008 American Institute of Physics ./H20851DOI: 10.1063/1.2838019 /H20852 I. INTRODUCTION The invention of the toggle-writing scheme by Savtch- enko et al. has helped solve the write margin problem of the conventional Stoner–Wohlfarth writing scheme in magne-toresistive random access memory /H20849MRAM /H20850, 1leading to a successful introduction of MRAM into the market.2How- ever, to further increase the memory density it is imperativeto reduce the switching ﬁeld. In toggle-MRAM, the freelayer consists of a synthetic antiferromagnet /H20849SAF /H20850. The word and digit ﬁelds are applied sequentially at /H1100645° with respect to the easy axis of the magnetic anisotropy of thememory element. According to the analytical/numerical/H20849A/N /H20850calculation based on the single domain model, 3–7this leads to a conclusion that the exchange coupling betweentwo ferromagnetic layers comprising SAF should be made tobe very small, or rather in the ferromagnetic range due to theexistence of the magnetostatic coupling. Moreover, the mag-netostatic coupling should also be reduced substantially byincreasing the thickness of the intermediate nonmagneticlayer in the SAF and/or keeping the aspect ratio /H20849length/ width /H20850of the shape of the memory cell relatively low. 3–7 For higher density memory, the magnetic layer thickness must be increased to prevent thermal instability. This willlead to generation of complex domain structures such as vor-tices and 2 /H9266wall during the switching process. In this ar- ticle, the toggle-switching behavior of circular and ellipticcylinder shaped memory cells with weakly coupled SAF isinvestigated by micromagnetic simulation in order to under-stand the deviation of the actual magnetization behavior ofthe memory cell from the single domain model. II. MODEL The Landau–Lifshitz–Gilbert equation is used for deter- mining local equilibrium of magnetizations in memory cells.We consider the memory cell of a magnetic tunnel junctionwith a SAF free layer structure combined with an ideal ref-erence layer structure designed to produce no stray ﬁeld.Thus, only the free layer structure is modeled here. The mag- netic layer of SAF is supposed to be CoFe with a saturationmagnetization of 1200 emu /cm 3, an uniaxial anisotropy of 1.8/H11003104erg /cm3, and an exchange constant of 1.6 /H1100310−6erg /cm. The interlayer exchange coupling of SAF is set to zero, assuming 5 nm thickness of an intermediate non-magnetic layer. 8The memory cells modeled are circular or elliptic cylinders. Each magnetic layer is discretized into asmall mesh of 5 /H110035/H11003tnm 3/H20849tis the thickness of the mag- netic layer /H20850using the commercial simulator “LLG.”9To compare the micromagnetic simulation results with the A/Ncalculation results, the simulations are quasistatic: the Gil-bert damping constant is 0.2 and relatively long ﬁeld pulsesof 3 ns duration and 1 ns rise/fall times are assumed for thetoggle-writing operation. The operation is supposed to beperformed by applying a series of a word ﬁeld pulse and adigit ﬁeld pulse with the same amplitude. Based on a single domain model, the critical ﬁelds for toggle switching, start ﬁeld H startand end ﬁeld Hend/H20849the minimum and maximum pulse durations that lead to switch-ing/H20850, can be expressed analytically as H start/H11015Hflop //H208812, Hend=Hsat//H208812 /H208491/H20850 with Hflop=/H20881HkHcouple +HkHk,tot /H20849Hk,tot/H333560/H20850, /H208492/H20850 Hsat=Hcouple −Hk,tot, /H208493/H20850 where Hk*is the effective anisotropy ﬁeld representing the thermal stability of the SAF memory cell, Hk,totis the total anisotropy ﬁeld which is the sum of the intrinsic uniaxialanisotropy ﬁeld and the shape anisotropy ﬁeld, and H couple is the coupling ﬁeld between two magnetic layers in SAF in-cluding both interlayer exchange coupling and magnetostaticcoupling. 7 III. RESULTS AND DISCUSSION In the simulation, the initial state is generated by ﬁrst saturating the magnetization of each layer at an angle of 80°a/H20850Electronic mail: fukuma@yamaguchi-u.ac.jp.JOURNAL OF APPLIED PHYSICS 103, 07A716 /H208492008 /H20850 0021-8979/2008/103 /H208497/H20850/07A716/3/$23.00 © 2008 American Institute of Physics 103 , 07A716-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.193.242.161 On: Mon, 08 Dec 2014 22:41:02with respect to the easy axis and then letting the magnetic moments relax to equilibrium without applying a ﬁeld. Then,a series of toggle-switch test pulse ﬁelds separated by anequilibrium time of 1.5 ns is applied consecutively, increas-ing the amplitude H pin increments of 5 Oe. After each toggle-switch testing, the magnetic moment of each layer ischecked to ﬁnd out whether the toggle switching has beensuccessful or not in determining H startand Hend. The span between HstartandHendis the operating ﬁeld margin. Figure 1shows HstartandHendas a function of a diameter dobtained for the circular cylinder shaped memory cell of CoFe /H208492.5 nm /H20850/Ru /H208495.0 nm /H20850/CoFe /H208492.4 nm /H20850. The thick- nesses of the bottom and top CoFe layers are taken to be slightly different to ensure that the ﬁrst relaxation processyields a unique initial state. The solid lines show the predic-tions of the A/N calculation. The results obtained by themicromagnetic simulations are quite different from the pre-dictions: the simulation gives higher H startandHend. Besides, Hendford=200 nm is lower than that for d=300 nm, while the single domain model shows a monotonic decrease withincreasing d. Figure 2shows how the magnetization conﬁgu- ration of the top and bottom layers changes during thetoggle-switching process for the case of d=300 nm /H20849H p =200 Oe /H20850. At the remanent state, the magnetic moments at both edges in the easy axis direction are tilted upward or downward, making an Sstate in both layers. The magnetiza- tion reversal is initiated, in this particular case, with the ro-tation of the magnetization at the center of the bottom layer/H20849with a negative x-directional net moment M x/H20850toward the word ﬁeld followed by or together with the switching of themoments at the edges. Note that those edge /H20849pseudo /H20850domain moments in the top and bottom layers are almost antiparallelto each other at the remanent state to reduce the magneto- static energy. This stabilizes the edge domains, causing thedelay of the response of the bottom magnetization to theword ﬁeld compared to a single domain. The deviation ofH startfrom the single domain model increases with decreas- ingdbecause the adverse effect of the edge domains for the net moment rotation is enhanced: the exchange interaction ofmoments between the edges and the middle of the cell be-comes looser as dincreases and, thus, the moments in the middle become easier to rotate ﬁrst leaving the switching ofthe edge moments to occur through a kind of wall motion.Actually, we observed that the magnetic moments around thecenter in the 200 nm cell were tilted clearly upward or down-ward to reduce exchange energy, demonstrating the effect ofthe exchange coupling throughout the cell. For H end, which gives the saturation ﬁeld of SAF along the easy axis direc-tion when both of the word and digit ﬁelds are applied, thesimulation values drastically increased compared to thesingle domain model due to the very high demagnetizingﬁeld H dat the edges. A higher ﬁeld is necessary to saturate the moments at the edges than to saturate the middle mo-ments. The decrease of H endfor the 200 nm cell is ascribed to the increases of the coherency to reduce the exchangeenergy. Thus, the aspect ratio dependence of H startandHendfor 200 nm width elliptic cylinder shaped cells of CoFe/H208492.5 nm /H20850/Ru /H208495.0 nm /H20850/CoFe /H208492.4 nm /H20850and CoFe /H208495.0 nm /H20850/Ru/H208495.0 nm /H20850/CoFe /H208494.9 nm /H20850has been studied. The results are shown in Fig. 3.H startfor the aspect ratio of 1.2 is almost consistent with the single domain model, while Hend is signiﬁcantly larger than that of the single domain model. As for Hstart, the difference increases with decreasing aspect ratio and with increasing thickness of the magnetic layer.Figure 4shows the deviation of H startfrom the single domain model and the normalized net moment Mxof the cells. These results suggest that the drastic increase of Hstartwith decreas- ing aspect ratio can be attributed to the formation of the edgedomains in the same manner as discussed in the previousparagraph. On the other hand, H endis less sensitive to the aspect ratio. This is because, in contrast to the magnetizationprocess determining H start, no switching process of the edge domains is involved in the ﬁnal stage of the saturation, that FIG. 1. /H20849Color online /H20850Diameter dependence of toggle start and end ﬁelds for circular cylinder shaped cells of CoFe /H208492.5 nm /H20850/Ru /H208495.0 nm /H20850/CoFe /H208492.4 nm /H20850. Closed circles and triangles show start and end ﬁelds, respectively, by simulation. Lines show the single domain result. FIG. 2. /H20849Color online /H20850Magnetic moment conﬁgurations of 300 nm circular cylinder shaped cell for toggle switching at 200 Oe. FIG. 3. /H20849Color online /H20850Aspect ratio dependence of toggle start and end ﬁelds for circular cylinder shaped elements of CoFe /H208492.5,5.0 nm /H20850/Ru /H208495.0 nm /H20850/CoFe /H208492.4, 4.9 nm /H20850with 200 nm width. Closed circles and tri- angles show start and end ﬁelds, respectively, by simulation. Lines show thesingle domain result.07A716-2 Fukuma, Fujiwara, and Visscher J. Appl. Phys. 103 , 07A716 /H208492008 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.193.242.161 On: Mon, 08 Dec 2014 22:41:02is, in aligning the edge magnetic moments in the easy axis direction against the edge demagnetizing ﬁeld. The remark-able increase of H start, especially for smaller aspect ratio and greater thickness, indicates that some measure, such as theuse of eye- or diamond-shaped memory cells, 10,11is neces- sary to prevent the formation of edge domains for futurehigh-density toggle-MRAMs. IV. CONCLUSIONS We have studied the toggle-switching behavior of circu- lar and elliptic cylinder shaped memory cells of weaklycoupled SAF by micromagnetic simulation. The critical ﬁelds are much larger than those predicted by a single do-main model due to the formation of end domains. As thediameter and the aspect ratio of the elliptic cylinder decrease,the moments at the edges participate in the switching to re-duce the exchange energy, causing higher start ﬁeld. Theelliptic cylinder with an aspect ratio of /H110221.2 is necessary to prevent the increase of the start ﬁeld. ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation MRSEC under Grant No. DMR-0213985. Y.F. issupported by JSPS Postdoctoral Fellowships for ResearchAbroad. 1L. Savtchenko, A. A. Korkin, B. N. Engel, N. D. Rizzo, M. F. Deherrera, and J. A. Janesky, US Patent No. 6,545,906 B1 /H208498 April 2003 /H20850. 2See; www.freescale.com 3H. Fujiwara, S.-Y. Wang, and M. Sun, Trans. Magn. Soc. Jpn. 4,1 2 1 /H208492004 /H20850. 4D. C. Worledge, Appl. Phys. Lett. 84,2 8 4 7 /H208492004 /H20850. 5D. C. Worledge, Appl. Phys. Lett. 84,4 5 5 9 /H208492004 /H20850. 6H. Fujiwara, S.-Y. Wang, and M. Sun, J. Appl. Phys. 97, 10P507 /H208492005 /H20850. 7S.-Y. Wang and H. Fujiwara, J. Appl. Phys. 98, 024510 /H208492005 /H20850. 8S. S. P. Parkin, Phys. Rev. Lett. 67,3 5 9 8 /H208491991 /H20850. 9M. Scheinfein /H20849http://llgmicro.home.mindspring.com /H20850. 10X. Zhu and J.-G. Zhu, J. Appl. Phys. 93, 8376 /H208492003 /H20850. 11W. L. Zhang, R. J. Tang, H. C. Jiang, W. X. Zhang, B. Peng, and H. W. Zhang, IEEE Trans. Magn. 41, 4390 /H208492005 /H20850. FIG. 4. /H20849Color online /H20850Aspect ratio dependence of deviation of toggle start ﬁeld from single domain model and normalized net moment for the identicalcells in Fig. 3.07A716-3 Fukuma, Fujiwara, and Visscher J. Appl. Phys. 103 , 07A716 /H208492008 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.193.242.161 On: Mon, 08 Dec 2014 22:41:02
5.0054459.pdf	Chaos 31, 063111 (2021); https://doi.org/10.1063/5.0054459 31, 063111 © 2021 Author(s).Local and network behavior of bistable vibrational energy harvesters considering periodic and quasiperiodic excitations Cite as: Chaos 31, 063111 (2021); https://doi.org/10.1063/5.0054459 Submitted: 19 April 2021 . Accepted: 24 May 2021 . Published Online: 07 June 2021 Karthikeyan Rajagopal , Arthanari Ramesh , Irene Moroz , Prakash Duraisamy , and Anitha Karthikeyan COLLECTIONS Paper published as part of the special topic on In Memory of Vadim S. Anishchenko: Statistical Physics and Nonlinear Dynamics of Complex Systems ARTICLES YOU MAY BE INTERESTED IN Extreme synchronization events in a Kuramoto model: The interplay between resource constraints and explosive transitions Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 063103 (2021); https:// doi.org/10.1063/5.0055156 Sparse optimization of mutual synchronization in collectively oscillating networks Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 063113 (2021); https:// doi.org/10.1063/5.0049091 ordpy: A Python package for data analysis with permutation entropy and ordinal network methods Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 063110 (2021); https:// doi.org/10.1063/5.0049901Chaos ARTICLE scitation.org/journal/cha Local and network behavior of bistable vibrational energy harvesters considering periodic and quasiperiodic excitations Cite as: Chaos 31, 063111 (2021); doi: 10.1063/5.0054459 Submitted:19April2021 ·Accepted:24May2021 · PublishedOnline:7June2021View Online Export Citation CrossMark Karthikeyan Rajagopal,1,a) Arthanari Ramesh,2,b)Irene Moroz,3,c)Prakash Duraisamy,1,d) and Anitha Karthikeyan4,e) AFFILIATIONS 1CenterforNonlinearSystems,ChennaiInstituteofTechnology, Chennai600069,India 2CenterforMaterialsResearch,ChennaiInstituteofTechnology, Chennai600069,India 3MathematicalInstitute,UniversityofOxford,AndrewWilesBuilding,Ox fordOX26GG,UnitedKingdom 4DepartmentofElectronicsandCommunicationEngineering,Prathy ushaEngineeringCollege,Thiruvallur,TamilNadu602025, India Note:ThispaperispartoftheFocusIssue,InMemoryofVadimS.Anishchen ko:StatisticalPhysicsandNonlinearDynamicsof ComplexSystems. a)Author to whom correspondence should be addressed: karthikeyan.rajagopal@citchennai.net b)ramesha@citchennai.net c)irene.Moroz@maths.ox.ac.uk d)prakash.duraisamy@citchennai.net e)mrs.anithakarthikeyan@gmail.com ABSTRACT Vibrational energy harvesters can exhibit complex nonlinear behavior when exposed to external excitations. Depending on the number of stable equilibriums, the energy harvesters are defined and analyzed. In this work, we focus on the bistable energy harvester with two energy wells. Though there have been earlier discussions on such harvesters, all these works focus on periodic excitations. Hence, we are focusing our analysis on both periodic and quasiperiodic forced bistable energy harvesters. Various dynamical properties are explored, and the bifurcation plots of the periodically excited harvester show coexisting hidden attractors. To investigate the collective behavior of the harvesters, we mathematically constructed a two-dimensional lattice array of the harvesters. A non-local coupling is considered, and we could show the emergence of chimeras in the network. As discussed in the literature, energy harvesters are efficient if the chaotic regimes can be suppressed and hence we focus our discussion toward synchronizing the nodes in the network when they are not in their chaotic regimes. We could successfully define the conditions to achieve complete synchronization in both periodic and quasiperiodically excited harvesters. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0054459 Vibration energy harvesters (VEHs) exhibit rich dynamical prop- erties when excited with periodic excitations as shown in the literature. Though such discussions are abundant, there is no lit- erature discussing the effect of quasiperiodic excitations. Also, we have discussed multistability and coexisting attractors in a bistable energy harvester and have shown that a quasiperi- odic excited energy harvester does not exhibit multistability. Though local dynamics of the VEHs are significant, it is its network behavior which is very important to investigate thesynchronization behavior as a periodic VEH completely synchro- nized will be efficient against a chaotic VEH exhibiting chimera states. I. INTRODUCTION Wireless sensors and other low powered health monitoring devices have been widely used in industrial, medical, military, Chaos31,063111(2021);doi:10.1063/5.0054459 31,063111-1 PublishedunderanexclusivelicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha engineering, and environmental health monitoring areas.1–3These devices need a continuous power supply with the least replacement cost and long-lifetime service. Their power supply requirements are not fully addressed using the traditional batteries.4Vibration energy is readily available in the environment in the form of wind, ocean waves, human motion, and mechanical vibration.5Among these, the mechanical energy is one of the most suitable one for structural health monitoring due to its availability.4As a result, energy har- vesting from mechanical vibration is a promising means to replace conventional power sources. Linear resonant energy harvesters have been used to exploit ambient vibration energy based on the linear resonant vibration principle. They are effective only if the excitation power is con- centrated in a narrow band of frequency, i.e., stationary excita- tion. But when the excitation frequency does not match the res- onant frequency, the harvesting efficiency reduced severely. Many researchers tried to solve this issue by expanding the bandwidth of frequency using techniques such as frequency upconversion, resonance tuning, and deliberate introduction of nonlinearity.6–12 Among these, the inclusion of nonlinearity to harvesters attracted much attention. Monostable and bistable harvesters found their significance due to various advantages under different excitation conditions and initial conditions. Monostable harvesters hold a sim- ple configuration and have one stable equilibrium state. Compared to linear systems, they are able to widen the frequency response range.8But their performance drops when they are subjected to real-world scenarios such as random excitation and have a similar output to linear harvesters when they are excited by white noise vibrations.13As an improvement to the drawback of monostable harvesters, researchers have focused on multistable harvesters. The bistable harvester has been explored widely.14–17It has two poten- tial wells with a barrier in between. If the excitation amplitude is high enough, the system will jump between the stable wells creating an interwell oscillation with high amplitude. The shape of the wells influences the response of the system. When the wells are shallow, the bandwidth of the harvester increases; however, the amplitude of the response is reduced. Compared to monostable harvesters, bistable shows improvement in harvesting, when the amplitude of the excitation is large enough to trigger an interwell oscillation.18,19 In addition to that, there is only minimal literature found for studying the network behaviors of energy harvesters but practi- cally, they should work in a network to obtain a significant output. The coupling effect plays an important role in network dynamics and shows significant results in output energy. If the network of harvesters is not properly tuned, there is chance of energy dissi- pation in nodes. Hence, there is a huge demand in finding proper coupling strength and periodic frequency. The study of an inter- esting phenomenon on synchronization20,21and incoherent oscil- lations of a horde non-locally coupled oscillators22is pronounced as “chimera states.” The dynamics of chimera states and different topologies to handle chimera has been investigated vigorously in recent literature.23–25The existence of chimera states and their con- trol are studied with experimental evidence in some literature.26–31 Hence, considering undeniable mutual effects of chimera states in dynamics of network and spatiotemporal nature, the intriguing characteristics of the system can be analyzed and it becomes a fruitful test ground for energy efficiency also.Motivated from the above discussion, in this paper, we for- mulated a Bistable Energy Harvester (BEH) supplied with higher order nonlinearity under quasi-periodic excitation. Stability analysis is carried out and presented in Sec. III. Bifurcation plots and the cor- responding Lyapunov spectrum are derived for different scenarios in Sec. IV. The major contribution of this work lies on the network dynamics of BEH under periodic and quasi-periodic excitations, the simulations are portrayed and interpreted in Sec. V. Finally, we pro- vided concluding remarks and highlighted the significance of the present study. II. MATHEMATICAL MODELING InFig. 1 , the configuration of a nonlinear energy harvester is shown. The configuration of a bistable energy harvester consists of a stainless-steel substrate with two lead zirconate titanate (PZT) piezoelectric layers positioned near the base and two tip magnets. There are two external magnets located with the required distance and angle. The equation of motion for the BEH configuration presented inFig. 1 is derived as follows: m¨r(t)+c˙r(t)+Fh−θV(t)=/Phi1(t), Cp˙V(t)+V(t) R+θ˙r(t)=0,(1) where mandcare the equivalent mass and damping, Cprefers to the equivalent capacitance, θrepresents the electromechanical coupling coefficient of the piezoelectric material, Rdenotes the load resis- tance, Vdenotes the voltage, ris the displacement, and /Phi1(t)refers to the external excitation. FIG.1.Conﬁgurationofbistableenergyharvester(BEH). Chaos31,063111(2021);doi:10.1063/5.0054459 31,063111-2 PublishedunderanexclusivelicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha For brevity, the equations of motion can be further nondimen- sionalized by using the following terms: x=r lc,t=tωn,V=Cp θlcV, (2) where lcis a length scale introduced to nondimensionalize the displacement and ωn=/radicalBig k mis the natural frequency of the har- vester. With these transformations, the nondimensional model can be expressed as ¨x+2ξ˙x+Fh−κ2V=/Phi1(t), ˙V+αV+˙x=0,(3) where ξ=c 2√k1m,δ=k3lc2 k1,κ=θ2 k1Cc,α=1 ωnCpR. Here, xrepresents displacement, ωnis the natural frequency, andξrefers to the damping ratio. In this paper, we introduced a quartic nonlinearity term as restoring force, and the expression is as follows: Fh= −x+βx2+δx3+γx4. (4) Most of the analyses done on the existing models are with peri- odic excitation. Literature on experimental studies of BEH show that quasi-periodic excitation affects the performance significantly; hence, we are introducing quasi-periodic excitation. Generally, quasi-periodic excitation will induce a strange nonchaotic attrac- tor (SNCA) and results in multistability. Hence, in this paper, we considered the system is supplied with quasi-periodic excitation and denoted as /Phi1(t)=A1[sin(ω1t)+A2sin(ω2t)]. The state space equation of the system can be written as ˙x=y, ˙y=A1[sin(ω1t)+A2sin(ω2t)]−2ξy+x−δx3 −βx2−γx4+κz, ˙z= −y−αz.(5) III. EQUILIBRIUM POINTS AND STABILITY ANALYSIS Fh(x)is the nonlinear restoring force, while /Phi1(t)is the external excitation: Periodic if A2=0 and quasi-periodic if A2/negationslash=0. We take the parameter values in Eq. (5)to be A1=0.5,ω1=1,ω2=√ 5−1 2, A2=1,ξ=0.0933, δ=0.5495, β=0.1,γ=0.1,α=0.4065, κ=0.001 85. Stanton et al.32performed a Melnikov analysis on a simplified bistable harvester by considering perturbations from a Hamiltonian limit. We follow part of their analysis here. We rewrite (5)as a perturbed Hamiltonian system, ˙x ˙y ˙z = y Fh(x) 0 −ε 0 −2ξy+κz+f(t) −y−αz , (6)where we have introduced a small parameter εto represent the non- Hamiltonian terms. When ε=0, Eq. (6)becomes ˙x=y, (7) ˙y=x−βx2−δx3−γx4, (8) which leads to the Hamiltonian E(t)=1 2y2+V(x). (9) With the potential energy function V(x) V(x)= −1 2x2+β 3x3+δ 4x4+γ 5x5. (10) The fixed points of (7)satisfy (x,y)=(0, 0),y=0 and the cubic roots of 1 −βx−δx2−γx3. The trivial fixed point (0, 0)is a saddle point. Values for the parameters, we find the remaining fixed points to be x1= −4.8675 (another saddle point), x2= −1.7810 and x3=1.1535 (both centers). Figure 2 shows the double homoclinc loop in black refers to the simplified model with γ=0, homoclinic loop (magenta) that passes through the saddle point (0, 0), as well as the large homo- clinic loop passes through the saddle point x1= −4.8675. For the double homoclinic loop, E(t)=0, while for the large homoclinic loop, E(t)=6.781. Solving (9)forywith E(t)=0 for the chosen parameter values, we get yDH= ±x[(x+5.8548 )(x+2.6347 )(1.6207 −x)], (11) so that the separatrix through the double homoclinic loop passes through xs2=1.6207 and xs1= −2.6347. There is also a separate branch (not shown) that passes through x= −5.8548, whose com- ponents tend to ±∞.Figure 2 shows the potential function V(x), including the two end points xs1andxs2of the separatrix for the double homoclinic loop. FIG. 2.The double homoclinic loop for the BEH (magenta) passing thro ugh the ﬁxedpoints (0,0)andenclosingthetwocenters x2andx3.Thereisalsoalarge homoclinicloop(green)passingthrough x1andenclosingtheotherﬁxedpoints. Chaos31,063111(2021);doi:10.1063/5.0054459 31,063111-3 PublishedunderanexclusivelicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha For the large homoclinic loop (green), we got x= −4.8677 (twice), x=2.6711 (the point at which the large orbit crosses the y-axis), and two other complex valued roots. If we set γ=0, the model considered by Wang et al.,33we no longer get the root x= −5.8548, and the result is the larger black double homoclinic loop, shown in Fig. 2 . Since E(t)=0, Eqs. (10) and(11)give y=dx dt= ±x/radicalbig g(x), (12) where g(x)=a2x2+a1x+a0 (13) =1−2βx 3−δx2 2(14) = −(x−xs1)(x−xs2). (15) Litak and Borowiec34investigated the case of an asymmetric double homoclinic loop for a potential function of the form Eq. (10)with γ=0. In Eq. (15), we have taken xs1andxs2to be the end points on the black and magenta separatrices of Fig. 1 , given in Eq. (11). Equation (14)uses the expressions for ajfrom Eq. (10). For(13), we obtain t=/integraldisplaydx x/radicalbig g(x)= −1√a0ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2a0+a1x+2/radicalbig a0g(x) x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (16) Provided a0>0. Since a0=0 [by rearranging Eq. (9)], this is certainly the case here. We can invert Eq. (17)to find an expression for xand then foryfor the double homoclinic loop. Defining E=exp t√a0, tak- ing the exponential of both sides of (17)and rearranging gives, after straightforward algebra, x=0 (the saddle point at the origin) and xdh=4a0E [(E−a1)2−4a0a2]. (17) The derivativedxdh dtgives the ycoordinate for the double homoclinic loop, ydh=4Ea3/2 0[a2 1−4a0a2−E2] [(E−a1)2−4a0a2]. (18) A. The unforced BEH system For the full BEH with γreinstated, we can find a solution for xdhin terms of Jacobi Elliptic functions of the first and third kinds and involving inverse trigonometric sine functions. We omit their expressions here as not being very instructive. There is, also, no sim- ple equation for the large green homoclinic loop, shown in Fig. 2 . Instead, some level sets, corresponding to a potential function V(x) for the full BEH system, are shown in Figs. 3 and4. In the absence of external forcing (so that /Phi1(t)=0), the equi- librium states are given by y=0,z=0, and x.sa solution to Fh(x)=0. We, therefore, obtain the trivial equilibrium x=0, together with the three nontrivial equilibrium states obtained in the Hamiltonian limit, namely, a saddle point and two centers. For the given set of parameter values, these are the roots of the RHS of FIG. 3.The potential function V(x) for the BEH with the end points xs1andxs2 oftheseparatrixofthedoublehomoclinicloop. Eq.(17). The linear stability of the equilibrium state is determined by the eigen spectrum of the characteristic equation, λ3+/Delta12λ2+/Delta11λ+/Delta10=0, (19) where /Delta12=2ξ+α, (20) /Delta11=2ξα+κ−Gx, (21) /Delta10= −αGx, (22) Gx=1−2βx−3δx2−4γx3. (23) FIG.4.Somelevelsetsforthepotentialfunction V(x)fortheBEH.Theseparatrix forthedoublehomoclinicloopandthelargehomoclinicloop areshownasblack curves. Chaos31,063111(2021);doi:10.1063/5.0054459 31,063111-4 PublishedunderanexclusivelicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha TABLEI. Equilibriumpointsanditsstability. Equilibrium points Eigen values Stability E0=[0; 0; 0] [0.9104; −1.0962; −0.4073] Saddle node E1=[−4.8675; 0; 0] [2.9155; −3.1020; −0.4066] Saddle node E2=[−1.781; 0; 0] [ −0.0935; ±1.2673 i;−0.4061] Stable focus E3=[1.1535; 0; 0] [ −0.0935; ±1.4252 i;−0.4061] Stable focus For the trivial equilibrium state x=0,Gx=α. This means there are no steady bifurcations from the trivial equilibrium. Moreover, by substituting λ=iωinto Eq. (19), it is straightforward to show that there are also no possible Hopf bifurcations, since this would require /Delta12+α=0: not possible when all parameters are positive. Codimension one steady state bifurcations can occur when /Delta10=0 for the nontrivial saddle point at γ=0.139 (keeping all remain- ing parameter values at their prescribed values). The condition for a Hopf bifurcation, /Delta10−/Delta11/Delta12=0, is not possible since this would require ω2=αGx 2ξ+α=2ξα+κ−Gx. (24) Numerical integrations show that these two criteria for ω2cannot be simultaneously satisfied. Moreover, numerical integrations for the unforced system also show that the only stable states are steady states; we found no evi- dence of periodic solutions. When the z-dependence is present, the saddle point x1remains a saddle, while the two centers x2andx3 become stable foci. For the parameter values ξ=0.0933; δ=0.5495; α=0.4065; β=0.1;γ=0.1;κ=0.001 84, the equilibrium points and corre- sponding eigen values are calculated and presented in Table I for understanding the stability. The system shows a chaotic attractor for the following param- eter values, A1=0.5,ω1=1,A2=1,ω2=√ 5−1 2,ξ=0.0933, δ=0.5495, β=0.1,γ=0.1,κ=0.001 84, α=0.4065 for the ini- tial condition {1,0,1}. The 2D phase portraits are given in Fig. 5 , where (a) represents the state phase portrait between Y and Z, (b)represents the state phase portrait between X vs Y, and (c) represents the state phase portrait between X vs Z, respectively. IV. BIFURCATION AND LYAPUNOV SPECTRUM We investigated the bifurcation property for two scenarios: case (1) the system under periodic excitation and case (2) the system under quasi-periodic excitation. We used Runge–Kutta numerical method for simulating the results. We considered the parameter values: A1=0.5,A2=1,ξ=0.0933, δ=0.5495, β=0.1,γ=0.1, κ=0.001 84, α=0.4065, and initial condition {1,0,1}. For inves- tigating the bistability nature of the system under mentioned exci- tation conditions, we provided the bifurcation plots using forward continuation (plotted with blue dots) and backward continuation (plotted with red dot), and the corresponding Lyapunov exponent spectrum is calculated using Wolf algorithm35and plotted for a finite time of 20 000 s. Case (1): Under periodic excitation First, we considered the system (5)is supplied with periodic excitation /Phi1(t)=Fsin(ωt)and the response of the system is noted and presented as a bifurcation plot. In Fig. 6 , we could observe the bistability phenomena which are considered dangerous for mechan- ical systems. We varied the parameter ωfor the range of 0.9–1.3. For detailed analysis, we provided the bistable regions in Fig. 6(b) and the corresponding Lyapunov exponent spectrum for forward and backward continuation also presented in Figs. 6(c) and6(d). From the bifurcation plots, we can observe the system holds the multistability property.36–39 Case (2): Under quasi-periodic excitation In this case, we considered the system (5)is supplied with quasi-periodic excitation; real systems that are mostly influenced with different frequencies for simplification purpose only are con- sidered as single frequency, i.e., periodic excitation. In order to analyze the system with quasi-periodic excitation, we considered the parameter values in Eq. (5)to be A1=0.5,ω1=1,ω2=√ 5−1 2, and A2=1. The influence of excitation frequency plays a vital role in behavioral analysis. We derived the bifurcation plots for frequency FIG. 5.2D phase portrait of system (6)for the parameter values A1=0.5,ω1=1,A2=1,ω2=√ 5−1 2,ξ=0.0933, δ=0.5495, β=0.1,γ=0.1,κ=0.001 84, α=0.4065andinitialcondition {1,0,1}(a)YvsZ,(b)XvsY,and(c)XvsZ. Chaos31,063111(2021);doi:10.1063/5.0054459 31,063111-5 PublishedunderanexclusivelicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG.6.BifurcationplotandthecorrespondingLyapunovexponents ofparameter ω1variation.(Periodicexcitation)(a)Representsthebifur cationplotof0.9 ≤ω1≤1.3,(b) representsthebifurcationplotof0.9 ≤ω1≤1,(c)representsLyapunovexponentsspectrumfor(a),and( d)representsLyapunovexponentsspectrumfor(b),respect ively. range 0 ≤ω1≤3. The corresponding Lyapunov exponent spec- trum also presented. Comparing with periodic excitation ( Fig. 6 ), we could observe that there is no limit cycle oscillation as in Fig. 7 . Chaotic attractor and torus were identified, and the corresponding Lyapunov exponent spectrum confirms it. We could observe that the multistability property is vanished during quasi-periodic excitation. V. NETWORK DYNAMICS OF BEH The local behavior of the BEH with periodic and quasi-periodic excitations shows some interesting dynamical behaviors but energy harvesters will normally be applied in large networks. Hence, we consider N coupled BEH whose mathematical model is shown in Eq.(25) ˙xi=yi+σN/summationdisplay j=1Cijxj, ˙yi=/Phi1(t)−2ξyi+xi−δx3 i−βx2 i−γx4 i+κzi, ˙zi= −yi−αzi.(25) The term σdefines the coupling constant and the con- nection between the nearby nodes is defined by the connection matrix Cij. The external excitation is defined by /Phi1(t)=A1[sin(ω1t) +A2sin(ω2t)], where ω1=1 is the frequency of the periodic term while ω2=√ 5−1 2is the golden mean contributing to the quasi- periodic excitation. We have considered two different cases fordiscussion depending on the type of excitation applied to the nodes in the network. A. Network behavior with periodic excitation In this case, we consider A2=0, we now apply a periodic excitation to the nodes in the network and for simulation we con- sider the parameters as A1=0.5,ω1=1,ξ=0.0933, δ=0.5495, β=0.1,γ=0.1,κ=0.001 84, α=0.4065 and random initial con- ditions are chosen. We have used the RK4 method to solve the system (25)with the step size of 0.01 and a total simulation time of 3000 s. In Fig. 8 , we have shown the spatiotemporal behavior of the network for coupling strengths σ≤0.015 for which we could see that the nodes are in complete incoherency. We have also plot- ted the instantaneous state variable value of xmeasured at the end of simulation. For σ=0.015, the nodes try to achieve synchronization and lead us to a clue to check for chimeras by increasing coupling. Now, we increased the coupling to σ=0.018 to identify the existence of coherent and incoherent nodes in the network. In Fig. 9 , we could see that most of the nodes try to achieve coherency and some nodes are still in the complete incoherent state. This confirms the existence of chimeras in the network. It should be noted that only if all the BEH nodes are in the coherent state, the energy effi- ciency will be more, and such chimeras will result in residue current in the BEH nodes which could damage the node permanently. Such chimeras are seen in the network for σ <0.03 but we have only shown spatiotemporal behavior of selected values of σ. Chaos31,063111(2021);doi:10.1063/5.0054459 31,063111-6 PublishedunderanexclusivelicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG. 7.Bifurcation analysis of system (5)under quasi-periodic excitation, (a) represents the bifur cation plot for 0 ≤ω1≤3 and (b) represents the Lyapunov exponent spectrumfor0 ≤ω1≤3. Further increasing the coupling strength to σ=0.03, the nodes try to achieve synchronization in small clusters. Though such phe- nomenon of cluster synchronization is not uncommon in spiking networks, its undesirable in such networks where complete synchro- nization is mandatory. In other words, we need all the BEH nodes in the network to operate in a single frequency or at least in coherency to maximize the energy efficiency of the harvesters. In Fig. 10 , we have shown several cluster synchronization conditions for various values of σ. We have used re-occurrence plots to identify different regimes of synchronization in the network. The re-occurrence plots are cal- culated by finding the Euclidean distance between xiandxjwhere i.j∈[1,N]. InFig. 11 , we have shown the re-occurrence plots and the absence of structures in the plot for σ=0.001 shows that thereare no coherent nodes in the network. For σ=0.01, we could see small structures formed in the network confirming the emergence of chimeras. For the remaining values of σ <0.03 shown in the plots, we could see majority of blue and red regions. The blue region shows coherent oscillators and the red shows the incoherent oscillators. The presence of other colors in these plots confirms the existence of multiple intermediate nodes which neither belong to the red and nor to the blue regions confirming different incoherent frequencies in the network. But when σ >0.03 we could note only blue and red dominant confirming different clusters of synchronizations. Though we could achieve cluster synchronization in the network as shown in Fig. 11 , we could not reach complete synchronization. For values of coupling σ >0.1, the network goes in to unbounded states and thus we could not use the coupling Chaos31,063111(2021);doi:10.1063/5.0054459 31,063111-7 PublishedunderanexclusivelicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG. 8.The collective behaviorofthe network (1)for differentvalues ofthe couplingcoefﬁcient ( σ)consideringa periodicexcitation. Theplots conﬁrmthat th e nodesare asynchronous. FIG.9.Thecollectivebehaviorofthenetwork (25)fordifferentvaluesofthecouplingcoefﬁcient( σ)consideringaperiodicexcitation.Theplotsconﬁrmthatth enodesare showingbothsynchronousandasynchronousnodesconﬁrming theexistenceofchimeras. Chaos31,063111(2021);doi:10.1063/5.0054459 31,063111-8 PublishedunderanexclusivelicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG.10.Thecollectivebehaviorofthenetwork (25)fordifferentvaluesofthecouplingcoefﬁcient( σ)consideringaperiodicexcitation.Theplotsconﬁrmsevera lclustersof synchronizedBEHnodesinthenetwork. strength to achieve synchronization. As the BEH is externally excited by the periodic force, our interest now is on the amplitude of the periodic force which can be properly tuned to achieve complete syn- chronization. In Fig. 12 , we have shown the spatiotemporal plots for the amplitude values A1=[0.4, 0.45, 0.5] and the plots confirm the emergence of chimeras in the network. Though for A1=0.5, the incoherent nodes are very less confirming that the nodes are moving to complete synchronization. InFig. 13 , we have shown the complete synchronization of the nodes achieved through the tuning of the amplitude of the external excitation. We could note that for A1<0.4 and A1>0.6 the nodes attain complete synchronization and achieve the same frequency of operation. Thus, we could achieve the maximum efficiency from the energy harvesters when connected to a network. But it has been shown in the literature that non-chaotic BEH34can be productive in energy harvesting but as seen from the re-occurrence plot ( Fig. 12 right most) for A1=0.3, the nodes are no more chaotic and show periodic behaviors and for A1=0.7 the nodes are chaotic and are completely synchronized as there Euclidean distance is very low in 10−13. Thus, we recommend an amplitude value of A1=0.3 to achieve complete synchronization in the periodically excited BEH network. B. Network behavior with quasi-periodic excitation In this case, we consider A1=0.5, we now apply a quasi- periodic excitation to the nodes in the network by consideringA2/negationslash=0,ω2=√ 5−1 2with the other parameters as ω1=1,A2=1, ξ=0.0933, δ=0.5495, β=0.1,γ=0.1,κ=0.001 84, α=0.4065. The other setting for simulations is similar to Sec. V A. As we have earlier shown that the coupling coefficient could not be tuned to achieve complete synchronization (Sec. V A), we verified the same for quasi and could again confirm that σcannot be tuned for synchronization in the quasi-periodic case. Hence, we have not provided the discussion on σas it will be redundant. We focus our discussion on the amplitude of the quasi-periodic term ( A2) and have kept σ=0.025. In Fig. 14 , we have shown the spa- tiotemporal plots for different values of A2and unlike Fig. 13 , we could not find periodic regimes for A2<1 and the network shows both coherent and incoherent nodes confirming the existence of chimeras. To find the amplitude values that could help us achieve com- plete synchronization with nodes are not in their chaotic regime, we further increase the value of A2=1.5 and could observe complete synchronization and the same is the case for A2=1.8, presented in Fig. 15 . But our interest is to check whether the nodes are not in their chaotic regime while achieving complete synchronization. As discussed, we must verify which of the amplitude (A2)can achieve complete synchronization while the BEH nodes are not chaotic. Hence, we use the re-occurrence plots as shown in Fig. 16 . While checking the re-occurrence plots for A2=1.5 we could see that the nodes are in synchronization with period-4 oscillations and forA2=1.8, the nodes whose different colors confirming they are chaotic but their Euclidian distance in the range of 10−13showing Chaos31,063111(2021);doi:10.1063/5.0054459 31,063111-9 PublishedunderanexclusivelicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG. 11.Re-occurrence plots of the BEH network for different values of the coupling coefﬁcient. The blue regions show the coheren t oscillators and the red show the incoherentoscillators. complete synchronization. Hence, when exposed to quasi-periodic excitation, we could recommend the amplitude combination as A1=0.5 and A2=1.5 for complete synchronization as for these two amplitude values the nodes are periodic and chaotic oscillations are suppressed. VI. CONCLUSION An energy harvester model with two stable equilibrium points is analyzed considering periodic and quasiperiodic externalexcitations. The dynamical properties of the model are analyzed and we could show coexisting hidden attractors in the system for peri- odic excitation. Though local behavior can help us understand the complex oscillations and bifurcation patterns of the bistable energy harvesters, our focus is to investigate its collective dynamics. A mathematical model of a 2D lattice network is constructed whose local dynamics is governed by the bistable energy harvesters. First, a periodic excitation is applied to the nodes and the spatiotempo- ral behavior is captured. We could observe regions of asynchronous nodes for very low coupling values, while increasing the coupling Chaos31,063111(2021);doi:10.1063/5.0054459 31,063111-10 PublishedunderanexclusivelicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG. 12.The collective behaviorof the network (1)for differentvalues of the amplitude of the periodicexcita tion considering σ=0.024.The plots emergenceof chimera stateswithsynchronousandasynchronousnodes. FIG. 13.The collective behavior of the network (25)for different values of the amplitude of the periodic excita tion considering σ=0.024. The plots conﬁrm the nodes achievingcompletesynchronization. Chaos31,063111(2021);doi:10.1063/5.0054459 31,063111-11 PublishedunderanexclusivelicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG. 14.The collective behavior of the network (25)for different values of the amplitude of the quasi-periodic excitation considering σ=0.025. The plots conﬁrm the emergenceofchimeras. FIG. 15.The collective behavior of the network (25)for different values of the amplitude of the quasi-periodic excitation considering σ=0.025. The plots conﬁrm nodes achievingcompletesynchronizationfor A2=1.5andA2=1.8. Chaos31,063111(2021);doi:10.1063/5.0054459 31,063111-12 PublishedunderanexclusivelicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG.16.Re-occurrenceplotsfordifferentvaluesof A2withtheblueregionsshowingcoherentnodeswhiletheother colorsshowingincoherentnodes. some nodes are synchronized while some remain incoherent. This confirms the emergence of chimeras. We tried different values of the coupling to observe complete synchronization, but we could observe only cluster synchronization. Hence, we shifted our investigation to the excitation amplitude, and we could observe two ranges of amplitude which could achieve complete synchronization. But our interest is on the amplitude for which the nodes are not in chaos, but the network achieved synchronization. To calculate this, we introduced the re-occurrence plots which is useful in understand- ing different regimes in a network. From the re-occurrence plots, we could show that for a certain amplitude the nodes are not in chaos, but the network achieved synchronization. Similar studies are conducted for quasiperiodically excited energy harvesters and again we could show some values of the amplitude that can achieve syn- chronization without disturbing the local complex behavior of the nodes. ACKNOWLEDGMENTS Arthanari Ramesh, Karthikeyan Rajagopal, and Prakash Duraisamy have been partially funded by the Research Grant of the Center for Nonlinear Systems, Chennai Institute of Technology with Reference No. CIT/CNS/2021/RP-017. The authors declare that they have no conflict of interest.DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Erturk, J. M. Renno, and D. J. 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1.468772.pdf	Is slow thermal isomerization in viscous solvents understandable with the idea of frequency dependent friction? Hitoshi Sumi and Tsutomu Asano Citation: The Journal of Chemical Physics 102, 9565 (1995); doi: 10.1063/1.468772 View online: http://dx.doi.org/10.1063/1.468772 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/102/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Isomerization dynamics in viscous liquids: Microscopic investigation of the coupling and decoupling of the rate to and from solvent viscosity and dependence on the intermolecular potential J. Chem. Phys. 110, 7365 (1999); 10.1063/1.478638 Solvent friction in polymer solutions and its relation to the high frequency limiting viscosity J. Chem. Phys. 89, 6523 (1988); 10.1063/1.455372 The effect of frequency dependent friction on isomerization dynamics in solution J. Chem. Phys. 78, 2735 (1983); 10.1063/1.444983 Breakdown of Kramers theory description of photochemical isomerization and the possible involvement of frequency dependent friction J. Chem. Phys. 78, 249 (1983); 10.1063/1.444549 Shear Stress Dependence of the Intrinsic Viscosity of Polymethylmethacrylate in an Extremely Viscous Solvent J. Chem. Phys. 38, 2315 (1963); 10.1063/1.1733978 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.176.129.147 On: Sun, 07 Dec 2014 14:41:42Is slow thermal isomerization in viscous solvents understandable with the idea of frequency dependent friction? Hitoshi Sumi Institute of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Tsutomu Asano Department of Chemistry, Faculty of Engineering, Oita University, Oita 870-11, Japan ~Received 2 November 1994; accepted 22 March 1995 ! Thermal Z/Eisomerization of substituted azobenzenes and N-benzylideneanilines takes place slowly after fast photoinduced E/Zisomerization. Its rate constant kobsis smaller than about 103s21 because of a high reaction barrier of about 50 kJ/mol. The pressure dependence of kobsmeasured in solvents as glycerol triacetate can well be understood in the framework of the transition state theory~TST!at low pressures. At high pressures, however, k obsbegins to steeply decrease as the pressure increases, to be more exact, as the solvent viscosity hincreases with the pressure, and the reaction enters the non-TST regime. Since the h-induced decrease of kobsat high pressures is slower than h21, it cannot be described by the Kramers theory which regards the reaction as the barrier surmounting by Brownian motions regulated by frequency independent friction. Next, it wasadjusted to the Grote–Hynes theory incorporating the idea of frequency dependent friction. Thesituation of k obsmentioned earlier enabled us to derive, without adjustable parameters, the correlation time tscamong random forces for friction due to solvent microscopic motions in the generalized Langevin equation on which the theory is based.At h;107Pa s, we obtained tsc;1 ms. It is too long to justify the theory, since such a long-time correlation cannot be realized amongrandom forces exerting on the isomerizing moiety with an angstrom dimension. It will also beshown that tscmust be so long unphysically as to be at least much longer than 1 ps even if kobsat low pressures is adjusted to the theory. © 1995 American Institute of Physics. I. INTRODUCTION The most traditional theory for chemical reaction rates is the transition state theory ~TST!established in the 1940’s. It has recently been disclosed, however, that the TST cannot beapplied to varieties of solution reactions. Examples can beseen in biological enzymatic reactions, 1electron or proton transfer reactions,2atom-group transfer reactions,3and iso- merization reactions.4Study of solution reactions is one of the most fundamental as well as the most traditional sub-jects in chemistry. The situation mentioned above, however,means that we have not yet established a general expressionon rates of solution reactions.Accordingly, many discussionshave been aroused on this subject. 5 In the TST, it is assumed a priorithat ﬂuctuations in the reactant state are so fast that the distribution of populationsin the reactant state is always maintained in thermal equilib-rium in the course of reactions. Then, the population of re-actants in the transition state is also always maintained inthermal equilibrium with those in the reactant state, and therate constant can be calculated from this population withoutknowledge on dynamics of ﬂuctuations in the reactant state.In this assumption, therefore, the rate constant should notdepend on how fast ﬂuctuations are in the reactant state.Molecular arrangements of the solute-solvent system cantake various conformations. They ﬂuctuate due to jolting anddamping by microscopic motions of solvent molecules.Since the speed of these conformational ﬂuctuations de-creases as the viscosity hof solvents increases, h21can be regarded as a measure of the speed. In the solution reactionsmentioned earlier, the rate constant observed decreases as hincreases, that is, as h21decreases. This means that the rate constant depends on the speed of conformational ﬂuctuationsin the solute-solvent system and, hence, that these reactionsare nonthermalized reactions located outside the framework of the TST. To be more exact, these reactions are controlledby slow speeds of these ﬂuctuations. The ﬁrst theory giving the h-induced decrease of the rate constant is the Kramers theory6presented as early as 1940. He explicitly treated dynamical processes of ﬂuctuations inthe reactant state, not assuming a priorithe thermal equilib- rium distribution therein. He envisaged solution reactions asoccurring as a result of surmounting over the transition-statebarrier along the reaction coordinate by diffusive Brownianmotions. These diffusive motions are excited by randomforces and damped by frictional forces, both of which arisefrom microscopic motions of solvent molecules. When thediffusive motions are sufﬁciently fast in his theory, the ther-mal equilibrium distribution in the reactant state is automati-cally obtained, and the rate constant reduces to that expectedfrom theTST.As hincreases, however, the diffusive motions become slow, and the reaction becomes limited by their slowspeed. In this non-TST regime, the rate constant given by histheory decreases in proportion to h21. Most often investigated experimentally, concerning the h dependence of the rate constant in solution, is photoinducedE/Z~trans!/~cis!isomerization of stilbenes. 4The isomeriza- tion takes place essentially by the surmounting over a tran-sition-state barrier on the excited-state potential surface. Thereaction is very fast with a rate constant on the order of 10 10 s21due to a small height ~about 15 kJ/mol !of the barrier. 9565 J. Chem. Phys. 102(24), 22 June 1995 0021-9606/95/102(24)/9565/9/$6.00 © 1995 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.176.129.147 On: Sun, 07 Dec 2014 14:41:42The observed rate constant kobsdecreases as hincreases. To be more exact, kobsoften decreases more slowly than h21, describable by a fractional-power dependence on h21as kobs}h2awith 0 ,a,1. The hdependence deviates from that expected from the Kramers theory. To understand the slow hdependence of kobs, an idea has often been employed that friction felt by the reactant candecrease when it surmounts over the transition-state barrierwith a high speed. In this case, the rate constant should notbe as small as expected from the Kramers theory, and henceits hdependence should be milder than h21expected from the theory.This idea of frequency dependent friction is basedon the Grote–Hynes ~GH!theory 7ﬁrst presented in 1980. Actual ﬁtting of the hdependence of kobswith the theory was made by several authors.8–11They all pointed out coin- cidentally that the ﬁtting could be made only by adopting anunphysically small value for the downward curvature of thepotential at the top of the transition-state barrier. The rate constant of photoinduced isomerization of stil- benes in solution always decreases as hincreases.This arises from a situation that the rate constant is very large because ofa low height of the transition-state barrier and hence the re-action is easily controlled by a slow speed of conformationalﬂuctuations in the solute-solvent system. This situation itselfis very interesting, but is not convenient from the standpointof investigating the general expression for the rate constantof solution reactions. More adequate is a situation that therate constant can be described by the TST in the small h region, but decreases with hin the large hregion, entering the non-TST regime. Such a situation was recently reportedby one of the present authors ~Asano !and his collaborators 12 for thermal Z/Eisomerization of substituted azobenzenes andN-benzylideneanilines. Although these molecules are similar in structure to stilbenes, the reaction investigatedwith these molecules is the recovering of the Eform from theZform obtained after the photoinduced E/Zisomeriza- tion, which was investigated for stilbenes. Quite differentlyfrom the E/Zphotoisomerization of stilbene, this reaction takes place slowly with a rate constant smaller than about10 3s21by surmounting over a high transition-state barrier ~with height of about 50 kJ/mol !on the ground-state poten- tial surface. The solvent viscosity was changed over about10 4times by pressures. In the low-pressure region, the rate constant of this reaction showed a pressure dependencewhich could be understood in the framework of the TST. Inthe high-pressure region, on the other hand, it steeply de-creased as the pressure increased, to be more exact, as thesolvent viscosity hincreased with the pressure. In this non- TST regime, the observed rate constant kobsshowed a frac- tional power dependence on h21, as observed for stilbenes. It was shown by the present authors13that the pressure ~or viscosity !dependence of kobsmentioned earlier could be understood by considering a two-step mechanism that thereaction took place as a result of sequential two steps in-duced ﬁrst by slow conformational ﬂuctuations in the solute-solvent system and then by fast intramolecular vibrationalﬂuctuations in the solute molecule. The former are regulatedby solvent ﬂuctuations, having a relaxation time proportionalto the solvent viscosity h, while the latter are not inﬂuencedby solvent ﬂuctuations, having a relaxation time on the order of a typical phonon period. To be more exact, it was experi-mentally shown that k obshad a form of kobs51/~kTST211kf21!, wherekTSTrepresents the rate constant expected from the TST, while kf~.0!represents the rate constant controlled by solvent ﬂuctuations, having a fractional-power dependenceon h21askf}h2awith 0 ,a,1. In this form, when kf@kTST in the low pressure ~or viscosity !region, we have kobs'kTST conﬁrming the TST. When kf!kTSTin the high pressure ~or viscosity !region, on the other hand, we have kobs'kf}h2a, as observed experimentally. The two-step mechanism mentioned above was origi- nally proposed by one of the present authors ~Sumi !and Marcus,14called the Sumi–Marcus model. Subsequently, it was shown by Sumi15that the rate constant in this model is given by the form of 1/ ~kTST211kf21!mentioned earlier. In this model, the reaction is directly induced by fast intramolecularvibrational ﬂuctuations, but, in order for this step to takeplace most effectively, the solute-solvent system must takeappropriate conformations induced by slow solvent ﬂuctua-tions. The fractional-power dependence of k fonh21is ob- tained when the appropriate conformations have a nonvan-ishing distribution. The two-step mechanism mentioned above is completely different from the reaction mechanism in the GH theory.Then, it is quite natural to ask what we can derive when k obs of the thermal isomerization of substituted azobenzenes and N-benzylideneanilines is adjusted to the GH theory. The pro- cess is performed in the present work. The plan of thepresent work is as follows. Characteristic features of the ther-mal isomerization are ﬁrst pointed out in Sec. II. The adjust-ing is performed in Sec. III. Remaining discussions are leftin Sec. IV. II. CHARACTERISTIC FEATURES OF THE THERMAL ISOMERIZATION As concrete examples in molecules investigated ex- perimentally in Refs. 12 and 13, we take 4- ~di- methylamino !-48-nitroazobenzene ~DNAB !andN-@4- ~dimethylamino !-benzylidene #-4-nitroaniline ~DBNA !, since these molecules showed widest ranges of variation in the rateconstant with the pressure increase. Solvents used were glyc-erol triacetate ~GTA!and Traction Fluid B ~TFB!, the latter of which was a commercial product ~of Nippon Oil Co.!whose main constituent was 2,4-dicyclohexyl-2-methyl- pentane. Molecular structures of DNAB and DBNA areshown in Fig. 1 together with the schematic proﬁle of theground ~S 0!- and the excited ~S1!-state potential surfaces. The photoinduced reaction cycle of these molecules pro- ceeds on these potential surfaces as follows. In the photoin-ducedE/Zisomerization process, denoted by 1 in Fig. 1, a molecule with the stable form Eon theS 0potential surface is photoexcited to the S1surface, from which, surmounting over a small potential barrier, the molecule drops to an inter-mediate state at the lowest point of the S 1surface. Then, the molecule makes a transition to the top of the S0surface, from which, along the S0surface, about half of it relax to the Z form and the remaining half relax to the original Eform. In this process 1, the surmounting over the potential barrier on9566 H. Sumi and T. Asano: Slow thermal isomerization in viscous solvents J. Chem. Phys., Vol. 102, No. 24, 22 June 1995 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.176.129.147 On: Sun, 07 Dec 2014 14:41:42theS1surface is rate limiting. Then, measuring the rise of population in the Zform, we can determine the rate constant of the barrier surmounting, as performed for stilbenes. SincetheZform is metastable on the S 0surface, higher than the E form, it gradually transforms itself to the Eform, surmount- ing over the potential barrier with height Dshown in Fig. 1. Process 1 is very fast because of a low barrier on the S1 surface. In stilbenes, the barrier height is about 15 kJ/mol and this process takes about 100 ps after photoexcitation,4as mentioned in Sec. I. Process 2 is, on the other hand, slowbecause of a high barrier on the S 0surface. In both DNAB and DBNA, the barrier height is about 50 kJ/mol and thisprocess takes at least about 1 ms. In the measurement withtime resolution longer than 1 ns, therefore, we observe as ifprocess 1 took place simultaneously after photoexcitation,and that process 2 follows slowly. In order for the reaction to take place with the mecha- nism in the GH theory as well as in the Kramers theory, thereactant must surmount over the transition-state barrier onlyby diffusional Brownian motions regulated by solvent ﬂuc-tuations. In the two-step mechanism of the Sumi–Marcusmodel, on the other hand, the surmounting over the trans-ition-state barrier is accomplished as a result of sequentialtwo steps. That is, the barrier is climbed ﬁrst by the diffu-sional Brownian motions only up to intermediate heights,from which much faster intramolecular vibrational motionsexcite the reactant to the transition state at the top of thebarrier. It is apparent, therefore, that differences betweenthese two mechanisms manifest themselves most drasticallywhen the transition-state barrier is much higher than the ther-mal energy k BTwhich is about 2.5 kJ/mol for T5300 K. In this respect, thermal isomerization of substituted azoben-zenes and N-benzylideneanilines on the S 0surface is moreinteresting than photoisomerization of stilbenes on the S1 surface, since the barrier height for the former12,13is about 50 kJ/mol while that for the latter4is only about 15 kJ/mol. The observed pressure dependence of the rate constant kobsfor thermal isomerization of DBNA in TFB and DNAB in GTAis shown, respectively, in Figs. 2 ~a!and 2 ~b!for four temperatures around room temperature. Detailed experimen-tal set up for the measurement was described in Refs. 12 and13. The viscosity hof these solvents was determined also in Refs. 12 and 13 by best ﬁtting to the values measured atseveral pressures with a formula that log his a linear func- tion of pressure. In Fig. 2 ~a!for DBNAinTFB, kobsis nearly independent of pressure PwhenP&100 MPa ~51000 bar !.This behavior can well be understood in the framework of the TST as asituation 16that the volume of the reactant is almost un- changed between the reactant and the transition states. InFig, 2 ~b!for DNAB in GTA, on the other hand, k obsin- creases as Pincreases when P&250 MPa ~52500 bar !. This behavior can also be understood well in the framework of theTST as a situation of a negative activation volume 17that the transition state with a polar structure has a volume smallerthan the reactant state. Fitting of the pressure dependence ofk obsby the TST with an appropriate activation volume in the low pressure region was performed in Refs. 12 and 13. The FIG. 1. Photocycle of isomerization of DBNA and DNAB, composed of photoinduced E/Zisomerization ~process 1 !and thermal Z/Eisomerization ~process 2 !in the ground ~S0!- and the excited ~S1!-state potential surfaces. FIG. 2. Pressure dependence of the rate constant kobsof thermal Z/E isomerization of DBNA in TFB @~a!#and DNAB in GTA @~b!#.9567 H. Sumi and T. Asano: Slow thermal isomerization in viscous solvents J. Chem. Phys., Vol. 102, No. 24, 22 June 1995 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.176.129.147 On: Sun, 07 Dec 2014 14:41:42ﬁtting was nearly complete for DBNA in TFB when P<90 MPa at 278 K, P<120 MPa at 288 and 298 K, and P<135 MPa at 308 K, while for DNAB in GTA when P<250 MPa at 278 K, P<300 MPa at 288 K, P<350 MPa at 298 K, and P<400 MPa at 313 K. Solvent viscosities hin these low- pressure regions in theTSTregime are shown in Fig. 3 ~a!for TFB and in Fig. 3 ~b!for GTA, where h/~1023Pa s!was shown on the ordinate with a logarithmic scale for later con-venience ~10 23Pa s51c P!. Expressing the rate constant observed in the low- pressure region in a form expected from the TST with anappropriate activation volume, we can analytically extrapo-late it to the high-pressure region, as performed in Refs. 12and 13. In this way we can obtain the rate constant expectedfrom the TST in the entire pressure region, and it is writtenask TST. In the low-pressure region, kobsis the same as kTST, but in the high-pressure region, kobsdeviates downwards fromkTST. The ratio kobs/kTSTbetween them in the high- pressure region was plotted in Fig. 4 ~a!for DBNA in TFB and in Fig. 4 ~b!for DNAB in GTA, where they were shown vs solvent viscosities hinstead of vs pressures P. The high- pressure region in Fig. 4 ~a!for DBNA in TFB is P>120 MPa ~h>1930 Pa s !at 278 K, P>150 MPa ~h>1310 Pa s ! at 288 K, P>150 MPa ~h>227 Pa s !at 298 K, and P>165 MPa ~h>99.7 Pa s !at 308 K. That in Fig. 4 ~b!for DNAB in GTA isP>350 MPa ~h>3050 Pa s !at 278 K, P>400 MPa ~h>775 Pa s !at 288 K, P>450 MPa ~h>201 Pa s !at 298 K, andP>500 MPa ~h>18.8 Pa s !at 313 K.III. ADJUSTING OF kobsTO THE GH THEORY Let us examine what we can derive by assuming that kobs is describable by the GH theory.7To this end, we ﬁrst need to review the essence of the GH theory, which is done below. Inthis theory the ratio k obs/kTSTis called the transmission coef- ﬁcient, and is related to the frequency ~that is, the speed !m with which the reactant passes, by diffusive Brownian mo- tions, through the transition-state barrier region, by kobs/kTST5m/vb, ~1! where vbrepresents the imaginary frequency, proportional to the square root of the downward curvature, at the top of thetransition-state barrier. The vbhas been estimated8,9to have a magnitude on the same order as typical phonon frequencieson the order of 10 13s21. Figure 4 gives kobs/kTSTdetermined experimentally as a function of the solvent viscosity hin the high-pressure region in the non-TST regime. Then, Eq. ~1! shows that the frequency mhas already been determined ex- perimentally as a function of hin the present systems if kobs is describable by the GH theory. In the GH theory, mis determined as a solution of m/vb5vb/@m1z~m!#, ~2! called the GH equation, where z~m!represents the frequency dependent friction, which is deﬁned by the Laplace trans-form of the frictional memory function zˆ(t) of time t,a s z~m!5E 0` zˆ~t!e2mtdt. ~3! FIG. 3. Solvent viscosity h~in units of 1023Pa s!in the low-pressure region in theTSTregime realized for DBNAinTFB @~a!#and DNAB in GTA @~b!#. FIG. 4. Viscosity dependence of the ratio of kobsto the TST-expected rate constantkTSTin the non-TST regime realized for DBNA in TFB @~a!#and DNAB in GTA @~b!#.9568 H. Sumi and T. Asano: Slow thermal isomerization in viscous solvents J. Chem. Phys., Vol. 102, No. 24, 22 June 1995 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.176.129.147 On: Sun, 07 Dec 2014 14:41:42The reaction coordinate in the GH theory is a coordinate connecting the two minimum points at the Zform and at the Eform on the S0surface in Fig. 1 through the transition-state barrier between them. It was written as Xin Fig. 1, and the double-well potential on the S0surface is written as W(X) alongX. The frictional memory function zˆ(t) regulates the time evolution of diffusive Brownian motions along the co-ordinateX, written as X(t) as a function of time t, through the generalized Langevin equation d 2X~t! dt252dW@X~t!# dX~t!2E 2`t zˆ~t2t8!dX~t8! dt8dt8 1R~t!, ~4! whereR(t) represents random forces in contrast to the sys- tematic force 2dW[X(t)]/dX(t) coming from the potential W(X) at the position X(t). The second term on the right- hand side of Eq. ~4!represents that the strength of frictional forces for the motion X(t) at time tcomes also from the magnitude of its velocity dX(t8)/dt8in the past at time t8,t through a nonvanishing tail of the frictional memory func-tion zˆ~t!att5t2t8~.0!. Since zˆ~t!.0a ta n y t.0, the frequency dependent friction z~m!deﬁned by Eq. ~3!is a decreasing function of m. This means that the faster is the passage of the reactant through the transition-state region,the smaller is the friction felt by the reactant. This was theoriginal starting point of GH for introducing the frequencydependence of friction to generalize the Kramers theorywithout it. In fact, the Fokker–Planck equation used byKramers can be derived from the Langevin equation for zˆ(t2t8)}d(t2t8) in Eq. ~4!.18 In general, both R(t) and zˆ(t) appearing in Eq. ~4!arise from microscopic motions of heat-bath modes interactingwith the reaction coordinate. 19–21To say speciﬁcally to isomerization in solvents, both of them arise from micro-scopic motions of solvent molecules interacting with theisomerizing moiety. Then they must be related to each other.This relation is known as the ﬂuctuation-dissipation theorem,and is written as ^R~t!R~t8!&5kBTzˆ~t2t8!fort.t8, ~5! where ^{{{&represents the statistical average over heat-bath modes at temperature T. In the present problem, the time of correlation between tandt8on the left-hand side of Eq. ~5! should be on the order of the inverse of the width of fre-quency distribution of microscopic solvent motions whosewavelength is comparable to or smaller than the dimensionof the isomerizing moiety of a solute molecule. 19–21The con- dition for the wavelength arises from the situation that themicroscopic solvent motions contributing to R(t) must inter- act with an appreciable strength with the isomerizing motiondescribed by the reaction coordinate X. Let us write this correlation time as tsc. It also gives the correlation time of the frictional memory function on the right-hand side of Eq.~5!. From the general principle for tscmentioned earlier, we can estimate tscto be at most on the order of 1 ps, since 1 ps corresponds to the width of frequency distribution of about30 cm 21'h/~1p s!wherehrepresents the Planck constant. Since the order of magnitude of tscthus estimated plays im-portant roles in the discussion developed later in the present work, it will be explained in more detail in the Appendix. Let us consider here a scaled correlation function given by^R(t)R(0)&/^R(0)2&and regard it as a function of a scaled variable t/tsc,a s f~t/tsc![^R~t!R~0!&/^R~0!2&5zˆ~t!/zˆ~0!, ~6! where the second equality is ensured by Eq. ~5!. Since zˆ(t) decays with the correlation time tsc, function f(x) deﬁned above satisﬁes f~0!51, andf(x)'1 forx!1 whilef(x)!1 forx@1. The Laplace transform of f(x) is written as F~l!, which is deﬁned by F~l![E 0` f~x!e2lxdxYE 0` f~x!dx5z~l/tsc!/z~0!, ~7! where the second equality is ensured by Eq. ~6!with Eq. ~3!. ThisF~l!is a decreasing function of l, since so is z~m!as a function of m, withF~0!51. Especially, it satisﬁes F~l!'1 for l!1 andF~l!'~al!21~!1!forl@1, ~8! with a[E 0` f~x!dx5O~1!, ~9! where the second relation in Eq. ~8!is obtained by approxi- matingf(x)b yf~0!~51!in the numerator in the central expression of F~l!in Eq. ~7!.As noted in Eq. ~9!, it is certain thatais a number of order unity, irrespective of the func- tional form of f(x), although amay have a slight depen- dence on the solvent viscosity hthrough a possible slight h dependence of the functional form of f(x). In the last expression of F~l!in Eq. ~7!,z~0!represents the friction at zero frequency. This is the friction used byKramers 6in his theory. In fact, if the time ( t) variation of dX(t)/dtis very small in the time interval of tsc, the second term on the right-hand side of Eq. ~4!can be approximated as E 2`t zˆ~t2t8!dX~t8! dt8dt8'z~0!dX~t! dt, ~10! since zˆ(t2t8)'0 whent2t8@tsc. In this case, therefore, the generalized Langevin equation of Eq. ~4!reduces to the Langevin equation, which can be converted to the Fokker–Planck equation used by Kramers. 18The zero-frequency fric- tionz~0!appearing in the Kramers theory has successfully been regarded as proportional to the solvent viscosity h, obeying the hydrodynamic Stokes–Einstein relation.10,22,23 To be more concrete, adopting the slip boundary conditionfor an uncharged rotating moiety or for nonpolar solvents,we can express z~0!as z~0!54pChr2d/I ~11! when the isomerization motion is modeled as the motion of a sphere of hydrodynamic radius drotated at a distance rwith a moment of inertia Iaround a ﬁxed axis, where Cis a number of order unity dependent on the actual shape andvolume of the rotating moiety. Equation ~11!has also been9569 H. Sumi and T. Asano: Slow thermal isomerization in viscous solvents J. Chem. Phys., Vol. 102, No. 24, 22 June 1995 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.176.129.147 On: Sun, 07 Dec 2014 14:41:42adopted as the zero-frequency value of the frequency depen- dent friction z~m!in its explicit calculation.8–11Combination of Eqs. ~7!and~11!enables us to express z~m!in terms of F~l!as z~m!5~4pCr2d/I!hF~mtsc!. ~12! The explicit adjusting procedure of kobsshown in Fig. 2 to the GH theory is started from here. Let us begin with theTST regime where k obsis describable by the TST. The TST regime for DBNA in TFB and DNAB in GTA is realized inthe low-pressure region which can be read in Fig. 3 by theprojection of a line segment for each temperature on theabscissa in Fig. 3 ~a!for DBNA in TFB and on that in Fig. 3~b!for DNAB in GTA. In order for theTSTto be recovered in the GH theory, m/vb'1 and z~m!!m, ~13! must be satisﬁed, respectively, in Eqs. ~1!and~2!. Applying Eq.~12!to Eq. ~13!,w eg e t bF~mtsc!!~1023Pa s!/h ~14! with b[4pC~1023Pa s!r2d/~Ivb!5O~1! ~15! as a relation which must be satisﬁed when the TST is recov- ered in the GH theory.As noted in Eq. ~15!,bis a number of order unity since I/r2has a magnitude on the order of the total mass of the rotating moiety ~;2310225kg!, and d;2.5310210m, while vb;1013s21as noted below Eq. ~1!. The value of h/~1023Pa s!, the inverse of which appears on the right-hand side of Eq. ~14!, can be read in Fig. 3 by the projection of a line segment for each temperature on theordinate in Fig. 3 ~a!for DBNA in TFB and in Fig. 3 ~b!for DNAB in GTA in the TST regime. Then we see that thevalue of ~10 23Pa s!/hhas a magnitude in the range of 1026–1022for DBNA in TFB and of 1025–1021for DNAB in GTA. Therefore, ~1023Pa s!/his much smaller than unity, and Eqs. ~14!and~15!require that F~mtsc!must be much smaller than unity, too. From Eq. ~8!, the situation of F~mtsc!!1 is obtained only when mtsc@1 and, in this case, F~mtsc!'~amtsc!21should be realized with adeﬁned by Eq. ~9!. Applying this relation to Eq. ~14!and utilizing the ﬁrst near equality in Eq. ~13!in the TST regime, we get cvbtsc@h/~1023Pa s! ~16! with c[a/b5O~1! ~17! as a relation which must be satisﬁed when the TST regime realized for DBNAin TFB and DNAB in GTAis describableby the GH theory. As noted in Eq. ~17!,cis a number of order unity since so are both aandbas noted respectively in Eqs.~9!and~15!. For other terms appearing in Eq. ~16!, vb has a magnitude on the order of 1013s21as noted below Eq. ~1!, while h/~1023Pa s!for DBNA in TFB, shown in Fig. 3~a!, is at least about 102at the low-pressure edge ~at ambi- ent pressure at 308 K !in the TST regime, and reaches about 33105at the high-pressure edge at 278 K in theTSTregime. For DNAB in GTA shown in Fig. 3 ~b!,h/~1023Pa s!is atleast about 10 at the low-pressure edge ~at ambient pressure at 313 K !, and reaches about 105at the high-pressure edge at 278 K. In order that the TST regime realized for DBNA inTFB is describable by the GH theory, therefore, the correla-tion time tscamong random forces in the generalized Lange- vin equation must be much larger than 10 ps at the low-pressure edge ~at ambient pressure at 308 K !in the TST regime, and be much larger than 30 ns at the high-pressureedge at 278 K in theTSTregime. For DNAB in GTA, it mustbe much larger than 1 ps at the low-pressure edge ~at ambi- ent pressure at 313 K !, and be much larger than 10 ns at the high-pressure edge at 278 K. These requirements cannot besatisﬁed in real systems, since tscmust be at most on the order of 1 ps as mentioned below Eq. ~5!.Therefore, theTST regime realized in these systems is not describable by the GHtheory. Let us next examine the non-TST regime where k obsde- viates downwards from the TST-expected rate constant kTST. It was shown above that the GH theory had a problem con-cerning tscin its applicability to the TST regime realized in these systems. It will be shown later that the problem be-comes more serious in its applicability to the non-TST re-gime also realized in these systems. If the non-TSTregime isdescribable by the GH theory, the transmission coefﬁcient m/vbmust be equal to kobs/kTSTplotted in Fig. 4 as a func- tion of the solvent viscosity hin the non-TST regime. In the GH theory, the frequency dependent friction z~m!divided by vbis given by ( m/vb)212(m/vb) from the GH equation of Eq.~2!. Then, Eq. ~12!enables us to get bF~mtsc!5~1023Pa s/h!@~kobs/kTST!21 2~kobs/kTST!#, ~18! withbdeﬁned by Eq. ~15!. It is a relation which must be satisﬁed when the non-TST regime is describable by the GHtheory. The value of the quantity on the right-hand side ofEq.~18!can easily be calculated from k obs/kTSTobtained in the non-TST regime. For DBNA in TFB, it is shown in Fig.5~a!as a function of the solvent viscosity hin the non-TST regime, while for DNAB in GTA, it is shown in Fig. 5 ~b!. We see in Fig. 5 that the value of the quantity on the right-hand side of Eq. ~18!is much smaller than unity in the non- TST regime realized in these systems. Then, from Eq. ~18!, the scaled correlation function F~ mtsc!must also be much smaller than unity in these systems, since bis a number of order unity as noted in Eq. ~15!. From Eq. ~8!, the situation ofF~mtsc!!1 is obtained only when mtsc@1, and in this case F~mtsc!'~amtsc!21should be realized with adeﬁned by Eq. ~9!. Applying this relation to Eq. ~18!and utilizing Eq. ~1!, we get cvbtsc5~h/1023Pa s!@12~kobs/kTST!2#, ~19! withcdeﬁned by Eq. ~17!. It is a relation which must be satisﬁed when the non-TST regime realized for DBNA inTFB and DNAB in GTA is describable by the GH theory.The value of the quantity on the right-hand side of Eq. ~19! can easily be calculated from k obs/kTSTobtained in the non- TST regime. For DBNAin TFB, it is shown in Fig. 6 ~a!as a function of the solvent viscosity hin the non-TST regime,9570 H. Sumi and T. Asano: Slow thermal isomerization in viscous solvents J. Chem. Phys., Vol. 102, No. 24, 22 June 1995 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.176.129.147 On: Sun, 07 Dec 2014 14:41:42while for DNAB in GTA, it is shown in Fig. 6 ~b!. We see in Fig. 6 ~a!that the value of the quantity on the right-hand side of Eq. ~19!is at least about 5 3105at the low-pressure edge ath'102Pa s at 308 K in the non-TST regime, and reaches about 5 31010ath'53107Pa s at 278 K for DBNAin TFB. We see in Fig. 6 ~b!that it is at least about 105at the low- pressure edge at h'20 Pa s at 313 K in the non-TST regime, and reaches about 7 3109ath'73106Pa s at 278 K for DNAB in GTA. These values give cvbtscfrom Eq. ~19!, wherecis a number of order unity as noted in Eq. ~17!and vbhas a magnitude on the order of 1013s21as noted below Eq.~1!. In order that the non-TST regime realized for DBNA in TFB is describable by the GH theory, therefore, the cor-relation time tscamong random forces in the generalized Langevin equation must be at least about 50 ns at the low-pressure edge at h'102Pa s at 308 K in the non-TST re- gime, and be as large as about 5 ms at h'53107Pa s at 278 K. For DNAB in GTA, it must be at least about 10 ns at thelow-pressure edge at h'20 Pa s at 313 K in the non-TST regime, and be as large as about 1 ms at h'107Pa s at 278 K. These requirements cannot be satisﬁed in real systems,since tscmust be at most on the order of 1 ps as mentioned below Eq. ~5!. Therefore, the non-TST regime realized in these systems is not describable by the GH theory. The problem mentioned here in the applicability of the GH theory to thermal Z/Eisomerization of DBNA in TFB and DNAB in GTA was derived, without adjustable param-eters, only from experimental data on the pressure ~or vis- cosity !dependence of the rate constant.The applicability of the Stokes–Einstein relation of Eq. ~11!has not yet been checked for viscosities has high as shown on the abscissa in Fig. 4, where the non-TST regimewas observed. It will be shown in Ref. 24 that the conclusionof the present paper remains unchanged even if Eq. ~11!is not maintained, as it is, for hin the non-TST regime. IV. DISCUSSION Thus, it was shown to be inappropriate to consider that thermalZ/Eisomerization of DBNA in TFB and DNAB in GTA with a transition-state barrier as high as about 50 kJ/mol took place only by diffusional Brownian motions in theframework of the GH theory. Then the interpretation pro-posed in Ref. 13 remains plausible following the Sumi–Marcus model 14whose rate constant was derived in Ref. 15. In this interpretation, the surmounting over the transition-state barrier is accomplished as a result of sequential twosteps, induced ﬁrst by diffusional conformational ﬂuctuationsin the solute-solvent system and then by much faster in-tramolecular vibrational ﬂuctuations in the solute molecule. In the analysis presented in this work for investigating the applicability of the GH theory, it was essentially impor-tant that the quantity on the right-hand side of Eq. ~14!was much smaller than unity in the TST regime realized forDBNA in TFB and DNAB in GTA. It was also essentially FIG. 5. Viscosity dependence of the scaled correlation function F~mtsc! multiplied by bof order unity in the non-TSTregime, with m/vb5kobs/kTST, obtained under an assumption that the non-TST regime is describable by theGH theory for DBNA in TFB @~a!#and DNAB in GTA @~b!#. FIG. 6. Viscosity dependence of the correlation time among random forces tscmultiplied by cof order unity and vb~;1013s21!in the non-TST regime, obtained under an assumption that the non-TST regime is describable by theGH theory for DBNA in TFB @~a!#and DNAB in GTA @~b!#.9571 H. Sumi and T. Asano: Slow thermal isomerization in viscous solvents J. Chem. Phys., Vol. 102, No. 24, 22 June 1995 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.176.129.147 On: Sun, 07 Dec 2014 14:41:42important in the non-TST regime realized in these systems that the quantity on the right-hand side of Eq. ~18!was much smaller than unity. The former situation enabled us to derivea requirement which must be satisﬁed by the correlation time tscamong random forces in the generalized Langevin equa- tion if the observed TST regime was describable by the GHtheory. The latter situation enabled us to derive, without ad-justable parameters, the value of the correlation time itself ifthe observed non-TST regime was describable by the GHtheory. It was also important that the TST-expected rate con-stant could be determined in the non-TSTregime by extrapo-lating the rate constant in the TST regime at low pressures. It is interesting to apply the analysis mentioned above also to photoinduced E/Zisomerization of stilbenes in solu- tion. As an example, let us take the reaction in liquidn-hexane solution. In this reaction, the observed rate con- stantk obsdecreases as the pressure is increased, obeying the fractional-power dependence on the inverse of the solventviscosity hin the high-pressure region,25as mentioned in Sec. I. In Fig. 3 of Ref. 25, however, it appears that the h dependence of kobsat 298 K reaches a saturation at two ex- perimental points at hof about 0.3 and 0.4 31023Pa s in the neighborhood of ambient pressure. If it is allowed to con-sider that k obsenters in theTSTregime in these two values of hat 298 K, Eq. ~14!must be satisﬁed there if theTSTregime is simultaneously describable by the GH theory.The quantityon the right-hand side of Eq. ~14!has a magnitude of about 2–3 at these values of h. Since it is not much smaller than unity, we cannot regard F~mtsc!on the left-hand side of Eq. ~14!as much smaller than unity.Then, we cannot derive such a requirement as Eq. ~16!. In this case, therefore, it is pos- sible that mtsc@'vbtscin the TST regime from Eq. ~13!#has a magnitude of order unity and it is possible from vb;1013 s21thattscbecomes a quantity, at most, on the order of 1 ps in the TST regime for stilbenes, as noted below Eq. ~5!. Concerning the non-TST regime realized for photoin- ducedE/Zisomerization of silbenes in solution, however, it is difﬁcult to perform an analysis based on Eq. ~18!since we do not know the TST-expected rate constant kTSTin the non- TSTregime. In fact, even in n-hexane at 298 K we have only two experimental points which can be located in the TSTregime at low pressures.Then, it is difﬁcult to derive k TSTby extrapolating kobsat these two points to the high-pressure region in the non-TST regime. APPENDIX Since random forces R(t) arise from microscopic mo- tions of solvent molecules, it can be expressed as a linearcombination of amplitudes of normal modes of these mo-tions, as R ~t!5( jcjxj~t!, ~A1! with appropriate coefﬁcient cj’s, where xj(t) represents the amplitude of the jth normal mode at time t. Let us denote the frequency of the jth normal mode by nj, and the distortion energy at its amplitude xj(t)b y1 2gjxj(t)2with a force con- stantgj~.0!of the normal mode. At temperature Taroundroom temperature, the thermal energy kBTcan be regarded as much larger than the energy quantum hnjof the normal mode, where hrepresents the Planck constant. In this situa- tion,xj(t) can be regarded as a classical variable whose sta- tistical properties are determined by ^xj~t!&50,^xi~t!xj~t!&5di,jkBT/gj and^x˙i~t!xj~t!&50 with x˙j~t![dxj~t!/dtJ, ~A2! where ^{{{&represents the statistical average. The second equality in Eq. ~A2!can be derived from the equipartition law of energy ^1 2gjxj(t)2&51 2kBT, while the third equality represents that amplitude xj(t) of a coordinate has no corre- lation with its velocity x˙j(t) as well as with velocity x˙i(t)o f any other coordinate on the average. Correlation betweenamplitudes at different times is determined by ^xi~t!xj~t8!&5di,j~kBT/gj!cos@2pnj~t2t8!#, ~A3! since it should oscillate in t2t8with frequency njunder the initial condition composed of the second and the third rela-tions in Eq. ~A2!. Then, Eqs. ~A1!and~A3!give ^R~t!R~t8!&5kBT( jsjcos@2pnj~t2t8!# withsj[cj2/gj. ~A4! This relation is essentially the same as shown in Refs. 19– 21. Whent2t850 on the right-hand side of Eq. ~A4!, each sj~.0!is summed up with a coefﬁcient of unity since cos@2pnj(t2t8)#51 for any mode in this case.As t2t8~.0! increases, however, differences in njamong different modes cause dephasing among 2 pnj(t2t8)’s.Ast2t8~.0!further increases, even the sign of cos @2pnj(t2t8)#begins to change among different modes and individual terms speci-ﬁed byjon the right-hand side of Eq. ~A4!begin to cancel among themselves in the summation in j. In this way, the correlation function on the left-hand side of Eq. ~A4!decays ast2t 8~.0!increases. Its average decay time was called the correlation time tscin the text. In the decay mechanism mentioned here, the average rate of the decay, 1/ tsc, should be nearly equal to the width of distribution of frequency nj’s of modes contributing to the summation on the left-hand sideof Eq. ~A4!. In this case, modes contributing to the summa- tion must have an appreciable value of s j. This means that these modes interact with an appreciable strength with mo-tions of the isomerizing moiety of the solute molecule. Theinteraction with an appreciable strength is realized by modeswhose wave length is comparable to or smaller than the di-mension of the isomerizing moiety at most of about severalangstroms. Inelastic neutron-scattering studies on the energyspectrum in liquids 26show that even if the wavelength is ﬁxed at a single value of about several angstroms, corre-sponding to a single wave vector of about 2 p/~several ang- stroms !;1Å21, microscopic solvent motions have a width of about several THz in their angular frequencies, that is, awidth of about 1 THz in their frequencies, or, a width ofabout 30 cm 21in their energies. This shows that frequencies9572 H. Sumi and T. Asano: Slow thermal isomerization in viscous solvents J. Chem. Phys., Vol. 102, No. 24, 22 June 1995 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.176.129.147 On: Sun, 07 Dec 2014 14:41:42of microscopic solvent motions contributing appreciably to the summation on the right-hand side of Eq. ~A4!are distrib- uted over a width at least larger than 1 THz. Then, it isreasonable to consider that the correlation time among ran-dom forces tscis a time at most on the order of 1 ps @5~1 THz!21#, or, much shorter than it. 1See, for example, N. G. Goguadze, J. M. Hammerstad-Pedersen, D. E. Khoshtariya, and J. Ulstrup, Eur. J. Biochem. 200, 423 ~1991!. 2See, as a review, M. J. Weaver and G. E. McManis III, Acc. Chem. Res. 23, 294 ~1990!. 3See, for example, P. J. Steinbach et al., Biochem. 30, 3988 ~1991!. 4See, as a review, D. H. Waldeck, Chem. Res. 91, 415 ~1991!. 5For example, Ber. Bunsenges. Phys. Chem. 95, No. 3 ~1991!. 6H. A. Kramers, Physica 7, 284 ~1940!. 7R. F. Grote and J. T. Hynes, J. Chem. Phys. 73, 2715 ~1980!;74, 4465 ~1981!. 8B. Bagchi and D. W. Oxtoby, J. Chem. Phys. 78, 2735 ~1983!; B. Bagchi, Int. Rev. Phys. Chem. 6,1~1987!. 9G. Rothenberger, D. K. Negus, and R. M. Hochstrasser, J. Chem. Phys. 79, 5360 ~1983!. 10S. K. Kim and G. R. Fleming, J. Phys. Chem. 92, 2168 ~1988!. 11N. Sivakumar, E.A. Hoburg, and D. H. Waldeck, J. Chem. Phys. 90, 2305 ~1989!. 12K. Cosstick, T. Asano, and N. Ohno, High Pressure Res. 11,3 7~1992!.13T. Anaso, H. Furuta, and H. Sumi, J. Am. Chem. Soc. 116, 5545 ~1994!. 14H. Sumi and R. A. Marcus, J. Chem. Phys. 84, 4894 ~1986!; see also, W. Nadler and R. A. Marcus, ibid.86, 3906 ~1987!. 15H. Sumi, J. Phys. Chem. 95, 3334 ~1991!. 16T. Asano, H. Furuta, H.-J. Hofmann, R. Cimiraglia, Y. Tsuno, and M. Fujio, J. Org. Chem. 58, 4418 ~1993!, and earlier papers cited therein. 17T.Asano and T. Okada, J. Org. Chem. 51, 4454 ~1986!, and earlier papers cited therein. 18N. G. van Kampen, Stochastic Processes in Physics and Chemistry, Re- vised and Enlarged Edition ~North Holland, Amsterdam, 1992 !. 19R. Zwanzig, J. Stat. Phys. 9, 215 ~1973!. 20E. Cortes, B. J. West, and K. Lindenberg, J. Chem. Phys. 82, 2708 ~1985!. 21B. J. Gertner, K. R. Wilson, and J. T. Hynes, J. Chem. Phys. 90, 3537 ~1989!. 22J. S. McCaskill and R. G. Gilbert, Chem. Phys. 44, 389 ~1979!. 23D. H. Waldeck, W. T. Lothshaw, D. B. McDonald, and G. R. Fleming, Chem. Phys. Lett. 88, 297 ~1982!. 24H. Sumi and T. Asano, Chem. Phys. Lett. ~to be published !. 25J. Schroeder, J. Troe, and P. Vo ¨hringer, Chem. Phys. Lett. 203, 255 ~1993!; see also, J. Schroeder and J. Troe, in Reaction Dynamics in Clus- ters and Condensed Phases , edited by J. Jortner et al. ~Kluwer, Nether- lands, 1994 !, p. 361. 26F. J. Bermejo, F. Batallan, J. L. Martinez, M. Garcia-Hernandez, and E. Enciso, J. Phys. Condensed Matter 2, 6659 ~1990!; E. G. D. Cohen, P. Westerhuijs, and I.M. de Schepper, Phys. Rev. Lett. 59, 2872 ~1987!.9573 H. Sumi and T. Asano: Slow thermal isomerization in viscous solvents J. Chem. Phys., Vol. 102, No. 24, 22 June 1995 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.176.129.147 On: Sun, 07 Dec 2014 14:41:42
1.4939441.pdf	Control of magnetic relaxation by electric-field-induced ferroelectric phase transition and inhomogeneous domain switching Tianxiang Nan , Satoru Emori , Bin Peng , Xinjun Wang , Zhongqiang Hu , Li Xie , Yuan Gao , Hwaider Lin , Jie Jiao , Haosu Luo , David Budil , John G. Jones , Brandon M. Howe , Gail J. Brown , Ming Liu, , and Nian Sun, Citation: Appl. Phys. Lett. 108, 012406 (2016); doi: 10.1063/1.4939441 View online: http://dx.doi.org/10.1063/1.4939441 View Table of Contents: http://aip.scitation.org/toc/apl/108/1 Published by the American Institute of Physics Control of magnetic relaxation by electric-field-induced ferroelectric phase transition and inhomogeneous domain switching Tianxiang Nan,1Satoru Emori,1BinPeng,2Xinjun Wang,1Zhongqiang Hu,1LiXie,1 Yuan Gao,1Hwaider Lin,1JieJiao,3Haosu Luo,3David Budil,4John G. Jones,5 Brandon M. Howe,5Gail J. Brown,5Ming Liu,2,a)and Nian Sun1,b) 1Department of Electrical and Computer Engineering, Northeastern University, Boston, Massachusetts 02115, USA 2Electronic Materials Research Laboratory, Xi’an Jiaotong University, Xi’an 710049, China 3Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai 201800, China 4Department of Chemistry, Northeastern University, Boston, Massachusetts 02115, USA 5Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson AFB, Ohio 45433, USA (Received 7 September 2015; accepted 19 December 2015; published online 5 January 2016) Electric-ﬁeld modulation of magnetism in strain-mediated multiferroic heterostructures is considered a promising scheme for enabling memory and magnetic microwave devices with ultralow power consumption. However, it is not well understood how electric-ﬁeld-induced strain inﬂuences mag- netic relaxation, an important physical process for device applications. Here, we investigate resonantmagnetization dynamics in ferromagnet/ferroelectric multiferroic heterostructures, FeGaB/PMN-PT and NiFe/PMN-PT, in two distinct strain states provided by electric-ﬁeld-induced ferroelectric phase transition. The strain not only modiﬁes magnetic anisotropy but also magnetic relaxation. In FeGaB/PMN-PT, we observe a nearly two-fold change in intrinsic Gilbert damping by electric ﬁeld, which is attributed to strain-induced tuning of spin-orbit coupling. By contrast, a small but measurable change in extrinsic linewidth broadening is attributed to inhomogeneous ferroelastic domain switchingduring the phase transition of the PMN-PT substrate. VC2016 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4939441 ] Electrical manipulation of the magnetization state is essential for improving the scalability and power efﬁciency of magnetic random access memory (MRAM).1–5A particularly p r o m i s i n gs c h e m er e l i e so na ne l e c t r i cﬁ e l dt oa s s i s to ri n d u c e magnetization switching with minimal power dissipation.4,6,7 Multiferroic magnetoelectric materials with coupled magnet- ization and electric polarization offer possibilities for electric- ﬁeld-driven magnetization switching at room temperature.8–14 Such magnetoelectric effects have been demonstrated with strain-,15–19charge-,20–23and exchange bias mediated cou- pling mechanisms.24–27For example, non-volatile magnetiza- tion switching with remarkable modulation of magneticanisotropy was realized using electric-ﬁeld-induced piezo- strain at the interface between ferromagnetic and ferroelectric phases. 28–31 On the other hand, a better understanding of the processes responsible for magnetic relaxation, especially at variousstrain states, is required for electric-ﬁeld-assisted MRAM or tunable magnetic microwave devices. Recent studies suggest that electric-ﬁeld-induced changes of magnetic relaxation arecorrelated to the piezo-strain state or effective magnetic ani- sotropy. 32–36A similar modulation of magnetic relaxation has also been observed in a charge-mediated magnetoelectric het-erostructure with ultra-thin ferromagnets. 37In general, the contributions to magnetic relaxation include intrinsic Gilbert damping due to spin-orbit coupling and extrinsic linewidthbroadening due to inhomogeneity in the ferromagnet. So far,the understanding of how a piezo-strain modiﬁes these intrin- sic and extrinsic contributions has been limited. 33 In this work, we quantify electric-ﬁeld-induced modiﬁ- cations of both intrinsic Gilbert damping and inhomogeneouslinewidth broadening in two ferromagnet/ferroelectric heter-ostructures: Fe 7Ga2B1/Pb(Mg 1=3Nb2=3)O3-PbTiO 3(FeGaB/ PMN-PT) with a strong strain-mediated magnetoelectric(magnetostrictive) coupling and Ni 80Fe20/Pb(Mg 1=3Nb2=3)O3- PbTiO 3with a negligible magnetoelectric coupling. The rhombohedral (011) oriented PMN-PT substrate provides twodistinct strain states through an electric-ﬁeld-induced phasetransformation. 38,39We conduct ferromagnetic resonance (FMR) measurements at several applied electric ﬁeld values todisentangle the intrinsic and extr insic contributions to magnetic relaxation. FeGaB/PMN-PT exhib its pronounced electric-ﬁeld- induced modiﬁcations of the resonance ﬁeld and intrinsicGilbert damping, whereas these parameters remain mostlyunchanged for NiFe/PMN-PT. These ﬁndings show that mag-netic relaxation can be tuned through a strain-mediated modiﬁ-cation of spin-orbit coupling in a highly magnetostrictiveferromagnet. We also observe in both multiferroic heterostruc- tures a small electric-ﬁeld-in duced change in extrinsic line- width broadening, which we attribute to the ferroelectric domain state in the PMN-PT substrate. 30-nm thick ﬁlms of FeGaB and NiFe were sputter- deposited on (011) oriented PMN-PT single crystal sub-strates buffered with 5-nm thick Ta seed layers. The FeGaBthin ﬁlm was co-sputtered from Fe 80Ga20(DC sputtered) and B (RF sputtered) targets. Both FeGaB and NiFe ﬁlms werecapped with 2 nm of Al to prevent oxidation. All ﬁlms werea)Electronic mail: mingliu@mail.xjtu.edu.cn b)Electronic mail: n.sun@neu.edu 0003-6951/2016/108(1)/012406/5/$30.00 VC2016 AIP Publishing LLC 108, 012406-1APPLIED PHYSICS LETTERS 108, 012406 (2016) deposited in 3 mTorr Ar atmosphere with a base pressure of /C201/C210/C07Torr. The thicknesses of deposited ﬁlms were calibrated by X-ray reﬂectivity. The magnetic hysteresis loop measurements were carried out by using a vibrating sample magnetometry (Lakeshore 7400) at different electricﬁelds. FMR spectra were measured using a Bruker EMX EPR spectrometer with a TE 102cavity operated at a micro- wave frequency of 9.5 GHz. Gilbert damping constant andinhomogeneous linewidth were carried out using a home- built broadband FMR system. The polarization domain phase images with various applied voltages were measured by apiezo-force microscope. The amorphous FeGaB thin ﬁlm was selected for its high saturation magnetostriction coefﬁcient of up to 70 ppm(Ref. 40) and large magnetoelectric effect when interfaced with ferroelectric materials. 19NiFe was chosen as the control sample with near zero magnetostriction; the thickness of30 nm is far above the thickness regime that shows high sur- face magnetostricion. 41Fig. 1shows magnetic hysteresis loops of FeGaB/PMN-PT and NiFe/PMN-PT, measured byvibrating sample magnetometry with an in-plane magneticﬁeld applied along the [100] direction of PMN-PT. An elec- tric ﬁeld was applied in the thickness direction of the PMN-PT substrate. Due to the anisotropic piezoelectric coefﬁcient of PMN-PT, an in-plane compressive strain is induced along the [100] direction, which results in uniaxial magnetic ani-sotropy along the same axis. In FeGaB/PMN-PT, the electric ﬁeld ( E¼8 kV/cm) increases the saturation ﬁeld by /C2540 mT, whereas only a small change is observed in NiFe/PMN-PT,conﬁrming the signiﬁcantly different strengths of strain- mediated magnetoelectric coupling for the two multiferroic heterostructures. Both ferromagnetic thin ﬁlms exhibit comparatively nar- row resonant linewidths, allowing for sensitive detection of the electric-ﬁeld modiﬁcation of spin relaxation. Electric-ﬁelddependent FMR spectra of FeGaB/PMN-PT and NiFe/PMN- PT were measured using a Bruker EMX electron paramagnetic resonance (EPR) spectrometer with a TE 102cavity operated at a microwave frequency of 9.5 GHz. The external magnetic ﬁeld was applied along the [100] direction of the PMN-PT sin- gle crystal. These spectra, shown in Figs. 2(a)and2(b),w e r e ﬁtted to the derivative of a modiﬁed Lorentzian function42to FIG. 2. (a) and (b) FMR (ﬁxed at 9.5 GHz) spectra at various electric ﬁelds with the magnetic ﬁeld appliedalong the [100] direction for FeGaB/ PMN-PT (a) and NiFe/PMN-PT (b). (c) and (d) Resonance ﬁeld HFMR as a function of the applied electric ﬁeld for FeGaB/PMN-PT (c) and NiFe/PMN-PT (d). Inset of (c) shows the piezo-strain as a function of electric ﬁeld for PMN-PT substrate along the [100] direction.FIG. 1. (a) and (b) Electric-ﬁeld de-pendent magnetic hysteresis loops with the magnetic ﬁeld applied along the [100] direction for FeGaB/PMN-PT (a) and NiFe/PMN-PT (b).012406-2 Nan et al. Appl. Phys. Lett. 108, 012406 (2016) extract the resonance ﬁeld HFMRand resonance linewidth W. In FeGaB/PMN-PT, upon applying E¼2k V / c m a l o n g t h e thickness direction of PMN-PT, a slight increase of HFMRby 10 mT is observed. A larger shift of 35 mT in HFMRis induced atE¼8 kV/cm. In comparison, NiFe/PMN-PT exhibits a much smaller HFMRshift of 1.5 mT at E¼8k V / c m ,a s s h o w n in Fig. 2(b). The shift of HFMRin FeGaB/PMN-PT and NiFe/PMN- PT as a function of E is summarized in Figs. 2(c) and2(d). Both samples show hysteric behavior that follows thepiezo-strain curve of PMN-PT (inset of Fig. 2(c)) measured with a photonic sensor. This can be understood by thestrain-mediated magnetoelectric coupling with the electric-ﬁeld-induced change of magnetic anisotropy ﬁeld ( DH k) expressed by DHk¼3kr 100/C0r0/C011 ðÞ l0Ms; (1) where r100and r0/C011are the in-plane piezo-stress and k andMsare the magnetostriction constant and the saturation magnetization, respectively. Considering an in-plane com- pressive strain along the [100] direction and a positive mag-netostriction coefﬁcient of both FeGaB and NiFe, a decreaseof the magnetic anisotropy ﬁeld H kis expected with a posi- tive electric ﬁeld. The drop of Hkresults in an increase of HFMRdescribed by the Kittel equation f¼c 2pl0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HFMRþHk ðÞ HFMRþHkþMef f ðÞq ; (2) where c=2p¼28 GHz =Ta n d Meffis the effective magnetiza- tion. At E<4 kV/cm, HFMRincreases linearly, which corre- sponds to the linear region of piezoelectric effect of PMN-PTwith a uniaxial compressive piezo-strain along [100] direction.The sudden change of H FMR atE¼4 kV/cm is attributed to the rhombohedral-to-orthorhombic (R-O) phase transitionof PMN-PT substrate. 39The PMN-PT substrate reverts to the rhombohedral phase upon decreasing the electric ﬁeld.Therefore, the R-O phase transformation with a large uniaxialin-plane strain induces two stable and reversible magneticstates at E¼0 and 8 kV/cm. This provides a reliable platform for studying magnetization dynamics in a controlled manner with the applied electric ﬁeld. The peak-to-peak FMR linewidth Wof FeGaB/PMN-PT and NiFe/PMN-PT, extracted from the same FMR measure-ments in Fig. 2, also exhibits a strong dependence on theapplied electric ﬁeld as shown in Fig. 3. For FeGaB/PMN- PT,Wremains unchanged within experimental uncertainty atE<4 kV/cm and abruptly increases from /C254.6 mT to /C255.6 mT across the R-O phase transition. By removing the applied electric ﬁeld, Wdecreases to the original value with the reversal to the rhombohedral phase. Comparing Figs. 2(c) and3(a), it is evident that the observed electric-ﬁeld- induced changes in H FMRandWin FeGaB/PMN-PT are cor- related, consistent with the recent studies.34–36The change in Windicates a modulation in spin-orbit coupling in the ferro- magnet; considering that spin-orbit coupling governs theintrinsic Gilbert damping, it is reasonable that we observesimultaneous modiﬁcation of WandH FMR by strain in the magnetostrictive FeGaB ﬁlm. Given the same sign of the magne tostriction coefﬁcient for FeGaB and NiFe,40,43one would expect to also observe a small increase in Wwith increasing electric ﬁeld across the R-O phase transition in NiFe/PMN-PT. However, NiFe/PMN-PT exhibits a decrease in Wacross the phase transition. This obser- vation indicates that the piezo-strain modiﬁes a different mag-netic relaxation contribution in NiFe. The FMR linewidth Wconsists of the intrinsic Gilbert damping contribution (parameterized by the damping con-stant a) and the frequency-independent inhomogeneous line- width broadening W 0 W¼W0þ4paﬃﬃﬃ 3p cf; (3) where fis the microwave excitation frequency. According to Eq.(3),aandW0can be determined simply by measuring the frequency dependence of W. For this purpose, we used a home-built broadband FMR system44with a nominal micro- wave power of /C05 dBm and f¼6–19 GHz. Just as in the single-frequency measurement using the EPR system (Figs. 2 and3), the external magnetic ﬁeld was applied along the [100] direction of the PMN-PT substrate. By ﬁtting the fre- quency dependence of HFMRto Eq. (1)(Figs. 4(a)and4(b)), we obtain anisotropy ﬁeld shift Dl0Hk/C2546 mT for FeGaB/ PMN-PT and Dl0Hk/C251 mT for NiFe/PMN-PT across the R- O phase transition, in agreement with the single-frequency FMR measurement (Fig. 2), while l0Mef fremains unchanged. Figs. 4(c)and4(d) plotWas a function of the frequency for FeGaB/PMN-PT and NiFe/PMN-PT, respectively. From the slope of the linear ﬁt (Eq. (3)), we ﬁnd that aof FeGaB/ PMN-PT increases from ð0:660:01Þ/C210/C02atE¼0t o FIG. 3. (a) and (b) Resonance line- width W at 9.5 GHz with the magnetic ﬁeld applied along the [100] direction as a function of the applied electric ﬁeld for FeGaB/PMN-PT (a) and NiFe/PMN-PT (b).012406-3 Nan et al. Appl. Phys. Lett. 108, 012406 (2016) ð1:0660:02Þ/C210/C02atE¼8 kV/cm, whereas ais unchanged atð1:2960:16Þ/C210/C02for NiFe/PMN-PT within experimen- tal uncertainty( a¼ð1:2760:2Þ/C210/C02atE¼8 kV/cm). The large change in afor FeGaB and negligible change for NiFe suggest a strong correlation between magnetostriction and theintrinsic Gilbert damping mechanism. In particular, a large in- plane uniaxial strain generated by the R-O phase transforma- tion induces an additional anisotropy ﬁeld in FeGaB that enhances the dephasing of the magnetization precession. 43 However, both FeGaB/PMN-PT and NiFe/PMN-PT show a decreased W0upon applying E¼8 kV/cm. This could be related to the ferroelectric domain state in the PMN-PT substrate that signiﬁcantly affects the homogeneity of the magnetic ﬁlm on top. The polarization domain phase images with various applied voltages are shown in Fig. 5by using a piezo-force microscope. For the unpoled state, as shown in Fig.5(a), the polarization state of PMN-PT surface is inho- mogenous, with polarization vectors oriented randomly along the eight body diagonals of the pseudocubic cell. By applying a voltage of 30 V within the gated area (dashed out-line in Figs. 5(b) and5(d)), the ferroelectric state becomes saturated within this area with all the polarization vectors pointing upward. This uniformly polarized state alters thesurface topology the PMN-PT substrate, 31thereby reducing the inhomogeneous linewidth broadening W0of the ferro- magnetic ﬁlm. We also measured frequency-dependent FMR spectra with an external magnetic ﬁeld applied along the ½0/C2211/C138direc- tion to examine the anisotropy of magnetic relaxation. For FeGaB/PMN-PT, aandW0are close to the [100] conﬁgura- tion at E¼0. At E¼8 kV/cm, we observed a non-linear rela- tion between Wand f, which might have resulted from a highly non-uniform magnetization state at low ﬁelds due tothe large electric-ﬁeld-induced Hk.45,46To extract areliably in this case, we would need to conduct FMR measurementsat higher frequencies. For NiFe/PMN-PT, aand the electric- ﬁeld dependence of W 0are identical for the ½0/C2211/C138and [100] directions. The parameters quantiﬁed in this study are sum-marized in Table I. In summary, we have quantiﬁed electric-ﬁeld-induced modiﬁcations of magnetic anisotropy and magnetic relaxa-tion contributions, namely, intrinsic Gilbert damping and in-homogeneous linewidth broadening, in multiferroicheterostructures. A large modiﬁcation of intrinsic dampingFIG. 4. (a) and (b) Frequency f as a function of resonance ﬁeld HFMR at different electric ﬁelds for FeGaB/ PMN-PT (a) and NiFe/PMN-PT (b). (c) and (d) Linewidth W as a function of frequency f at different electric ﬁelds for FeGaB/PMN-PT (c) and NiFe/PMN-PT (d). The magnetic ﬁeldwas applied along the [100] direction. FIG. 5. (a) and (b) The out-of-plane vertical PFM (VPFM) phase imagesupon applying different voltages to the square area outlined by a red dashed line. (c) and (d) Corresponding amplitude images at different voltage biases.012406-4 Nan et al. Appl. Phys. Lett. 108, 012406 (2016) arises from strain-induced tuning of spin-orbit coupling in the ferromagnet and is correlated with the magnitude of mag-netostriction. A small change in the extrinsic linewidth con-tribution is attained by controlling the ferroelectric domainstates in the substrate. These ﬁndings are not only of technol-ogy importance for the application on low-power MRAMand magnetic microwave devices but also permit investiga-tion of the structural dependence of spin-orbit-derived phe-nomena in magnetic thin ﬁlms. 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FeGaB/PMN-PT NiFe/PMN-PT E(kV/cm) 0 8 0 8 l0Mef fðTÞ 1.4860.01 1.46 60.01 0.96 60.04 0.96 60.04 l0HkðmTÞ½100/C138 5.860.5 /C041:360:3 1.67 60.2 0.27 60.2 að10/C02Þ½100/C138 0.660.01 1.06 60.02 1.29 60.16 1.27 60.2 W0ðmTÞ½100/C138 2.460.05 1.8 60.07 0.66 60.06 0.35 60.07 l0HkðmTÞ½0/C2211/C138 3.2460.4a1.5460.2 3.1 60.3 að10/C02Þ½0/C2211/C138 0.660.02 1.21 60.12 1.29 60.15 W0ðmTÞ½0/C2211/C138 2.860.05 5.9 60.08 2.9 60.05 aThe values were not obtained due to the frequency constraint and the ﬁeld- dragging effect at measured low frequencies.012406-5 Nan et al. Appl. Phys. Lett. 108, 012406 (2016)
1.4938549.pdf	Modeling of hysteresis loops by Monte Carlo simulation Z. Nehme , Y. Labaye, , R. Sayed Hassan , N. Yaacoub , and J. M. Greneche Citation: AIP Advances 5, 127124 (2015); doi: 10.1063/1.4938549 View online: http://dx.doi.org/10.1063/1.4938549 View Table of Contents: http://aip.scitation.org/toc/adv/5/12 Published by the American Institute of PhysicsAIP ADV ANCES 5, 127124 (2015) Modeling of hysteresis loops by Monte Carlo simulation Z. Nehme,1Y. Labaye,1,aR. Sayed Hassan,1,2N. Yaacoub,1 and J. M. Greneche1 1Université du Maine, Institut des Molécules et Matériaux du Mans, IMMM, UMR CNRS 6283, F-72085, Le Mans, France 2MDPL, Université Libanaise, Faculté des Sciences Section I, Beyrouth, Liban (Received 21 October 2015; accepted 9 December 2015; published online 18 December 2015) Recent advances in MC simulations of magnetic properties are rather devoted to non-interacting systems or ultrafast phenomena, while the modeling of quasi-static hysteresis loops of an assembly of spins with strong internal exchange interactions remains limited to specific cases. In the case of any assembly of magnetic moments, we propose MC simulations on the basis of a three dimensional classical Heisenberg model applied to an isolated magnetic slab involving first nearest neighbors exchange interactions and uniaxial anisotropy. Three di fferent algorithms were successively implemented in order to simulate hysteresis loops: the classical free algorithm, the cone algorithm and a mixed one consisting of adding some global rotations. We focus particularly our study on the impact of varying the anisotropic constant parameter on the coercive field for di fferent temperatures and algorithms. A study of the angular acceptation move distribution allows the dynamics of our simulations to be charac- terized. The results reveal that the coercive field is linearly related to the anisotropy providing that the algorithm and the numeric conditions are carefully chosen. In a general tendency, it is found that the e fficiency of the simulation can be greatly enhanced by using the mixed algorithm that mimic the physics of collective behavior. Consequently, this study lead as to better quantified coercive fields measurements resulting from physical phenomena of complex magnetic (nano)architectures with different anisotropy contributions. C2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http: //dx.doi.org /10.1063 /1.4938549] I. INTRODUCTION During the last decades, the study of magnetic nanostructures including as-prepared and func- tionalized nanoparticles, multilayers and nanostructured powders has been the center of interest among the scientific community because of their increasing number of applications in various fields including catalysis, biotechnology,1biomedicine (Magnetic Resonance Imaging MRI2), recording media.3,4The physical properties of these nanostructures are strongly a ffected by the enhanced role of surfaces, symmetry breaking and finite size, etc. In addition to experimental approach, numerical methods become an alternative route to study and to understand the influence of di fferent parame- ters on the structural and magnetic properties of the nanostructures. The Monte Carlo (MC) method seems to be a suitable numeric tool that allows a detailed microscopic description of an assembly of spins which does correspond to a magnetic material.5In the MC simulation, the procedure consists in first creating a model system and then modifying it through a variety of states randomly chosen from initial thermodynamic conditions in order to reach the equilibrium state. The physical quantities such as magnetization, susceptibility, magnetic energy, are estimated by averaging those thermodynamic quantities over a sequence of states accepted according to the Boltzmann distri- bution. The interest of the MC method is that it allows detailed interactions of systems and can allow local homogeneities (frustration, non-collinear behavior) to be treated. As a second positive aCorrespondence should be addressed to Y . Labaye (yvan.labaye@univ-lemans.fr). 2158-3226/2015/5(12)/127124/11 5, 127124-1 ©Author(s) 2015. 127124-2 Nehme et al. AIP Advances 5, 127124 (2015) advantage, it gives rise to the thermal dependence of the magnetic properties of the system (critical temperature, temperature dependence of magnetization).5,6Some properties as thermal activation and switching behavior (from coherent rotation to nucleation) in systems with continuous degree of freedom have been yet successfully investigated using the MC method.7,8 It is important to emphasize that the hysteresis loops have a major role in the understanding of magnetic properties, the main characteristics of which is coercive field H C. It comes in the study of various physical phenomena such magnetic anisotropy and exchange bias (EB) coupling.9Indeed, it is well established that the EB coupling induces an additional anisotropy in some materials. The main indicator of the EB is the shift of the hysteresis loop along the field axis and an increase of the coercive field H C(probably in most nanostructures) after a field cooling procedure.10–13However, as the hysteresis loop describes a series of metastable states, the MC method is not a priori the best route to simulate it since it provides a way to find the equilibrium state of the system even if this equilibrium is reached very slowly. The real system is generally blocked in a metastable region where it may remain for a long time in a local energy minimum. The energy barrier between this metastable state and the equilibrium state can be overwhelmed by the temperature e ffect and /or the presence of an external magnetic field. Such features are, for example, observable in the case of ultrafine parti- cles where the magnetization can fluctuate spontaneously from one direction to the opposite one at temperature well below the critical temperature T c. This phenomenon known as superparamagnetism can be modeled by MC, following the magnetization versus the simulation “time” (i.e. the Monte Carlo Steps MCS, if the system contains N sspins, one MCS corresponds to N stries to change one spin). But one should emphasize that algorithmic conditions strongly influence the characteristics of the computed hysteresis loop. In addition to intrinsic physical parameters of the magnetic material, the computed value of the coercive field is also a ffected by the simulation conditions: essentially the algorithm and the field scan speed (FSS).14–17Consequently, it is necessary to discriminate the e ffects due to the simulation conditions from those attributed to the physical parameters. The classical MC method is a stochastic process having no physical time associated with each MCS, thus the obtained dynamics depends on the algorithm. The detailed balance of the MC procedure says that, at equilibrium, the transition probability between states is independent of time (stationary state). Thus, the true time evolution of the system does not appear in any of the used equations.18However, a significant contribution involving the Time Quantify Monte-Carlo (TQMC) procedure was proposed by Nowak et al. 12 years ago.19This algorithm resembles the classical one except in the way of choosing a new configuration from the initial one. The new state has to be chosen within a cone around the initial spin direction. The time associated with one MCS is then related to a real time and is proportional to R2, where R is the reduced cone radius. This cone radius is expected to be ≪1 in order to fulfill the validity of the approximations made to establish the relation between real time and R2. The TQMC results were successfully compared to those obtained from Landau–Lifshitz–Gilbert (LLG) dynamic calculations of switching times in the case of non-interacting assembly of spins20,21and spin waves at finite temperatures. Extension to wider range of damping factor was proposed in Refs. 22 and 23 by introducing a precessional term in the choice of new spin state to be applied to 2D arrays of interacting spins of max size 40x40 as well as to spin waves on linear chain of spins. In order to apply the TQMC to simulate superparamagnetic phenomena of non-interacting magnetic moments with interesting anisotropy values (experimental ones), Melenev et al.24propose first to modify the relation (R2∝∆t) established by Nowak et al. in Ref. 19; then they successively apply their model to simulate dynamic hysteresis loops on a randomly oriented ensemble of 106moments. But all those calculations are devoted to ultrafast (ns) phenomena and can hardly be applied to quasi-static or static cases. The idea to perform MC quasi-static calculations of hysteresis loops was proposed in Ref. 25. Indeed, it results in the analysis of each energy landscape of the possible orientation change of mag- netic moment belonging to a randomly oriented anisotropic assembly of non-interacting moments. The flipping probability of one magnetic moment depends on the location of both the initial and final states in the energy landscape as well as the value of the saddle point energy (assuming that only two minima in the energy landscape are present). One does conclude that the simulation of hysteresis loops in the case of interacting spins remains somehow more di fficult by means of the methods listed above. The di fferent reasons are the following:127124-3 Nehme et al. AIP Advances 5, 127124 (2015) - the first one is that the collective behavior of an assembly of interacting moments is not taken into account (global rotation of the whole system with e ffective field) rather than by a huge number of MCS in the case of single spin flip algorithm. The energy barrier seen by a magnetic moment in- teracting with neighbors during single spin changes can be very high compared to k BT (high modi- fication of exchange energy with small acceptation probability); but it can be very small or negative if a global rotation is performed (no exchange energy needed in case of isotropic exchange). At low temperature the critical slowing down will also lower this acceptation probability; - the second constraint comes from the real time a ffected to 1 MCS (order of 10−12s) which is too small to perform quasi-static simulations; - the third problem is concerned by the idea of Du and Du25where a breakdown occurs if more than two minima and /or saddle points are found in the energy landscape. The increased complexity of such algorithm with number of minima /saddle points seems to be tricky to implement; in addition, such an algorithm is not relevant for an assembly of magnetic moments interacting by exchange. In this context, our objective is to investigate the di fferent algorithms available in order to find the best compromise between physically sound results and e fficiency for simulations of hysteresis loops. To perform such simulation one has first to choose dynamics, i.e. a rule for changing one magnetic state into another. The possible changes used in modeling spin systems can be divided into two main algorithms: single spin flip (SSF) and cluster spin flip (CSF) algorithms. SSF algorithm consists in selecting randomly a new state for a single randomly selected spin. This new state can be either restricted to a part of the space around the previous spin direction (cone algorithms) or free to explore the whole space (free algorithm). Both have the advantage to be able to keep metastable states but the last one shows poor e fficiency to reproduce the hysteresis loop. The main explanation for this poor e fficiency is that, in the SSF method, the collective change of the magnetization with applied external field is not assumed and can be reproduced only by a large, nearly uncorrelated, number of trial moves. CSF algorithm (Swendsen-Wang algorithm or Wol ffalgorithm)26–28includes by essence those collective behaviors preventing the metastable state to be reached. Those collective changes algo- rithms simply wipe out metastable states and consequently are not a good choice to simulate hysteresis loops. This fact justifies why the SSF algorithm, despite its low e fficiency, should be used to reproduce hysteresis loop. Consequently, the present computer modeling of hysteresis loop will be based on the SSF algorithm but the validity of such an approach has to be carefully checked in order to be applied for finite temperatures. At zero temperature and for a coherent rotation reversal mechanism (the whole system be- haves as a macro-spin), the magnetic field at which the reversal occurs is well described by the Stoner-Wohlfarth model. It suggests that the relation between reversal field H rand e ffective anisot- ropy constant K effis given by H r=2 K eff/Ms, where M sis the saturation magnetization. But a more complex relationship could link H rto K effwhen the above conditions are not fulfilled (non-coherent reversal, finite temperature, surface e ffects). For example, the e ffect of surface anisotropy on hyster- esis loops of ferromagnetic particles was performed by MC simulations. It results a non-trivial relation between the coercive field and surface anisotropy e ffects. Simulating hysteresis loop by Monte Carlo at a given temperature consists in spending N (MCS) (number of MCS) at a fixed field and then increasing (or decreasing) this field by an increment ∆H and repeating those two steps for fields ranged in the scanning window [+Hmax,−Hmax,+Hmax]. The results obtained are strongly dependent on field scan speed (FSS) defined as FSS =∆H N(MCS ). It is important to note that a fixed FSS can be obtained by changing ∆H and N (MCS) by the same factor and gives rise to the same hysteresis loop. As in real systems, the coercive field depends strongly on the speed at which the field is swept as illustrated in Fig. 1(a). Such results are perfectly consistent with previous comments. An example of the estimated H cfor di fferent N (MCS), is illustrated in Fig. 1(b) on a system without anisotropy, H cshould be zero. The power law plotted in Fig. 1(c) shows that infinitely slow sweeping field has to be used to reach the zero theoretical value of H c.127124-4 Nehme et al. AIP Advances 5, 127124 (2015) FIG. 1. (a) Average (20 cycles) hysteresis loops calculated with Ku=0 and T/Tc=0.007 at di fferent N (MCS) using the cone algorithm, (b) variation of Hc versus N (MCS) and (c) log plot of Hc versus N (MCS). In the following calculations, we will use quite low N (MCS) (faster calculations) leading to overestimate H cvalues but leading to correct behavior. The aim of the present study is to understand the impact of the simulation dynamics on the estimation of the coercive field and try to figure out its dependence on the e ffective anisotropic constant. This could be applied to the study of induced anisotropies such as those coming from exchange bias coupling e ffects. Understanding the impact of the e ffective anisotropy on the hysteresis loop can allow to predict variation of the coercive field when the exchange bias coupling constants are modified. II. MODELANDSIMULATION The simulated structure consists of a ferromagnet composed of 2160 atoms forming a paral- lelepiped box, where the [111] crystallographic direction of the BCC cell is pointing in the OZ direction of the simulation box. The limits of the box are ±5√ 2,±3√ 6 and±3√ 3 along x, y and z directions, respectively. The implemented model is based on three-dimensional classical Heisenberg Hamiltonian: H=−1 2 <i,j>Jij⃗Si.⃗Sj−CH∗⃗Hext∗ i⃗Si+KU i(1−(⃗ui⃗Si Si)2) Here, a classical vector spin Si=1µBis associated for each atom i. The first term is the exchange energy between spins ⃗Siand⃗Sj, Jijis the coupling constant while index j is related to the first nearest magnetic neighbors of atom i. The second term corresponds to the Zeeman energy, C His a unity conversion factor equal to (µB kB0.672) used to insure that the energy is given in Kelvin when ⃗Hext, the external applied magnetic field, is given in Tesla. The third term deals with the uniaxial volume anisotropy energy with K Uthe effective anisotropy constant, where ⃗uiis a unit vector along the easy127124-5 Nehme et al. AIP Advances 5, 127124 (2015) magnetization axis. For the present work, neither internal dipolar interactions nor surface anisotropy are taken into account since internal dipolar interactions of a 2160 µBisolated system are negligible. We performed our simulation using Metropolis Monte Carlo algorithm with periodic bound- ary conditions along x, y and z directions to avoid surface e ffects. Temperature T and exchange coupling constant J ijare expressed in Kelvin per square Bohr magneton, field H extis in Tesla, the magnetic moments are in µBand the anisotropic constant K Uis in K /atom. The exchange coupling constant and the temperature are normalized to the transition temperature which is about 700 K. In our simulations, J /Tcis equal to 0.65. The anisotropy is supposed to be uniaxial along the OZ direction considered as the magnetization easy axis. The anisotropic constant varies as follows: KU={0.024; 1; 5; 10; 15; 20 K /atom}. Our numerical process starts from a random spin configuration at high temperature T i(above Tc). The temperature of the system is slowly cooled obeying the law T N=αNTiwhere αis the lowering coe fficient and N the number of temperature steps. The initial temperature is T i/Tc=2.14 withα=0.97 and N =380 to reach a final temperature around T f. We use 5000 MCS for each temperature step and apply a small external magnetic field along the anisotropy direction to ensure that the simulation parameters used converge to the lowest attainable energy state. The resulting spin configuration is then used as an input for the hysteresis loop calculations. The external field will be swept in the anisotropic direction (i.e. Z) and at fixed T =Tfof the equilibrium state calculated before. The hysteresis loop will be recorded at a fix Field Scan Speed and averaged over 20 cycles. The value of the coercive field is defined as H c=H+c−H−c 2where H+ cand H− care the points corresponding to the intersection of the hysteresis loop with the field axis. III. RESULTSANDDISCUSSION A. The“freealgorithm” As mentioned below, the “free algorithm” (FA) is a sampling method of the Single Spin Flip (SSF) dynamics where the new direction of the spin is randomly chosen without any space constraint. In order to check the dependence of the coercive field on the e ffective anisotropic constant, hysteresis loops were collected after changing the value of K Uunder a maximum applied magnetic field of 300 T. This unrealistic huge field is necessary to close the cycle due to the low (3000) MCS value used for each field step. The study was done at di fferent temperatures T /Tcranged from 0.007 to 0.43. Fig. 2(a) and 2(c) illustrates the simulated hysteresis loops at T /Tc=0.007 and T/Tc=0.43. For each case, we plotted the variation of H cas function of K U: a linear dependence can be thus expressed as H c=H0 c+αH∗KUwhere H0 cis the measured coercive field when K U =0 K/atom and αHthe slope. The di fferent tests show that αHdepends slightly on the temperature, whereas H0 cis strongly a ffected by the temperature. According to the Stoner-Wohlfarth model at T=0 K, the reversal field should be proportional to the e ffective anisotropy with a slope equal to 2 (if Ms =1). The slope measured at T /Tc=0.007 is found to be 1.92, as shown in Fig. 2(b), close to theoretical value of 2. For non-zero temperatures, the slope should decrease due to thermal e ffect facilitating thus the crossing of the energy barrier. We found such decrease for T /Tc=0.43 where αHis 1.05 as shown in Fig. 2(d). At fixed anisotropic constant value the coercive field decreases with temperature as expected as seen in Fig. 3. It is important to emphasize that η, the acceptance rate, defined as η=n∗100 N(MCS )∗Nswith n the number of accepted flips and Ns the total number of sites, remains rather small at low temper- ature and does not exceed 12 % at T /Tc=0.43 considered as high temperature. A large part of the simulation time is wasted in considering configurations energetically improbable and conse- quently rejecting most of them. According to the Metropolis algorithm, the probability of accepting new configurations becomes very small at low temperatures. Therefore, the FA is ine fficient at low temperatures and gives rise to non realistic values of coercive fields due to under-sampling conditions.127124-6 Nehme et al. AIP Advances 5, 127124 (2015) FIG. 2. Hysteresis loops at T/Tc=0.007 (a) and T/Tc=0.43 (c) for di fferent anisotropy constants. Relation between the coercive field and the anisotropic constant at T/Tc=0.007 (b) and T/Tc=0.43 (d). The di fferent curves were obtained by the “FA”. B. TheAngularAcceptanceDistribution As it is well established that the physical time is not included in classical MC simulation, the present dynamics is mainly due to the algorithm used. So, we focused our interest in characterizing the dynamics originating from our “free algorithm”. For this purpose, it is necessary to study the probability of accepting new configurations P( θ) as a function of θ, the angle between initial and accepted new spin direction. The angular acceptance distribution is defined as P (θ)=Nacc(θ−δθ 2,θ+δθ 2) MCS∗NSwhere N accrepresents the number of accepted flips in the interval [θ−δθ 2,θ+δθ 2]. P(θ) allows to visualize the θ-space FIG. 3. Variation of the coercive field as a function of temperature resulting from the “FA” where KU=0.024 K /atom.127124-7 Nehme et al. AIP Advances 5, 127124 (2015) FIG. 4. Variation of probability distribution P (θ)as a function of the accepted angle at T/Tc=0.007 (a) and T/Tc=0.43 (b) for KU=20 K/atom by the “FA” with 3000 MCS. region where our sampling gives non zero acceptance. It should be noted that the following proba- bility distributions result from an average over all measured distribution probabilities obtained for each field step of the hysteresis loop. Fig. 4 shows how probability distribution P( θ) drops down to 0 for an angle close to 20◦at T/Tc=0.007 (a), whereas it vanishes at a value close to 150◦at T/Tc=0.43 (b). The low temper- ature P( θ) shape allows the small acceptance rate of the ”free algorithm” to be explained : we are wasting time in scanning configuration space between 0◦and 180◦, while the dynamics is restricted to much smaller angles contrarily to high temperature. C. The“conealgorithm” In order to accelerate the dynamics and to increase the acceptance rate of simulations, we modi- fied the sampling method of the configuration space by implementing a restricted algorithm: the “cone algorithm” (CA). This algorithm generates SSF dynamics within a fixed cone radius R cone. A new state is obtained by adding a random vector of norm R coneto the initial direction of the spin and then normalize. It is important to note that the cone radius is normalized to the spin modulus and is dimensionless. R conevalue has a strong influence on the generated dynamics as previously proved in TQMC method.19 The cone radius is related to a maximum angle θmaxwhich has to be chosen carefully in order to provide a good sampling of the phase space. Consequently, the angle where the P (θ)freevanishes is the optimum value of θmax. According to results reported in Fig. 4, at T /Tc=0.007, the optimum value of R conemust lead to θmaxaround 20◦, giving R cone=0.4. Therefore, we run a simulation with a R cone value of 0.4 and record the P (θ)restricted in order to check how our configuration space will be dependent127124-8 Nehme et al. AIP Advances 5, 127124 (2015) FIG. 5. Variation of probability distribution P (θ)as a function of the accepted angle at T/Tc=0.007 for Rcone=0.4 and KU=20 K/atom. on those sampling conditions. The results are reported in Fig. 5: one observes that P (θ)restricted leads to the same shape as P (θ)freebut an enhancement of the acceptance rate is well achieved. The study is focused on the influence of the anisotropy constant variation on the coercive field. The calculations were performed at di fferent values of K Uand for di fferent temperatures. The FSS was kept constant (2 .10−3T/MCS) and equal to that used in “FA” calculations by applying a maximum field of about 200 T. Indeed, this field is necessary to close the cycle due to the rather low number of MCS used for each field step (here 2000 MCS). For all temperatures studied, a linear dependence of the coercive field with the anisotropy constant was observed. An illustrative case is given in Fig. 6 where the hysteresis loop (a) and the linear relationship (b) obtained by the cone algorithm with a cone radius R coneof 0.4 at T/Tc=0.007 for di fferent K Uis shown. It should be noted that the values of H care smaller than those of the FA, but still unrealistic (high FSS). At low temperature, the CA is more e fficient than the FA as mentioned below. The determination ofθmaxwith a FA is needed to check the validity of the calculation and this for the temperature used for the simulation. It is important to emphasize that certain choice of θmax(i.e R cone) can lead to wrong results. For example, a fixed R coneof 0.2 was used for simulations of the same system in a range of T/Tcfrom 0.007 to 0.43. The dependence of H cwith K Uwas always linear but the variation of H c versus T for a fixed value of K Uwas not physically realistic as shown in Fig. 7. The explanation of this effect is provided by the TQMC relation R2∝T.∆t. It means that when we want to compare dynamic properties the rule cited above has to be respected (i.e. R2has to be changed proportionally to T). This algorithm can speed up calculations and gives better results only if parameters are chosen with great care. FIG. 6. Hysteresis loops for di fferent values of KU (a) and the relation between the coercive field and the anisotropy constant (b) at T/Tc=0.007. The di fferent curves were obtained with the ”CA” for Rcone=0.4.127124-9 Nehme et al. AIP Advances 5, 127124 (2015) FIG. 7. Relation between the coercive field and the anisotropy constant at T/Tc=0.43 obtained with the “CA” (a) and variation of the coercive field as a function of temperature (b) for Rcone=0.2 and KU=0.024 K /atom. D. The“GlobalRotationConeAlgorithm” For low temperatures it is well established that SSF algorithms are of rather poor e fficiency. When applying a magnetic field to the system this e fficiency drops down dramatically. This is due to the fact that global rotation of the magnetization is only reproduced when a very large number of SSF is achieved. The case of spin wave calculation with TQMC shows the ability of the algo- rithm to treat collective behavior. In order to increase the e fficiency of the procedure one can think about introducing an amount of global rotation to the system. This algorithm Global Rotation Cone Algorithm (GRCA) will then be a mixture of restricted SSF moves and a small amount of restricted global rotation. The global rotation had to be chosen using a cone algorithm method. As mentioned in previous sections, the P (θ)has to be computed in order to reduce the angular domain for the CA trial. For the global rotation step a cone radius RGRCA cone has to be chosen. This cone radius value is a ffected by the angular distance between the current global magnetization direction of the system and that corresponding to the minimum of its total energy. When performing hysteresis loops, the magnetic field usually increases by small steps leading to a narrow global P( θ) distribution. In this case, a small cone radius for the GRCA has to be chosen. A random vector of norm R coneis added to the unitary vector in the direction of the mean global magnetization ( ⃗Ui) and then normalized. This procedure defines a new direction in space ⃗Ujand we rotate each spin by an angle corresponding to the angle between ⃗Uiand ⃗Ujperpendicularly to the plane formed by the two vectors. The acceptance rule will still be the Metropolis one and then only small rotation of the global magnetization has a non-zero chance to be accepted. As an example, we repeat the hysteresis loop simulations at T /Tc=0.007 for di fferent values of effective anisotropy constant K U. The FSS was kept constant (2.10−3T/MCS) and equal to the FIG. 8. Hysteresis loops for di fferent values of KUobtained with GRCA (a) and comparison between the relations of the coercive field and the e ffective anisotropy constant obtained with di fferent algorithms (b) at T/Tc=0.007; in addition, the theoretical curve (green line).127124-10 Nehme et al. AIP Advances 5, 127124 (2015) FIG. 9. Log plot of Hc versus FSS obtained with three algorithms for KU=0 K/atom at T/Tc=0.007 (the error bars are similar to the size of the dots). one used in the previous simulations with a maximum field of 200 T. The ratio of SSF MCS to the GRCA steps was chosen to be 1%. The GRCA radius (here RGRCA cone =Rcone/20=0.4/20=0.02) gives an acceptance of ½. Ideally if we move one magnetic moment µwithin a cone R cone, a macro-magnetic moment of (N.µ)had to be moved within a radius cone RGRCA cone =√ N.Rconein order to keep the same reel time step for the dynamics. Results obtained with this algorithm are illustrated in Fig. 8(a). The hysteresis loops remain nearly square-shaped, as for the FA and CA, as typically expected for hysteresis measurements in the easy anisotropy direction. The coercive fields are plotted against the e ffective anisotropy constant K U. The linear dependence of the coercive field is still present with a slope of 1.92 (a value of 2 is expected at T =0 K). In Fig. 8(b), we plot the variation of H cagainst K Uestimated from the three algorithms cited before (FA, CA and GRCA). All simulations were done in the same conditions of temperature (T /Tc=0.007) and FSS. One can see that the GRCA decreases the H0 c values by a third compared to the CA and by 8 when compared to the FA. The numerical cost of this global step is proportional to the SSF /GRCA ratio. In the present case (1%) it is negligible. The GRCA allows hysteresis loops on a wide range of temperatures to be simulated (when the temperature is quite high, the SSF cone algorithm had to be replaced by a free SSF algorithm). In order to compare the e fficiency of the three algorithms listed before, we simulate hysteresis loops at the same conditions (K U=0 K/at, constant FSS) at T /Tc=0.007. The theoretical value of H0 chas then to be zero (Stoner-Wohlfarth model), but as aforementioned the MC procedure requires infinite N (MCS) to reach this value. Fig. 9 compares the di fferent coercive field values estimated from the three algorithms and for di fferent values of FSS. As we can see, the three algorithms give different values of H0 c. For the same FSS (i.e. calculation time) the log /log slope is nearly the same for the FA and CA ( ≈0.52), however this value increases to 0.66 using the GRCA which accelerates the simulations. In figure 8(b), are plotted the results of simulation performed using the GRCA with a slower FSS (2.10−4T/MCS). As a reference, the theoretical evolution is also reported (green line): the discrepancy between this theoretical curve and the results obtained with low FSS and GRCA leads us to feel confident with quantitative measurements of Hc. Indeed, one can estimate H0 C(Ku=0)with a better precision using appropriate simulation conditions leading to low FSS (i.e. N(MCS) su fficiently large, reduced value of H max). To highlight the behavior of the three algorithms, we report in Tab. I the values of calculation time needed to reach the same H cvalue, assuming GRCA as reference. TABLE I. Ratio of calculation time required to reach the same H cby the three algorithms, the GRCA is taken as reference. Algorithm GRCA CA FA calculation time ratio 1 ∼866 ∼10000127124-11 Nehme et al. AIP Advances 5, 127124 (2015) IV. CONCLUSION We investigated the dynamics of di fferent algorithms in order to simulate hysteresis loops using Monte Carlo based method. We compared the coercive field (found at di fferent temperatures, at constant field scan speed FSS) with an increase of the volume anisotropy. The comparison of “free” and “restricted” single-spin flip algorithms allows us to validate the restricted condition (i.e. cone radius) that gives the best acceleration of the calculation code while keeping the sampling accurate. The results give clear evidence of a linear dependence of the coercive field with the e ffective anisot- ropy constant for non-biased conditions (free or correctly adapted cone algorithms). As we have demonstrated the values of the measured H cby Monte Carlo method are strongly dependent on the FSS and the implemented algorithm. A high value of FSS leads to large /unphysical values of H c. In addition, we show that at very low temperature (T /Tc≈0K) both the “free algorithm” and “cone algorithm” become poorly e fficient when they are used for low field scan simulations. This case (T /Tc≈0K with strong spins exchange interactions) remains the most di fficult situation for MC algorithms. A small amount of global rotation strongly helps to reduce the part of the coercive field due to the dynamics of the algorithm while keeping the H cversus K Uslope close to the theoretical one (Fig. 8). To reach theoretical values of H cat low temperature a huge number of MCS is needed, however our global rotation cone algorithm leads to an increased e fficiency compared to that of the classical free algorithm. In the case of ideal coherent rotation of the magnetic mo- ments, as one observe a linear dependence of H cversus K u(obtained by MC) providing the same simulation conditions (algorithm and parameters), coercive field can be estimate by subtracting the intrinsic contributions when K u=0. 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1.5127262.pdf	J. Appl. Phys. 126, 214304 (2019); https://doi.org/10.1063/1.5127262 126, 214304 © 2019 Author(s).On the micromagnetic behavior of dipolar- coupled nanomagnets in defective square artificial spin ice systems Cite as: J. Appl. Phys. 126, 214304 (2019); https://doi.org/10.1063/1.5127262 Submitted: 12 September 2019 . Accepted: 21 November 2019 . Published Online: 05 December 2019 Neeti Keswani , and Pintu Das On the micromagnetic behavior of dipolar- coupled nanomagnets in defective square artiﬁcial spin ice systems Cite as: J. Appl. Phys. 126, 214304 (2019); doi: 10.1063/1.5127262 View Online Export Citation CrossMar k Submitted: 12 September 2019 · Accepted: 21 November 2019 · Published Online: 5 December 2019 Neeti Keswani and Pintu Dasa) AFFILIATIONS Department of Physics, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110016, India a)Electronic mail: pintu@physics.iitd.ac.in ABSTRACT We report here the results of micromagnetic simulations of square arti ﬁcial spin ice (ASI) systems with defects. The defects are introduced by the misaligning of a nanomagnet at the vertex. In these defective systems, we are able to stabilize emergent monopolelike state byapplying a small external ﬁeld. We observe a systematic change of dipolar energies of the systems with varying misalignment angle. The ﬁelds at which the emergent monopoles are created vary linearly with the dipolar energies of the systems. Our results clearly show that the magnetization reversal of the ASI systems is intricately related to the interplay of defects and dipolar interactions. Published under license by AIP Publishing. https://doi.org/10.1063/1.5127262 I. INTRODUCTION Artiﬁcial spin ice (ASI) systems are lithographically patterned arrangements of interacting magnetic nanostructures that were introduced for investigating the e ﬀects of geometric frustration in a controlled manner. 1In particular, nanomagnets in an arti ﬁcial 2D square and kagome array that mimic the spin ice behavior haveemerged as the subject for intensive investigation in recent years. 1–6 Square ASI can be considered as composed of two orthogonal sub- lattices of identical nanomagnets owing to their easy axes alignedalong the [10] and [01] directions. The recent progress in nanoli-thography techniques enables us to tune various parameters suchas interaction strength between nanomagnets, their geometry, as well as the introduction of arti ﬁcial defects. 7–10In 2008, Castelnovo et al. realized that excitations above the degenerate ground states in spin ice systems, where ice rule is violated, could be interpreted asemergent magnetically charged quasiparticles that behave like mag-netic monopoles. 11An important aspect of vertex frustration is its fascinating relation to the lattice topology and defects.3,12–16This makes it possible to tune the complex dynamics of the magneticallycharged vertices. 10,12,17–22During the fabrication of a large array of such nanostructures, it may be possible that a nanoisland whichcan be considered as a macrospin may be misaligned or it may lose its magnetically single-domain character due to an unintentional structural defect occurring during the fabrication steps. In arti ﬁcialspin ice systems, the impact of such defects in the overall spin ice behavior can be studied in a controlled way. Moreover, the creationor annihilation of excited states is connected to the magnetizationreversal of the nanostructures at the vertex. In order to understand this aspect of ASI systems, we carried out detailed micromagnetic simulation studies for individual square ASI vertices. In a recentpaper, we reported the observation of stable emergent monopole-like state in an even numbered vertex with vacancies at speci ﬁc square lattice sites at the edges. 23In this paper, we report the ener- getics of individual vertices as defects are introduced in the form of controlled misalignment of a vertex island. The magnetizationreversals of the magnetic nanostructures in the form of ellipticalshaped nanoislands of the vertices exhibit an angle dependentbehavior. An emergent monopolelike state was stabilized in these structure with defects. Figure 1(a) shows the schematics of types of possible vertices for macrospins arranged in square geometry inincreasing energy (E typeI,/C1/C1/C1,EtypeIV ). The schematics of the ASI structure used for the simulations is shown in Fig. 1(c) . II. METHODS Elliptical nanomagnets of Ni 80Fe20of aspect ratio 3 with dimensions of 300 /C2100/C225 nm3were used in square ASI geom- etry. The lattice constant for the lattice consisting of these elliptical nanomagnets is de ﬁned as the separation between edge-to-edge ofJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 214304 (2019); doi: 10.1063/1.5127262 126, 214304-1 Published under license by AIP Publishing.the nanomagnets. The lattice constant is 150 nm [see Fig. 1(a) ]. For micromagnetic simulations of the individual vertices, ﬁnite di ﬀer- ence based three-dimensional solver of the Object Oriented Micro Magnetic Framework (OOMMF), which is an openware software from the National Institute of Standards and Technology, is used.24 The time dependent magnetization dynamics in the nanoisland isgoverned by the Landau-Lifshitz-Gilbert (LLG) equation, dm dt¼/C0j γjm/C2Heffþαm/C2dm dt/C18/C19 , where γis the gyromagnetic ratio and αis the Gilbert damping constant. The e ﬀective magnetic ﬁeld is given by Heff¼/C01 μo(δW/δm), where μois the vacuum permeability and W is the magnetic energy of the system, which consists of exchange, anisotropy, anddipolar energy. For simulations, the nanoislands are discretizedinto cubic cells with each side of dimension 5 nm, which is less than the exchange length ( /difference5:3 nm). For the calculations, typical experimentally reported values of saturation magnetizationM s¼8:6/C2105A/m, the exchange sti ﬀness constant A¼13 pJ/m, and the damping constant of 0.5 for Ni 80Fe20are used.25The anisotropy is dominated by the shape ( Kshape/C257:3/C2104Jm/C03)o f the nanomagnets, and, therefore, the magnetocrystalline anisotropy is neglected in the computation. For these dimensions, the nano-magnets are magnetically in single-domain state, which was veri ﬁed from simulations as well as experiments (not shown). Starting from a randomized state, the system was allowed to reach its minimum energy. Thereafter, the state was saturated by applying a magneticﬁeld of 200 mT along the [10] direction. The magnetization reversal was studied while sweeping the ﬁeld between +200 mT at T¼0K . III. RESULTS AND DISCUSSION In the earlier work of magnetization reversal behavior of a regular vertex with closed edges, it was observed that the remanentstate follows a two-in/two-out spin ice state. 23Detailed analysis of the micromagnetic behavior showed the speci ﬁc way in which the reversal of the vertex magnetization proceeds. Here, a deformity is introduced by misaligning the easy axis of one of the vertex nanois-lands in a sublattice with respect to the applied magnetic ﬁeld direction. We de ﬁne the angle between the misaligned easy axis and the applied ﬁeld direction as misalignment angle θ [see Fig. 1(b) ]. If such a misalignment (defect) exists in a large array of the square ASI system, the defect may have a signi ﬁcant role in overall behavior of the array. Therefore, primarily in thiswork, we simulate such individual defective vertex structures andcarry out systematic investigations of their micromagnetic behavior for varying misalignment angle θ, where 20 /C14/C20θ/C2085/C14.F o r the nanomagnet under consideration, θ¼90/C14corresponds to the applied ﬁeld direction oriented along its hard axis. Thus, for smaller values of θ, the orientation of the ﬁeld approaches toward the easy direction of the nanomagnet. Initially at θ¼85/C14, the system is saturated by applying a positive magnetic ﬁeld of 200 mT along the [10] direction as indicated in Fig. 2(a) . Thereafter, while sweeping the ﬁeld between +200 mT, the micromagnetic calcula- tions are carried out at di ﬀerent external ﬁelds. The static equilib- rium magnetization at remanence evolves into a two-in/two-out magnetic state as depicted in Fig. 2(c) . The edge magnetic FIG. 1. Illustration of 16 possible spin states at a vertex of the two- dimensional square arti ﬁcial spin ice system. Degenerate states are groupedinto four different types of increasinglyhigher energies. The different colors represent the net magnetic charge at the vertex (a). Schematics of a regular(without defect) vertex in square ASIgeometry (b) and a defective vertex with misaligned nanoisland (c).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 214304 (2019); doi: 10.1063/1.5127262 126, 214304-2 Published under license by AIP Publishing.conﬁgurations at remanence and beyond are identi ﬁed as consisting of two onion and two horse-shoe type chiral states as illustrated inFigs. 2(c) and2(d), respectively. The onion state refers to two sub- lattices of nanoislands with parallelly aligned magnetizationwhereas horse-shoe state corresponds to parallelly aligned magneti- zation of one sublattice and antiparallelly aligned magnetization of the other sublattice as illustrated in Figs. 2(b)(i) and 2(b)(ii) .O n the other hand, the microvortex state consists of sublattices ofnanoislands with opposite magnetizations 23,26[see Fig. 2(b)(iii) ]. Considering the macrospin model27for these single-domain nano- magnetic islands, four head-to-tail magnetic con ﬁgurations form amicrovortex loop whereas both onion and horse-shoe states have two head-to-tail, one head-to-head, and one tail-to-tail con ﬁgura- tions [see Fig. 2(b) ]. Calculations based on the macrospin model show that the energy hierarchy of these states followsE microvortex ,Ehorse/C0shoe,Eonion.23As shown in Fig. 2(a) , the mag- netization of the system is found to reverse in four distinct steps where sharp jumps are observed within a small ﬁeld range of 124–140 mT. To understand the origin of these sharp jumps, we investigated the exact micromagnetic states of the system at every2 mT during the reversal. Figure 2(d) shows the magnetic con ﬁgu- ration just before the ﬁrst jump (hereafter switching), showing two-in/two-out magnetic con ﬁguration at the vertex. It is observed that the ﬁrst sharp jump at μ oH¼/C0124 mT corresponds to the simultaneous reversals (switching) of magnetization of two nano-magnets 1 and 2 situated at the diagonally opposite positions at 1stand 3rd quadrants [see Fig. 2(e) ]. These reversals convert the two horse-shoe type loops to two lower-energy microvortex loops. The next jump in hysteresis occurs at μ oH¼/C0126 mT, where the other diagonally opposite nanomagnets 3 and 4 switch simultaneously,thereby converting two onion states to two lower-energy horse-shoestates as shown in Fig. 2(f) . As the ﬁeld is further increased in the negative direction, the nanomagnets 5 and 6 in two di ﬀerent sublattices switch simultaneously [see Fig. 2(g) ]. Interestingly, we observe that due to the misalignment angle of 85 /C14, a small compo- nent of the ﬁeld ( μoHcosθ) along the easy axis direction of the nanomagnet assists the switching of magnetization of the nano- magnet to one of its minimum energy states. Such a switching inthe similarly positioned nanomagnets (i.e., easy axis orthogonal tothe applied ﬁeld) in the nondefective ASI vertices was not observed before. 23This clearly demonstrates the tunability of the switching behavior by introducing such defects in ASI vertices. This also leads to the creation of a type I state at the vertex. Withfurther increase in ﬁeld, the nanomagnet 7 ﬁnally reverses at μ oH¼/C0140 mT. As shown in Fig. 2(h) , this ﬁnal reversal changes the vertex from two-in/two-out state to three-in/one-out state. According to the dumbbell model proposed by Castelnovo et al. ,11 a magnetic dipole (macrospin) can be assumed to represent mag- netic charges of þQmand/C0Qm. Thus, the vertex as shown in Fig. 2(h) has charge þ2Qmmagnetic state, which is an emergent magnetic monopole state. Remarkably, the emergent monopolelike state during reversal has not been observed for a regular vertex (without defects) with closed edges23so far. Though the misaligned defect does not modify the remanent state of the vertex, it leads toa drastic change in the reversal mechanism with the creation of emergent monopolelike (type III state) from the lowest energy type I state. Interestingly, the edge loops now change to two onion andtwo horse-shoe type loops. As the misalignment angle is changed from 85 /C14to 80/C14,w e observe again a two-in/two-out (type II) spin ice state at remanence as shown in Fig. 3(b) . After saturation, the ﬁrst sharp jump in the hysteresis is observed at μoH¼/C0124 mT as shown in Fig. 3(a) . Simulation results show that this corresponds to the simultaneousreversal of nanomagnets 1 and 2 of di ﬀerent sublattices [see Fig. 3(d) ]. In this case, the ﬁeld has a stronger component along the easy axis direction of the misaligned nanomagnet. Thus, this nano- magnet switches at a relatively smaller ﬁeld. This reversal converts one horse-shoe loop to one microvortex and the other horse-shoe to FIG. 2. (a) Hysteresis loop for a vertex with closed edges for misaligned angle θ¼85/C14. (b) Schematics of (i) onion state (ii) horse-shoe state, and (iii) micro- vortex state. (c) –(h) Magnetic states at intermediate ﬁelds showing magnetic switchings as shown in (a). The external ﬁeld direction is shown in (c).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 214304 (2019); doi: 10.1063/1.5127262 126, 214304-3 Published under license by AIP Publishing.an onion type edge loops. Furthermore, the vertex now turns to an emergent monopolelike state with magnetic charge þ2Qm. With further increase in Zeeman energy, the magnetization of nanomag-net 3 reverses so that a higher energy onion state is converted to alower-energy horse-shoe state at μ oH¼/C0126 mT [see Fig. 3(e) ]. The next jump is observed at μoH¼/C0128 mT, which corresponds to the simultaneous switchings of nanomagnets 4 and 5, thereby converting the two onion loops to two horse-shoe type loops. Theﬁnal switching takes place at μ oH¼/C0132 mT, where the two nano- magnets at the vertex, indicated by 6 and 7, reverse simultaneously as shown in Fig. 3(g) . It is remarkable that the vertex remains at theþ2Qmcharged state during the entire reversal process. Thus, astable emergent magnetic monopole state is generated for this defect conﬁguration. With further reduction of the misalignment angle by 5/C14, i.e., forθ¼75/C14, interesting changes are observed in the micromagnetic behavior of the nanomagnets. The hysteresis shows ﬁve distinct jumps for θ¼75/C14as seen in Fig. 4(a) . The remanent state still follows type II spin ice state as for the other cases [see Fig. 4(b) ]. Theﬁrst jump in the hysteresis loop —which takes place at lower ﬁeldμoH¼/C0116 mT —corresponds to a change in the detailed micromagnetic pattern in the misaligned nanomagnet 1 fromsingle domain to magnetic vortex state. The core of the magnetic vortex lies at the edge of the elliptical nanomagnet, which is close to the ASI vertex as shown in Fig. 4(d) . The chirality of magnetic FIG. 3. (a) Hysteresis loop for a system with misaligned angle θ¼80/C14. (b)–(g) Corresponding magnetic states at intermediate ﬁelds showing magnetic switch- ings as shown in (a). FIG. 4. (a) Hysteresis loop for a vertex with a closed edge with misalignment angle of 75/C14. (b)–(h) Magnetic states after corresponding magnetic switchings of individual nanomagnets as shown in (a).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 214304 (2019); doi: 10.1063/1.5127262 126, 214304-4 Published under license by AIP Publishing.vortex in this nanomagnet appears to depend on the direction of theﬁeld sweep. Vortex of clockwise chirality was observed while downsweeping the ﬁeld from 200 mT to −200 mT as shown in Fig. 4(d) –4(f). Vortex of opposite chirality is observed while upsweeping the ﬁeld (not shown). Such a magnetic vortex structure is purely due to the energy minimization as a result of a complex interplay of the defect, dipolar interactions, and the magnetic ﬁeld. The change of single-domain state to a magnetic vortex state is asigniﬁcant result, which clearly underlines the role of the defect in creating new states in the nanomagnets. As a result of this magneticvortex-type magnetic structure, the two-in/two-out spin ice rule at the vertex is violated, thereby converting the vertex to an anoma- lous state where the nanomagnet does not retain the single-domainbehavior. Considering the magnetic vortex in the nanomagnet asmagnetically chargeless, the net charge at this vertex is + Q m. As the negative ﬁeld is further increased, the second jump is observed at μoH¼/C0124 mT, which corresponds to switching of nanomagnet 2 [see Fig. 4(e) ]. The next jump is observed at μoH¼/C0126 mT, where the magnetization of the two nanomagnets 3 and 4 reversessimultaneously. This is shown in Fig. 4(f) . The fourth and ﬁfth jumps occur at μ oH¼/C0128 mT and μoH¼/C0130 mT, respec- tively. The jump at μoH¼/C0128 mT is due to the reversal of mag- netization of nanomagnet 5 as shown in Fig. 4(g) . With the reversal of nanomagnet 5, the charge of 3 Qmis generated at vertex. This additional magnetic charge observed at the vertex is due to the creation of a magnetic vortex in the misaligned nanomagnet. The jump at μoH¼/C0130 mT corresponds to the simultaneous reversals of magnetizations of nanomagnets 6 and 7, respectively,as illustrated in Fig. 4(h) . With the ﬁnal reversals, the magnetic state of the misaligned nanomagnet again changes back to a single-domain state, thereby creating a charge of þ2Q m, i.e., an emergent monopolelike state at the vertex. After the complete reversal of nanomagnets, themagnetic state constitutes one horse-shoe and three onion states[see Fig. 4(h) ]. Thus, for θ¼75 /C14, we observe the creation of an emergent monopolelike state with three-in/one-out magnetic orien- tation at the vertex via an anomalous state (of charges Qmand 3Qm). When carefully calculated, we ﬁnd that such a vortex-type structure in the misaligned nanomagnet appears for angle at leastuntil θ¼73/C14; however, it disappears for θ/C2070/C14. The calculations are performed for various angles until θ¼20/C14. In general, forθ/C2070/C14, the reversal mechanism exhibits similar behavior as described for misalignment angle θ¼80/C14. The emergence of monopolelike state appears for all the misalignment angles studied in this work. Thus, we observe that such emergent monopolelikemagnetically charged states can be predictably created at the verti-ces due to the interplay of defects and dipolar interactions. Thecorresponding magnetic states for all the misalignment angles studied in this work are summarized in Table I . In order to clearly demonstrate the role of such defects in stabilizing the observedmagnetic states, we plot the ﬁelds ( μ oHmp) at which emergent monopoles are observed against the dipolar energies ( Edip) of the respective defective structures (see Fig. 5 ). The results as obtained from the micromagnetic simulations are used for the plots. Asshown in the inset of Fig. 5 , we observe that the e ﬀective dipolar energy of the vertex system increases linearly as a function of themisalignment angle of a vertex island. Thus, these results underline the role of the interplay of defects and dipolar interactions in stabilizing the charged emergent monopolelike states. The resultsindicate that a misalignment may be used to predictably create acharged vertex state at a desired ﬁeld in a square ASI system. IV. CONCLUSION In summary, we investigated in detail the micromagnetic behavior of ASI vertices with defects in the form of a misalignednanomagnet in the vertex. Systematic studies of the switching behavior exhibit sharp jumps in the hysteresis loops. Detailed investigations of the jumps show that they correspond to switchingTABLE I. T able summarizes the magnetic states at intermediate ﬁelds correspond- ing to different misalignment angles. The numbers in bracket indicate the ﬁelds at which the corresponding magnetic states are observed. Misalignment angle ( θ)Remanent stateMagnetic states at intermediate field 85° Type II state Type I (136 mT) and Type III (140 mT) 80° Type II state Type III (124 mT) 75° Type II state Anomalous state (116 mT) and Type III (128 mT) 50° Type II state Type III (92 mT)45° Type II state Type III (88 mT)30° Type II state Type III (90 mT) 20° Type II state Type III (98 mT) FIG. 5. Field ( μoHmp) at which a monopole is generated vs the dipolar energy (Edip) of the system. Error bars indicate the ﬁeld separation at which magnetic states are calculated. The inset shows the variation of dipolar energy ( Edip)a sa function of the angle of misalignment ( θ) in the misaligned vertex.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 214304 (2019); doi: 10.1063/1.5127262 126, 214304-5 Published under license by AIP Publishing.of individual nanomagnets or simultaneous switchings of two nanomagnets in the ASI system. While switching, indirect cou- plings of two nanomagnets are observed in certain cases due to thestrong dipolar interaction among the involved nanomagnets. Forthe misalignment angle of 75 /C14, an interesting change of the mag- netic state of the misaligned nanomagnet from single domain to magnetic vortex state is observed. The dipolar energy of the system is observed to increase linearly with the misalignment. The ﬁelds at which emergent monopolelike states are observed exhibit a linearrelationship with the dipolar energy. Thus, we ﬁnd a clear role of defects and dipolar interactions in stabilizing an emergent mono- polelike state in such systems. 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1.367726.pdf	Magnetic ordering in Co films on stepped Cu(100) surfaces S. T. Coyle and M. R. Scheinfein Citation: Journal of Applied Physics 83, 7040 (1998); doi: 10.1063/1.367726 View online: http://dx.doi.org/10.1063/1.367726 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/83/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Growth-induced uniaxial magnetic anisotropy in Co/Cu(100) J. Appl. Phys. 95, 7300 (2004); 10.1063/1.1687611 Effect of growth temperature on Curie temperature of magnetic ultrathin films Co/Cu(100) J. Appl. Phys. 89, 7153 (2001); 10.1063/1.1358827 Magnetic phase diagram of ultrathin Co/Si(111) film studied by surface magneto-optic Kerr effect Appl. Phys. Lett. 74, 1311 (1999); 10.1063/1.123534 Effect of surface roughness on magnetic properties of Co films on plasma-etched Si(100) substrates J. Appl. Phys. 83, 5313 (1998); 10.1063/1.367357 Growth, morphology, and magnetic properties of ultrathin epitaxial Co films on Cu(100) J. Vac. Sci. Technol. A 15, 1785 (1997); 10.1116/1.580870 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sat, 20 Dec 2014 19:59:29Magnetic ordering in Co ﬁlms on stepped Cu 100surfaces S. T. Coyle and M. R. Scheinfein Dept. of Physics and Astronomy, PSF 470 Box 871504, Arizona State University, Tempe, Arizona 85287 Ultrathin ﬁlms of Co were grown on Cu ~100!and characterized by nanometer resolution secondary electron microscopy, Auger electron spectroscopy, and the surface magneto-optic Kerr effect. Anunexpected out-of-plane remanence was detected in many ﬁlms. The anisotropy of atoms neardefects along the Co/vacuum interface calculated via the Ne ´el model indicates that atoms at the bottom corner of a step edge are canted out-of-plane. Full three-dimensional micromagneticssimulations which incorporate site speciﬁc anisotropy ~including step edges, kinks, and voids !have been performed. Simulations with unidirectional arrays of @11I0#steps, such as vicinal surfaces, do not exhibit out-of-plane remanence. Simulations with facets consisting of connected @110#and @11I0#steps exhibit out-of-plane remanence of 0.03. This is lower than the experimental value of 0.11. © 1998 American Institute of Physics. @S0021-8979 ~98!37911-6 # I. INTRODUCTION Magnetic surface anisotropies play a key role in deter- mining the magnetic properties of thin ﬁlms andmultilayers. 1Recently the anisotropy of steps has been found to be important in understanding the magnetic behavior ofsome systems. 2–4Since the roughness of the Co/Cu interface plays a key role in determining GMR properties,5character- izing the Co/Cu interface including the effect of defects isimportant. Ultrathin ﬁlms of Co were grown on Cu ~100!in order to study the morphology and the resulting magneticproperties at early stages of growth. The ﬁlms were charac-terized by nanometer resolution secondary electron micros-copy, Auger electron spectroscopy, and the surface magneto-optic Kerr effect ~SMOKE !. 6An unexpected out-of-plane remanence was detected in many ﬁlms. The cause of this out-of-plane component of the magne- tization could be related to ﬁlm morphology at the earlystages of growth. One possible mechanism which may pro-duce out-of-plane magnetization is defect related anisotropyon imperfect surfaces. The anisotropy of atoms near defectsalong the Co/vacuum interface has been calculated. Atomsalong the bottom corner of a ^110&step which have strong uniaxial anisotropy canted out-of-plane, may couple to thespins of nearby atoms. A signiﬁcant out-of-plane componentto the magnetization may occur for some critical density ofthese sites. This short article will address the feasibility ofthis mechanism for the origin of the perpendicular compo-nent to the observed magnetization. Co grown on substrates with high defect densities re- sulted in dramatic faceting of step edges and the creation ofrectangular pits. 6The anisotropy of atoms of low coordina- tion created by this morphology may signiﬁcantly affect themagnetization of the ﬁlm, and may thus affect the GMRproperties of multilayers. To evaluate the equilibrium mag-netic microstructure in such ﬁlms, and to determine if the anisotropy at sites with low symmetry may be responsiblefor the observed out-of-plane remanence, full three-dimensional micromagnetics simulations were performed in-corporating the calculated site speciﬁc anisotropies.II. EXPERIMENTAL RESULTS Morphological characterization with concurrent magne- tization measurements was obtained from Co grown on bulksingle crystal Cu ~100!samples. 6Cu substrates were cleaned by repeated Ar1ion sputter and anneal ~600 C !cycles. Co was grown by electron-beam evaporation at rates between0.05 ML/min and 0.2 ML/min at pressures ,5310 210 mbar (1 ML 51.5331015atoms/cm2). Samples were trans- ferredin situinto the SMOKE chamber for magnetic char- acterization, then transferred in situinto an ultrahigh vacuum scanning transmission electron microscope for nanometerresolution secondary electron ~SE!imaging. 7SE micro- graphs revealed complex growth morphologies which variedbetween different ﬁlms. Many ﬁlms contained high densitiesof steps, kinks, and facets. Co/Cu ~100!ﬁlms in this study became ferromagnetic at room temperature at about 1.7 ML. Zero ﬁeld susceptibilityin the paramagnetic regime and remanence in the ferromag-netic regime generally increased with coverage. In manyﬁlms a second magnetic phase was detected with out-of-plane remanence and a coercivity 5–10 times the in-planevalue which increased with Co coverage. Figure 1 containssuch Kerr hysteresis loops taken in the longitudinal @Fig. 1~a!#and polar @Fig. 1 ~b!#geometries 7f r o ma2M L thick ﬁlm. As a result of the 45° incident scattering angle, polarsignals were ﬁve times stronger than the longitudinalsignals. 8The out-of-plane component of the magnetization in the ﬁlm in Fig. 1 is therefore ;0.11. FIG. 1. Kerr hysteresis loops from a sample exhibiting out-of-plane rema- nence. Part ~a!was taken in the longitudinal geometry and ~b!was taken in the polar geometry. The Kerr signal is given in arbitrary units, and the scalesin~a!and~b!are the same.JOURNAL OF APPLIED PHYSICS VOLUME 83, NUMBER 11 1 JUNE 1998 7040 0021-8979/98/83(11)/7040/3/$15.00 © 1998 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sat, 20 Dec 2014 19:59:29III. MICROMAGNETICS SIMULATIONS The anisotropy of face centered cubic atoms has been calculated for reduced symmetry structures such as steps andkinks following the Ne ´el model of anisotropy, 9and including the effects of strain.10In agreement with Chuang et al.,11the anisotropy of atoms along the bottom corner of a step edge~step corner !was found to be uniaxial and canted out-of- plane, while the anisotropy of atoms along the top corner ofa step ~step edge !was uniaxial in-plane along the step direc- tion. The anisotropy of bulk atoms and surface atoms wasbiaxial in-plane along ^110&. The anisotropy of kink edge atoms was biaxial in-plane along ^110&and that of kink cor- ner atoms was uniaxial in-plane along the kink direction. Asummary of these results is given in Table I, and a schematicof a Co surface identifying individual sites is presented inFig. 2. Strain due to the misﬁt of face-centered-cubic ~fcc! Co and fcc Cu has been included in all anisotropy calcula-tions except when noted. The anisotropy terms ~Table I !proportional to cos 2u extract a high penalty for magnetization out of the plane. Of the remaining terms, those proportional to L(r0) are larger by a factor of 102than terms containing e0orQ(r). The step corner, kink-in corner and kink-out corner sites hold promisefor out-of-plane magnetization due to the term proportionalto sin ucosu. The step corner has an energy minimum which is out-of-plane, while the kink-in corner and kink-out cornersites have energy minima which are in-plane. These atomic, site speciﬁc anisotropy energies have been incorporated into micromagnetics simulations 12of Co on stepped Cu ~100!. The simulation searches for solutions to the Landau–Lifshitz–Gilbert equation. The following energieswere included: exchange energy, site-speciﬁc anisotropy en-ergy, magnetostatic self-energy, and external magnetostaticﬁeld energy. The saturation magnetization, exchange stiff-ness, gyromagnetic frequency gamma, and damping con-stant alpha were set to the bulk values for Co. This is a continuum model which has been discretized at atomiclength scales. The micromagnetic structure of two monolayer ~ML! ﬁlms has been simulated where the top layer consists of aterrace one half the width of the bottom layer. The two step edges in the top layer were aligned along @11I0#or@100#.I n some simulations kinks were inserted into the step edges atregular intervals and the terrace widths were varied. Thesimulations used periodic boundary conditions in both in-plane directions. The system was discretized into cells withsides of length a 0/A2 on a simple cubic lattice, where a0is 0.361 nm. This insured that the volume of the region with FIG. 2. Schematic representation of atomic sites in the vicinity of a kinked ^110&step. The anisotropy of these sites is given in Table I. TABLE I. Anisotropy energies for fcc ~100!sites described in the text and shown schematically in Fig. 2. DerivationsofanisotropyenergyandthevalueoftheconstantshavebeengiveninRef.9.Note r,u,and fhave been deﬁned in the usual way. The step direction is @11I0#. For @110#steps, change wtot2w. Site/constant Anisotropy energy Bulk unstrained ( Ebu) Q(r)/4(sin22u1sin22wsin4u) Bulk strained ( 26e0L(r0)2e0]L/]rr0)cos2u1Ebu Surface ( 21/2L(r0)23e0L(r0)2e0]L/]rr0)cos2u1Ebu Step edge ( 21/4L(r0)23e0L(r0)23/4e0]L/]rr0)cos2u 1(21/2L(r0)21/2e0]L/]rr0)sin2usinwcosw1Ebu Step corner ( 21/4L(r0)29/2e0L(r0)2e0]L/]rr0)cos2u 21/2L(r0)sinucosu(sinw1cosw)1Ebu Kink-in edge ( 21/2L(r0)23e0L(r0)2e0]L/]rr0)cos2u1Ebu Kink-out edge ( 23e0L(r0)21/2e0]L/]rr0)cos2u1Ebu Kink-in corner ( 21/4L(r0)211/2e0L(r0)2e0]L/]rr0)cos2u 1(21/4L(r0)21/2e0L(r0))sin2ucos2w 21/2L(r0)sinucosucosw1Ebu Kink-out corner ( 21/4L(r0)27/2e0L(r0)2e0]L/]rr0)cos2u 1(1/4L(r0)11/2e0L(r0))sin2ucos2w 21/2L(r0)sinucosucosw1Ebu Q(r0) 21.23106erg/cm3 L(r0) 21.53108erg/cm3 ]L/]rr0 5.53108erg/cm3 e0 0.0197041 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 S. T. Coyle and M. R. Scheinfein [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sat, 20 Dec 2014 19:59:29step anisotropy matched the volume occupied by a step ori- ented along ^110&. The system was allowed to relax with no applied ﬁeld in order to determine the equilibrium magneti-zation distribution in the ﬁlm. No signiﬁcant out-of-plane ( ^Mz&.0.01) component of the magnetization was present in the equilibrium magnetiza-tion distribution. The spins of the step corner atoms wereexpected to couple to the spins in the terraces and perhapscause them to be somewhat canted out-of-plane. This did notoccur as a result of the balance between the anisotropy en-ergies of neighboring atoms and the exchange energy whichcouples them. The out-of-plane anisotropy of the step cornersite occurs via the term proportional to cos u~sinusinw 1sinucosw!@i.e.,mz(mx1my)#. The minimum energy oc- curs when 2mz5mx1my. This equals zero for w53p/4 (@11I0#), which is the uniaxial anisotropy axis for the step edge site. Although the anisotropy energy of the step edgeand step corner sites are about equal, the coupling via ex-change to nearby surface and bulk sites ~biaxial ^110&!en- sures that the step edge site is dominant. For any initial con-dition on the magnetization the ﬁnal minimized energyconﬁguration has the spins aligned along the step direction,and the out-of-plane component vanishes with ~sin w 1cosw!. The magnetization conﬁguration is somewhat different for faceted steps and square islands. At the corner joining a@110#step to a @11I0#step, the magnetization of each step will be forced away from ^110&by the ﬁeld due to the other step, resulting in a nonzero out-of-plane component. If thedensity of facets is large enough or the size of islands smallenough a signiﬁcant out-of-plane remanence will exist. Thisconﬁguration has been simulated via a square island~3n m 33n m !o na5n m 35 nm square layer with periodic boundary conditions. The length of the sides of the islandwas chosen to approximate the length of facets observed inﬁlms which exhibited out-of-plane remanence. The out-of-plane component of the calculated average equilibrium mag-netization was ;0.03. This was signiﬁcantly less than the results from Kerr loops shown in Fig. 1. IV. CONCLUSION It is apparent from these micromagnetics simulations that the anisotropy of step atoms can not be responsible forthe out-of-plane remanence we observed experimentally. For surfaces with a high density of ^110&facets, this anisotropy may be a contributing factor. This micromagnetics resultfrom semi-inﬁnite parallel ^110&steps agrees with the experi- mental results from Co deposits on vicinal Cu ~111 3 ! surfaces. 4The anisotropy switches to biaxial in-plane at in- creased temperatures.13This may be due to Cu atoms deco- rating the step edges,14or to restructuring of the step edges with rectangular protrusions perpendicular to the originalstep. 15In the latter case, micromagnetics simulations re- ported here predict a small ~;3%!out-of-plane component to the magnetization. ACKNOWLEDGMENTS The authors would like to acknowledge Dr. G. G. Hem- bree for collaboration in the experimental work. This work issupported by ONR under Grant No. N00014-93-1-0099. 1U. Gradmann, J. Magn. Magn. Mater. 54/57, 733 ~1986!. 2M. E. Buckley, F. O. Shumann, and J. A. C. Bland, J. Phys. 8, L147 ~1996!. 3M. Albrecht, T. Furubayashi, M. Przybylski, J. Korecki, and U. Grad- mann, J. Magn. Magn. Mater. 113, 207 ~1992!. 4A. Berger, U. Linke, and H. P. Oepen, Phys. Rev. Lett. 68, 839 ~1992!. 5W. F. Egelhoff, Jr., P. J. Chen, R. D. K. Misra, T. Ha, Y. Kadmon, C. J. Powell, M. D. Stiles, R. D. McMichael, C.-L. Lin, J. M. Sivertsen, and J.H. Judy, J. Appl. Phys. 79, 282 ~1996!. 6S. T. Coyle, G. G. Hembree, and M. R. Scheinfein, J. Vac. Sci. Technol. A15, 1785 ~1997!; S. T. Coyle, J. L. Blue, and M. R. Scheinfein, J. Vac. Sci. Technol. A ~in press !; S. T. Coyle and M. R. Scheinfein, Appl. Phys. Lett. ~in press !. 7K. R. Heim, S. D. Healy, Z. J. Yang, J. S. Drucker, G. G. Hembree, and M. R. Scheinfein, J. Appl. Phys. 74, 7422 ~1993!. 8Z. J. Yang and M. R. Scheinfein, J. Appl. Phys. 74, 6810 ~1993!. 9L. Ne´el, J. Phys. Radium 15, 225 ~1959!. 10H. Fujiwara, H. Kadomatsu, and T. Tokaunaga, J. Magn. Magn. Mater. 31–34, 809 ~1983!. 11D. S. Chuang, C. A. Ballentine, and R. C. O’Handley, Phys. Rev. B 49, 15 084 ~1994!. 12M. R. Scheinfein ~LLG Micromagnetics Simulator, © 1997 !. 13W. Wulfhekel, S. Knappmann, B. Gehring, and H. P. Oepen, Phys. Rev. B 50, 16 074 ~1994!. 14W. Weber, C. H. Back, A. Bischof, D. Pescia, and R. Allenspach, Nature ~London !374, 788 ~1995!. 15M. Giesen, F. Schmitz, and H. Ibach, Surf. Sci. 336, 269 ~1995!.7042 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 S. T. Coyle and M. R. Scheinfein [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sat, 20 Dec 2014 19:59:29
1.2829775.pdf	Current-induced motion of narrow domain walls and dissipation in ferromagnetic metals M. Benakli, J. Hohlfeld, and A. Rebei Citation: Journal of Applied Physics 103, 023701 (2008); doi: 10.1063/1.2829775 View online: http://dx.doi.org/10.1063/1.2829775 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/103/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electrical current induced torque in the one and two dimensional domain wall AIP Conf. Proc. 1461, 104 (2012); 10.1063/1.4736877 Currentinduced coupled domain wall motions in a twonanowire system Appl. Phys. Lett. 99, 152501 (2011); 10.1063/1.3650706 Track heating study for current-induced domain wall motion experiments Appl. Phys. Lett. 97, 243505 (2010); 10.1063/1.3526755 Current-induced domain wall motion in a nanowire with perpendicular magnetic anisotropy Appl. Phys. Lett. 92, 202508 (2008); 10.1063/1.2926664 Current-induced switching in a single exchange-biased ferromagnetic layer J. Appl. Phys. 97, 10C709 (2005); 10.1063/1.1852437 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Wed, 03 Dec 2014 00:38:51Current-induced motion of narrow domain walls and dissipation in ferromagnetic metals M. Benakli, J. Hohlfeld, and A. Rebeia/H20850 Seagate Research Center, Pittsburgh, Pennsylvania 15222, USA /H20849Received 17 August 2007; accepted 4 November 2007; published online 16 January 2008 /H20850 Spin transport equations in a nonhomogeneous ferromagnet are derived in the limit where the sd exchange coupling between the electrons in the conduction band and those in the dband is dominant. It is shown that spin diffusion in ferromagnets assumes a tensor form. The diagonal termsare renormalized with respect to that in normal metals and enhance the dissipation in the magneticsystem while the off-diagonal terms renormalize the precessional frequency of the conductionelectrons and enhance the nonadiabatic spin torque. To demonstrate what additional physics isincluded in the theory, we show that self-consistent solutions of the spin diffusion equations and theLandau-Lifshitz equations in the presence of a current lead to an increase in the terminal velocity ofa domain wall which becomes strongly dependent on its width. We also provide a simpliﬁedequation that predicts damping due to the conduction electrons. © 2008 American Institute of Physics ./H20851DOI: 10.1063/1.2829775 /H20852 I. INTRODUCTION Dynamics of magnetic domain walls /H20849DWs /H20850is a classic topic1–3that recently received a lot of attention due to new fabrication and characterization techniques that permit theirstudy at the nanometer scale. Moreover, the subject of spindynamics in the presence of large inhomogeneities is cur-rently of great interest experimentally and theoretically dueto the potential applications in various nanodevices, espe-cially magnetic storage. 4One particular area that is still not well understood is the interaction of DWs with polarizedcurrents. The question here is how best to represent the con-tribution of the spin torque to the dynamics of themagnetization. 5–15So far attention has been focused on wide DWs where it was shown that terminal velocities are inde-pendent of the DW width. 9,14 This paper extends previous treatments to the case of thin, less than 100 nm, DWs. One of the main objectives ofour work is to expose the interplay between linear momen- tum relaxation and spin relaxation as the conduction elec-trons traverse a thin DW. This interplay originates from thestrong exchange interaction between the conduction selec- trons and the localized dmoments, and makes the terminal velocities as well as the transport parameters of the conduc-tion electrons dependent on the conﬁguration of the localmagnetization. This leads to an enhancement of the nonadia-batic contribution of the spin torque to the DW motion andopens the way to study spin torque-induced magnetizationdynamics in thin DWs in greater depth by measurement ofDW velocities. Moreover, we show that the interaction of theconduction electrons and the dmoments is also relevant for homogeneously magnetized metallic systems, where it is atthe origin of intrinsic damping. Our work can be easilyadapted to magnetic multilayer structures and hence theequations derived here are capable to treat noncollinear mag-netization geometries as opposed to that in Ref. 16, which deal only with collinear conﬁgurations. Narrow DWs canexist either naturally 17,18or artiﬁcially19,20and we hope the results discussed here show the potential beneﬁts of studyingdissipation in DW-like structures. II. GENERAL THEORY: OFF-DIAGONAL CONTRIBUTION To derive the spin coupling of the selectrons to the magnetization, we adopt the sdpicture which has been the basis for most of the studies in DW motion.9In the follow- ing, we use /H20849l,m,n/H20850for moment indices and /H20849i,j,k/H20850for space indices. In addition, the transverse domain wall is assumed to extend in the xdirection, with magnetization pointing in the zdirection. We start from the Boltzmann equation satisﬁed by the 2 /H110032 distribution function of the conduction electrons, f=fe+fs·/H9268, where /H9268l/H20849l=1,2,3 /H20850are Pauli matrices, in the presence of the magnetization Mof the system and an exter- nal electric ﬁeld E: /H11509tf+v·/H11612f+e/H20849E+v/H11003H/H20850·/H11612pf+i/H20851/H9262B/H9268·Hsd,f/H20852 =−fe−f0e /H9270p−f−f0s /H9270sf. /H208491/H20850 The sdexchange ﬁeld is Hsd/H20849x,t/H20850=JM/H20849x,t/H20850//H9262Bwith J /H110150.2 eV, and /H9270p,/H9270sfare the momentum and spin relaxation times, respectively.15,21,22The variables v,e, and/H9262Bare the velocity, the charge, and the magnetic moment of the selec- trons, respectively. fe0andfs0are the equilibrium charge and spin distribution. The conduction electrons have a polarization m =/H9262B/H20848dp//H208492/H9266/H208503Tr/H9268fand carry a charge current jc =e/H20848dp//H208492/H9266/H208503vTrf, as well as a spin current js=/H20885dp /H208492/H9266/H208503vTr/H9268f. /H208492/H20850a/H20850Electronic mail: arebei@mailaps.org.JOURNAL OF APPLIED PHYSICS 103, 023701 /H208492008 /H20850 0021-8979/2008/103 /H208492/H20850/023701/4/$23.00 © 2008 American Institute of Physics 103 , 023701-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Wed, 03 Dec 2014 00:38:51In the following, we use normalized deﬁnitions of the moments, i.e., /H20648M/H20648=/H20648m/H20648=1. The delectrons will be assumed to satisfy a Landau-Lifshitz-Gilbert /H20849LLG /H20850equation dM dt=−M/H11003/H20873/H9253Heff+1 /H9270exm/H20874+/H9251pdM/H11003dM dt, /H208493/H20850 where /H9270exis the inverse of the precessional frequency, /H9275c =J//H6036, of the conduction electrons due to the exchange ﬁeld. Heffis the total ﬁeld acting on the magnetization which in- cludes the exchange ﬁeld between the dmoments, the de- magnetization ﬁeld, and the anisotropy ﬁeld. In metals, themain source of dissipation is believed to be due to the con-duction electrons which in our theory is accounted for ex-plicitly within the limitations of the sdmodel. 23Hence, the damping constant /H9251pdis assumed to be due to dissipation caused by channels other than the conduction electrons suchas phonons or defects.In inhomogeneous magnetic media, the sdexchange term becomes comparable to that of the Weiss molecularﬁeld and hence the effect of the conduction electrons on themagnetization should be taken beyond the linear responseapproach. Going beyond the linear theory will allow us tosee how the presence of the background magnetization af-fects the transport properties of the conduction electrons. Webelieve this is especially true in transition metal nanomag-netic devices where the hybridization of the sanddelectrons is strong. Using standard many-body methods, 15the diffu- sion contribution to the spin current can be found jsli/H20849t,x/H20850=−Dln/H20849t,x/H20850/H11612imn/H20849t,x/H20850, /H208494/H20850 where Dis a diffusion tensor with effective relaxation time /H9270 which will be assumed equal to the momentum relaxation /H20849/H9270/H11015/H9270p/H20850. The Dtensor obeys the reduced symmetry of the ferromagnetic state and is15 D=D/H11036/H209001+/H92702/H9024x2/H9270/H9024z+/H92702/H9024x/H9024y−/H9270/H9024y+/H92702/H9024z/H9024x −/H9270/H9024z+/H92702/H9024y/H9024x 1+/H92702/H9024y2/H9270/H9024x+/H92702/H9024y/H9024z /H9270/H9024y+/H92702/H9024z/H9024x−/H9270/H9024x+/H92702/H9024y/H9024z 1+/H92702/H9024z2/H20901, /H208495/H20850 where /H9024=JM //H6036,D/H11036=D0/1+/H20849/H9270/H9275c/H208502with D0=1 3vF2/H9270pbeing the diffusion constant of the electron gas with Fermi velocity vF. It should be observed that in the presence of spin-orbit coupling, the symmetry of the diffusion tensor will be thesame as given here but the separation of the relaxation timesin independent channels of momentum and spin relaxationwill not be valid. In the following, the effect of the electricﬁeld is taken only to ﬁrst order. The symmetry of the spin current is best revealed by going to a local frame where the magnetization lies in the z direction. In this frame, one obtains for E=0, j /H11036=−Deffdm dx,jz=−D0dmz dx, /H208496/H20850 where m/H20849x/H20850=mx/H20849x/H20850−imy/H20849x/H20850, and Deff=D/H11036+iDxyis an effec- tive diffusion coefﬁcient with Dxy=D/H11036/H9270/H9275c. From the diver- gence of the spin current we get the steady-state equation forthe spin accumulation, d 2m dx2=m /H9261eff2,d2mz dx2=mz−m0 /H9261sdl2, /H208497/H20850 where /H9261eff2=/H9270effDeffwith/H9270eff=1 //H20851/H208491//H9270sf/H20850−i/H9275c/H20852,m0is the equi- librium spin density, and /H9261sdlis the longitudinal spin diffu- sion length typically in the range of 5–100 nm. The generalsolutions for the complex accumulation are of the formm/H20849x/H20850=Aexp /H20851−x//H9261 eff/H20852+Bexp /H20851x//H9261eff/H20852, i.e., they show an ex- ponential decrease /H20849or increase /H20850and oscillations from a local inhomogeneity in M. In the limit of a large sdexchange ﬁeld the period of the oscillations is vf//H9275cwhich corresponds tothe coherence length 1 //H20841k↑−k↓/H20841in the ballistic approach, where k↑is the spin-up momentum. Our expressions for the spin current generalize those used currently in the literature.9We ﬁnd that the diffusion constant D0is now renormalized by 1 //H208511+/H20849/H9270/H9275c/H208502/H20852which means that precession in the exchange ﬁeld reduces diffu- sion. Moreover, the precession gives rise to off-diagonalterms in the diffusion tensor which reﬂect the local two-dimensional /H208492D/H20850rotational symmetry around M. The origin of the off-diagonal term D xycan be under- stood qualitatively in terms of ﬂux. First, we rewrite it in thefollowing form: D xy=/H208731 3vF2/H9270p/H20874/H9270p/H9275c 1+ /H20849/H9270p/H9275c/H208502=1 3vF2/H9275c /H92632+/H9275c2, /H208498/H20850 where /H9263=1 //H9270p. In the limit of fast precession, /H9263/H11270/H9275c,w e have Dxy=1 3vF2//H9275c. Next, if we set vF//H9275c=Lm, then Lmis the distance a spin typically goes before it “converts” into thespin at 90° to that which it started with. The correspondingcontribution to the ﬂux has an obvious interpretation—thesource of spin x,m x, is particles coming from a distance Lm away where they had spin y,my. The ﬂux can be derived from a simple “kinetic” argument. A distance Lmupstream, the density is my=my0+Lmdmy/dxand a distance Lmdown- stream, my=my0−Lmdmy/dx. The ﬂux of particles with spin x, mx, crossing a point, coming from upstream, is my↑vF/3 and from downstream it is my↓vF/3. The difference is then /H20849my↑ −my↓/H20850vF/3=2Lmdmy/dxvF/3 which, within a factor of 2, is our off-diagonal ﬂux. In short, the off-diagonal terms are the corrections induced by precession on the diffusion process.023701-2 Benakli, Hohlfeld, and Rebei J. Appl. Phys. 103 , 023701 /H208492008 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Wed, 03 Dec 2014 00:38:51To get the effective equation for M, we use Eq. /H208493/H20850to express min terms of the magnetization M, then use the implicit solution back into the equation for m, Eq. /H208491/H20850.W e ﬁnd that the equation of motion for the magnetization be-comes /H9252dM dt=−/H9253M/H11003Heff+a·/H11612M+/H20849/H9251pd−/H9264/H20850M/H11003dM dt −/H9264/H9253M/H11003/H20849M/H11003Heff/H20850+/H11612·D/H11612m, /H208499/H20850 where a=P/H20849/H9262B/e/H20850Trj,/H9252=1+ m0+/H9251pd/H9264,/H9264=/H9270ex//H9270sfis the ratio of the precessional time to the spin relaxation time of the conduction electrons, and Pis the spin current polarization which we set to 0.4.9The equilibrium polarization is taken m0=0.01 M. The second term on the right is the adiabatic spin torque while the last term is the diffusion contribution.The third and the fourth terms are equivalent to the nonadia-batic spin torque with the original damping /H9251pdfrom Eq. /H208493/H20850 added to the third term. For uniform magnetization, /H11612M =/H11612m=0, and damping constant /H9251pd=0, Eq. /H208499/H20850reduces to dM dt=−/H9253/H92642+/H9252 /H92642+/H92522M/H11003Heff−/H9253/H9264m0 /H92522+/H92642M/H11003/H20849M/H11003Heff/H20850. /H2084910/H20850 Hence, we are able to predict damping due to conduction electrons, quantify the corresponding damping constant /H9251el =/H9253/H9264m0//H20849/H92522+/H92642/H20850, and identify its origin as the spin torque contribution of the conduction electrons. We have written Eq. /H2084910/H20850in the LL form, but it equally can be written in the LLG form. We have already shown in Ref. 15that the mag- netization dynamics of a thin ﬁlm embedded between twonormal conductors and subjected to an electric ﬁeld is notwell described by closed LL /H20849or LLG /H20850equations. Next we discuss qualitatively the effect of the diagonal and off-diagonal terms of the diffusion tensor on the velocityof a domain wall, of width /H9261. If we ignore the spatial depen- dence of the diffusion tensor elements and replace the La-placian in the diffusion equation by 1 //H9261 2, then we recover equations similar to those discussed by Zhang and Li9but with renormalized spin ﬂip scattering rate, 1 //H9270sf→1//H9270sfN =1 //H9270sf+D0//H92612, and renormalized precessional frequency, 1//H9270ex→1//H9270exN=1 //H9270ex−Dxy//H92612. Therefore, the velocity and the effective damping of the DW are dependent on the size ofthe inhomogeneities in the magnetization. This can be under-stood qualitatively from the results in Ref. 9which showed that the DW velocity vfor a wide DW, i.e., /H11612m/H110150, is in- versely proportional to the damping /H9251el/H20849in the case /H9251pd=0/H20850, v/H11015/H20849Pj/H9262B/e/H20850/H20851/H208491+/H92642/H20850//H20849/H9264m0/H20850/H20852. Then, ignoring the renormal- ization of the diffusion coefﬁcient D0, the velocity is ex- pected to take a similar form as in the case which does notaccount for the diffusion but with /H9264replaced by /H9264N=/H9270exN//H9270sfN. The damping /H9251will be also affected by this renormalization as is expected, since broadening due to inhomogeneities iswell known to occur in ferromagnetic resonance measure-ments.III. APPLICATION: 1D CASE Now, we turn to the discussion of the results of the above theory for a one-dimensional /H208491D/H20850DW conﬁguration. We solve numerically the coupled equations of motion forthe conduction electrons and that of the magnetization. Weinclude the d-dexchange between the local moments, the anisotropy along the direction of the current, and the dipoleﬁeld. Pinning is neglected but can be easily included in thesimulations. Besides varying the width of the DW, we alsovary the other parameters in the sdmodel since there is no universal agreement on their exact values. For example, it isgenerally believed that spin relaxation times are about twoorders of magnitude longer than momentum relaxation times.While this may be true in paramagnets, we already know thatin Ni 80Fe20they are comparable.24In Permalloy, the spin diffusion length, ls=vF/H20881/H9270sf/H9270p, is of the order of 5 nm which is of the same order as the mean free path, lp=3 nm. Figure 1shows the effect of introducing the /H20849unnormal- ized /H20850diffusion term D0in the equations of motion of the magnetization. For DW width larger than 100 nm our resultrecovers that of Ref. 9. The variations of the domain wall velocity vwith/H9261are found to depend strongly on D0. This is expected since vis, to ﬁrst order, a function of D0//H92612/H20849see inset /H20850. Moreover, the velocity peaks when the mean free path of the conduction electrons, lp, is of the same order as the DW width, since for lp/H11271/H9261 there is almost no scattering while for lp/H11270/H9261there is only slow diffusion. In Fig. 2, we show the effect of the corrections intro- duced by the off-diagonal terms in the diffusion tensor. Thisnonadiabatic effect actually appears to suppress the DW ve-locity or the effect of diffusion as we explained earlier. Oth-erwise, the functional behavior of the velocity remains simi-lar to the one discussed in Fig. 1. Finally, in Fig. 3, we extract the contribution of the con- duction electrons to the effective damping of the magnetiza-tion. First, we observe that the off-diagonal diffusion termshave little effect on the relaxation of Mwhich is mainly determined by the spin relaxation time /H9270sf. These results are FIG. 1. Domain wall velocity as a function of domain wall width for jc =108A/cm2,/H9270sf=1.10 /H1100310−13s,/H9251pd=0.01, and three different diffusion co- efﬁcients, D=D0I, given in the ﬁgure in units of m2/s. A value of D0 =10−2m2/s corresponds to /H9270p/H1101510−14s. The solid thick line is that of Zhang and Li /H20849Ref. 9/H20850. The inset shows the DW velocity vs D0//H92612.023701-3 Benakli, Hohlfeld, and Rebei J. Appl. Phys. 103 , 023701 /H208492008 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Wed, 03 Dec 2014 00:38:51also not strongly dependent on the DW width and the ex- tracted electronic damping has the correct order of magni-tude for metals. IV. CONCLUSION In summary, we have solved the conduction electron- magnetization problem in the presence of a current self-consistently. We found that the diffusion term provides alarger contribution to the drive torque than to the dampingprocess, leading to an overall increase of domain wall veloc-ity. We also showed that the additional off-diagonal terms ofthe diffusion tensor enhance the DW velocities which be-come at least one order of magnitude larger than previouslyfound. Moreover, the dependence of the DW velocity on the width of the DW was found to be nonlinear and stronglydependent on the nonadiabatic behavior of the conductionelectrons through the nondiagonal corrections of the diffu-sion tensor. We have been also able to determine the contri-bution of the conduction electrons to the damping in ferro-magnetic metals with domain wall conﬁguartions which wefound to be of the same order as the typical measured valuesof /H9251. Therefore, our treatment allows us to include electronic damping in micromagnetic calculations in a more rigorousway than is currently done by simply accounting for it by asimple /H9251parameter. ACKNOWLEDGMENTS We are very grateful to W. N. G. Hitchon and E. Si- manek for important early discussions related to this work.We also thank P. Asselin for useful comments. M.B. thanksL. Berger for discussions. 1L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8,1 5 3 /H208491935 /H20850;T . Gilbert, Phys. Rev. 100, 1243 /H208491955 /H20850. 2A. A. Thiele, Phys. Rev. Lett. 30, 230 /H208491973 /H20850. 3A. A. Thiele and P. Asselin, J. Appl. Phys. 55, 2584 /H208491984 /H20850. 4L. Thomas, M. Hayashi, X. Jiang, C. Rettner, and S. S. P. Parkin, Nature /H20849London /H20850443, 197 /H208492006 /H20850. 5A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 /H208492004 /H20850. 6M. Hayashi, L. Thomas, Ya. B. Bazaliy, C. Rettner, R. Moriya, X. Jiang, and S. S. P. Parkin, Phys. Rev. Lett. 96, 197207 /H208492006 /H20850. 7G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 97, 057203 /H208492006 /H20850. 8A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69,9 9 0 /H208492005 /H20850. 9S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 /H208492004 /H20850. 10G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 /H208492004 /H20850. 11E. Simanek and A. Rebei, Phys. Rev. B 71, 172405 /H208492005 /H20850. 12S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 /H208492005 /H20850. 13L. Berger, Phys. Rev. B 73, 014407 /H208492006 /H20850. 14R. A. Duine, A. S. Nunez, and A. H. MacDonald, Phys. Rev. Lett. 98, 056605 /H208492007 /H20850; Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. Bauer, Phys. Rev. B 74, 144405 /H208492006 /H20850. 15A. Rebei, W. N. G. Hitchon, and G. J. Parker, Phys. Rev. B 72, 064408 /H208492005 /H20850. 16T. Valet and A. Fert, Phys. Rev. B 48, 7099 /H208491993 /H20850. 17A. Kubetzka, O. Pietzsch, M. Bode, and R. Wiesendanger, Phys. Rev. B 67, 020401 /H20849R/H20850/H208492003 /H20850. 18H. Tanigawa, A. Yamaguchi, S. Kasai, and T. Ono, J. Appl. Phys. 99, 08G520 /H208492006 /H20850. 19S. Khizroev, Y. Hijazi, R. Chomko, S. Mukherjee, R. Chantrell, X. Wu, and R. Carley, Appl. Phys. Lett. 86, 042502 /H208492005 /H20850. 20A. Aziz, S. J. Bending, H. G. Roberts, S. Crampin, P. J. Heard, and C. H. Marrows, Phys. Rev. Lett. 97, 206602 /H208492006 /H20850. 21L. L. Hirst, Phys. Rev. 141, 503 /H208491966 /H20850. 22J. I. Kaplan, Phys. Rev. 143, 351 /H208491966 /H20850. 23A. Rebei and J. Hohlfeld, Phys. Rev. Lett. 97, 117601 /H208492006 /H20850. 24S. Dubois, L. Piraux, J. M. George, K. Ounadjela, J. L. Duvail, and A. Fert, Phys. Rev. B 60,4 7 7 /H208491999 /H20850. FIG. 2. Domain wall velocity as a function of domain wall width with the correct diffusion tensor taken into account. The solid /H20849open /H20850symbols are without /H20849with /H20850off-diagonal corrections of the diffusion tensor. Parameters are identical to those in Fig. 1. FIG. 3. The electronic damping /H9251elas a function of spin ﬂip scattering /H9270sf for a 10 nm domain wall. The solid /H20849open /H20850symbols are for off-diagonal terms included /H20849not included /H20850. The diffusion constant is D0=10−2m2/s and /H9251pd=0.023701-4 Benakli, Hohlfeld, and Rebei J. Appl. Phys. 103 , 023701 /H208492008 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Wed, 03 Dec 2014 00:38:51
1.1663819.pdf	Theory of magnetic domain dynamics in uniaxial materials J. A. Cape, W. F. Hall, and G. W. Lehman Citation: Journal of Applied Physics 45, 3572 (1974); doi: 10.1063/1.1663819 View online: http://dx.doi.org/10.1063/1.1663819 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/45/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Calculation of the magnetic stray field of a uniaxial magnetic domain J. Appl. Phys. 97, 074504 (2005); 10.1063/1.1883308 Erratum: Theory of magnetic domain dynamics in uniaxial materials J. Appl. Phys. 46, 2338 (1975); 10.1063/1.322276 THEORY OF THE STATIC STABILITY OF THICKWALLED CYLINDRICAL DOMAINS IN UNIAXIAL PLATELETS AIP Conf. Proc. 5, 140 (1972); 10.1063/1.3699410 UNIAXIAL MAGNETIC GARNETS FOR DOMAIN WALL ``BUBBLE'' DEVICES Appl. Phys. Lett. 17, 131 (1970); 10.1063/1.1653335 Theory of the Static Stability of Cylindrical Domains in Uniaxial Platelets J. Appl. Phys. 41, 1139 (1970); 10.1063/1.1658846 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.63.180.147 On: Sat, 22 Nov 2014 20:36:48Theory of magnetic domain· dynamics in uniaxial materials J. A. Cape and W. F. Hall Science Center, Rockwell International, Thousand Oaks, California 91360 G. W. Lehman University of Kentucky, Lexington. Kentucky 40506 (Received 10 December 1973; in final form II March 1974) We have calculated the total Lagrangian and Rayleigh dissipation functions for an isolated domain of arbitrary cross section in an infinite plate with perpendicular anisotropy. Variation of these functions yields a set of coupled equations describing the motion of the center of mass and the boundary R (<1» (in general noncircular) of the domain. We neglect z dependence and assume aiR 0(1 where a is the wall "thickness". The theory is applicable for applied field variations of arbitrary speed and magnitude. For uniform field pulses, the equations reduce to the Callen-Josephs theory in the weak-pulse limit. For pulses > 27rM, la, where a is the Gilbert parameter, the behavior again tends to be linear with, generally, a greatly reduced apparent mobility, while in the transition region 27r M, a < Hp < 27r M, I a, the predicted behavior is highly nonlinear with an oscillatory substructure which causes an alternating sequence of collapse-noncollapse regions in the conventional plot of inverse pulse length vs pulse height. Translatory motion of the domain in a field gradient is also highly nonlinear reducing to "effective mass" behavior only when R H' <47r M a. An approximate prediction of the theory is that regardless of the magnitude of the pulse gradient (i) the net displacement is given by x 0 '" {twR (dHldx)t 0' where {tw is the wall mobility and (ii) the minimum elapsed time for a displacement is "" T A = (M ,12K )(R l/al)(yar' , where K is the anisotropy constant and y the gyromagnetic ratio. Finally, the theory predicts a finite displacement in a direction transverse to the sense of the field gradient. I. INTRODUCTION Since the discovery and elucidation of most of the static properties of "bubble domains", their dynamical properties have been discussed largely within the framework of two distinct models. The simplest, yet in many ways most successful, has been what might be called the quasistatic wall-energy model or simply wall model (WM) for short. Here the domains are assumed to consist of regions uniformly magnetized to saturation, separated by dimensionsless "walls" from their oppositely magnetized surroundings, also uniform ly saturated. The walls are assigned a surface energy density to account for exchange energy which must ac company the magnetization reversal. The WM has been eminently successful in explaining the static properties of domains including sizes, shapes, critical points of stability; 1-9 and even much of the complex many-domain structures which arise from interaction. 9.10 WM theory has been adapted to describe the dynamics of isolated domains, essentially by associating a wall-dissipation mechanism with time variation of the "static" WM energy. 11-13 That is, if UwM(R,xo,He) is the total energy (including surface energy, external field, and demag netization field terms) of a domain of dimensions R at rest at the point Xo in a static external field He' the time rate of change of U WM accompanying a time variation in R or Xo is assumed to be dissipated (neglecting coercive effects) according tol3 dU 1 ~ =- <l!oV~ dA, dt wall area (1) where v n is the component of the wall velocity along the normal to the wall. The phenomological constant <l!o is related to the "wall mobility" Ilw by <l!o=2Ms/llw, and coercive effects may be included, if desired. Employing the assumption embodied in Eq. (1), Thiele11 first considered the 3572 Journal of Applied Physics, Vol. 45, No.8, August 1974 motion of a rigid cylindrical bubble domain in an exter nal field gradient and obtained the result 5-_ P dH., dt -Ilw,o dx (2) where Ro is a constant radius. In a similar vein, Callen and Josephs12 derived an equation of motion for an iso lated bubble domain subjected to a time-varying uniform external bias field. Their result can be expressed in the form where FWM(R) is an "effective field" deriving from the magnetostatic energy and the wall energy. (3) The common hypothesis underlying Eqs. (2) and (3) was pointed out by Cape13 who also derived a formula, similar to Eq. (2), for strip domains in the course of an experimental study of the motion of bubbles and strips in garnet films subjected to oscillatory and pulsed gradient fields. In what follows, we shall see that Eq. (1) can be obtained as a very crude approximation to a complete Lagrangian-Rayleigh dissipation theory from which all kinetic effects have been omitted. The second popular theoretical approach to domain dynamics is that based on what Brown14 has called "micromagnetics". For our purpose this means the equations of motion are derived from variational principles applied to the energy of a system in which the local magnetization vector M(r) is considered to have constant magnitude Ms while itsdirection cosines are continuous func;tions of the coordinates r. The do main walls are now not imaginary dimensionless boundaries but finite regions across which the magneti zation reorientation is accomplished in a continuous fashion. The application of micromagnetics thus far, totaling a rather large number of papers, has been Copyright © 1974 American Institute of Physics 3572 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.63.180.147 On: Sat, 22 Nov 2014 20:36:483573 Cape, Hall and Lehman: Magnetic domain dynamics directed almost entirely towards the dynamics of an isolated domain-wall segment, 15-20 i. e., the wall does not enclose a finite region of spin reversal (for example, a bubble domain) nor is there a pair of walls so as to clearly define a striplike domain. The static limit of an open single-wall configuration is not a stable state of any homogeneous medium, except in the unusual cir cumstance that the imposed external field is strongly graded from large negative values to large positive values. A single wall may then form along the line of zero field. 21 Our principal criticism of the micromagnetics wall dynamics formulations to date is that they are funda mentally incomplete insofar as they do not describe the actual ground or excited states of the system [contrast with Eq. (3), the solution of which for dR/dt=O gives precisely the equilibrium static bubble diameter]. In particular these calculations have not treated adequately the magneto static self-energy (the demagnetization field) which depends specifically on the actual configuration of the domain. On the other hand, the wall model must be faulted at a more fundamental level. Having structure less walls, it cannot provide for changes in the wall structure with varying velocity (and cannot, therefore, account for complex metastable structures such as "hard" bubbles). Moreover, the wall model is an es sentially linear theory inasmuch as it assigns a strict proportionality to the viscous drag (dissipative) force and therefore cannot provide for nonlinear high-velocity effects. On the positive side, the wall model treats correctly the magneto static effects and yields the correct gross features in the static limit. Not surprisingly, the wall model is in good agreement with experiment as long as the wall velocities are not too great. In this connection may be cited the low-field bubble-collapse experiments originated by Bobeck et al. 22 The low-field reciprocal collapse times are linear in the collapse field amplitude as predicted by Callen and Josephs12 on the basis of their analYSis of Eq. (3). Similarly, Cape13 has shown that the response of a bubble and a strip domain to an oscillatory gradient field is consistent with Eq. (2) if reasonable values for the mobility are assumed, and if an allowance is made for the coercivity. On the other hand, bubble-collapse experiments carried to high pulsed field values exhibit a strong nonlinear saturation with, in some instances, a region of negative differ ential mobility. 23 This is not explained by the wall model and presumably the explanation must be sought in micromagnetics. As already mentioned, the micro magnetics problem has not been solved for a finite domain, but a great deal of insight has been gained by the studies performed thus far. Doring's16 early calcu lation ascribed an effective mass to a moving wall. More recently, Lehman24 has calculated the first-order correction term to Eq. (2), i. e., an additional term in x reminiscent of the effective mass result. Walker's17 well-known result predicts an upper velocity limit for uniform wall mption. Slonczewski's20 extensive calcu lations have brought out the possibilities of nonuniform motion, oscillatory behavior, and negative differential mobility. His results, as well as those of Walker and Doring, apply to isolated domain walls and as such can- J. Appl. Phys., Vol. 45, No.8, August 1974 3573 not be compared with the experiment. What is needed is a complete micromagnetics theory formulated for an idealized "true" domain structure and reducing-in the limit of small velocities-to Eqs. (2) and (3). This is what we set out to achieve in the following. Some of our results, specialized to the case of stationary circu lar "normal" bubble domains, have been presented previousl y25 in connection with the bubble-collapse problem. II. LAGRANGIAN FORMULATION Consider a uniaxial ferromagnet of saturation moment lVIs in the form of a flat plate of thickness h and of in plane cross-sectional area OJ>' The magnetization at the point r and time t is given in cartesian and cylindri cal coordinates, respectively, by (see Fig. 1) lVI x=lVI. sinU cos V, lVIp=lVIs sinU cos(V -cp), lVIy=lVI. sinU sinV, lVIq, =Ms sinU sin(V -cp), (4) M .=lVI. cosU, lVI.=Ms cosU, where in general U and V are functions of p, z, and t in a cylindrical coordinate system fixed in the plate and R(cp, z, t) is the locus [defined by MI/(r, t)=O] of the do main wall in the body coordinate system moving with the domain and centered at Po(t) [defined by lVI .(Po' z, t) =lVI.]. Our prescription for calculating the equations of motion follows the concepts evolved by Gilbert26 and elaborated in detail by Brown14 in his books and papers. The starting point is the Lagrangian density14 L(r, t) = T(r, t) -G(r, f), (5) where the kinetic function ) lVI. a T(r, f :; -y' V(r, t) at cosU(r, t) (6) contains the inertial properties of the spin system (y is the gyromagnetic ratio) and the potential function G, for a z-uniaxial medium, can be written G(r, t) =A[(VU)2 + sin2U(VV)2] + K sin2U -~M· Hm -M· He' (7) where A and K are, respectively, the exchange and anisotropy constants, H. is an externally applied field, and Hm(r, t), the "demagnetization" field, is given by Hm=-VCP; where the integration extends over the entire plate volume OJ>. If damping of the spin system is ignored, variation of the time-averaged total Lagrangian 1t2 r . L = dt Jo dp [T(U, V, U) -G(U, V, VU, VV, t, p)] tl P leads to the equations of niotion (8) (9) (10) (11) It is straightforward to show that Eqs. (10) and (11) are [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.63.180.147 On: Sat, 22 Nov 2014 20:36:483574 Cape, Hall and Lehman: Magnetic domain dynamics FIG. 1. Schematic of a magnetic domain viewed along the uni axial direction. A cylindrical coordinate system (p, </», the "bubble" coordinates. moves with the domain following the trajectory PoW in the laboratory (plate) coordinate system. The domain boundary presented by the dashed curve is given by P = R(</>, t), and the domain wall is of "thickness" 6 (r!> , t). equivalent to the familiar torque equation (12) where Heff is the variational derivative of G with respect to M.14 Dissipative effects are incorporated by intro ducing the Rayleigh dissipation !unction14.26 (13) where a is the "Gilbert damping parameter". The equations of motion are then modified by the addition of (a/aiT)j(r, I) to the right-hand side of Eq. (10) and a similar term from a/af to Eq. (11). While in principle the domain configuration can be determined from the Euler equations, we proceed by assuming suitable analytic forms for sinU(r, t) and cosU(r, t). A great simplification is achieved by as suming M to be independent of z. We shall make this approximation for the remainder of the present dis cussion, leaving such matters as "twisted" walls27 to a subsequent discussion. By analogy with the familiar structure for long planar Bloch walls, we assume sinU = sech[A(r, cp, t)], (14) where A", [p -R(cp, l)]ja(cp, f). (15) For simplicity, we further assume V to be independent of p for fixed cp. The "radius" R( cp, t) defines the boundary, i. e., the locus of the domain wall, in general noncircular, and a(cp, f) measures the wall "thickness". Equation (15) leads to an unphysical singularity in the exchange energy at p = O. De Bonte28 has discussed how this singularity may be removed for circular domains. For more general domain structures, the success of such a procedure requires the existence of a point Po( t) J. Appl. Phys., Vol. 45, No.8, August 1974 3574 in the (x, y) plane where Mq"Mp=O and M.=Ms' The point Poll) we take to be the origin (see Fig. 1) of a co ordinate system moving with the domain. Inasmuch as R, a, and V are functions only of cp and t, we will make use throughout of the convenient notation oR and R' ",-(jcp (16) with similar notation for the derivatives of a(cp, t) and V(cp, t). A. The demagnetization energy The demagnetization field Hm is given by Hm=-Vr<I>=-vri dr' M(r',t)· V" Ir-r' 1-1. (17) Since M = M(p) by assumption, we may write <I> = inp dp' M .(p' , t) x {[(z .... W + (p + p')2]-1/2 _ [Z2 + (p _ p' )2]-1/2} + ~ dp' ML(p', t)· VL t dz' [(z -Z,)2 + (p _p,)2]-I/2. p utilizing Eq. (18), one may show that -t i dr M· Hm = W. + WL, where Wz'" I-dp J, dp' M.(p, f)Mz(p', t) rip rip x[lp_p' I-I_(lp-p' 12+h2)-I/2], (18) (19) (20) which reduces to the demagnetization energy as calcu lated in the wall model9 when Mz=M., and WL "'t J dp f, dp' [ML(p, t)· vJ rip rip (21) x [ML(p', t). V~lK( Ip -p' I), which derives from the in-plane magnetization and is, therefore, absent in the wall model. In Eq. (21), K(p) '" 2{h 10gJh/p + (h2 + p2)1 /2/p] + p _ (h2 + p2)1/2} (22) and o V = L -op' a V'",-· LOp' For circular bubbles with R »a, we show in the Appendix that WL = 21Th in dp [M/ cp, f)]2 + O{[(a/R)2lnR la-I}. p utilizing Eqs. (14) and (15), from which Mp =M. sechA cos( V -cp) we obtain for circular bubbles (23) (24) WL = 41TM;hR tv dcp a( cp, t) sin2</J( cp, f), (25) o where, for convenience, we have introduced the variable 1]J '" cp -V + t1T (see Fig. 2). A more general, but less rigorous derivation of WL, valid presumably for noncircular as well as circular bubbles, can be obtained by the following argument. If the magnetization varies slowly along the domain wall over a region large compared to a but small compared to the wall curvature, it is intuitive that only the in- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.63.180.147 On: Sat, 22 Nov 2014 20:36:483575 Cape, Hall and Lehman: Magnetic domain dynamics plane component normal to the wall, M~(p, </>, t), con tributes to the demagnetization field, and that the contribution is Hm:" -41TM~(p, </>, t). To put it another way, all "poles" due to Ml tend to cancel except those associated with the magnetization component normal to the wall. Accordingly, we take W1 = -t f dp f dz M1• Hml :::::21Th f dpM~. (26) To evaluate M~, we note that~, the unit vector nor mal to the domain boundary at p = R( </>, t), is given by il = [1 + (R' /R)2]-1/2 lip -(R' /R)iq,], (27) where ip and i0 are the radial and azimuthal unit vectors in circular cylindrical coordinates. We then obtain M,=il·{ioM p +~M(!>} with Mo and M(!> given by Eq. (4). Performing the in dicated operations we obtain the result W1 = 41ThM! f~ d</> o( </>, t)R( </>, t) x [1 + (R' /R)2]-1 [sinlji -(R' /R) cosl/J)2 which reduces to Eq. (25) for circular bubbles. (28) B. Exchange and anisotropy terms To evaluate the second term in the exchange energy and the anisotropy term, we make use of ('1U)2 = sech2 A('1A)2 = 0-2 sech2A[1 + (R' + M,)2 (R + Mt2], (29) hence, .( dp p('1U)2::::: 0-1 I: dAsech2 A(R + M +Wl(R' + M')2]. (30) In writing Eq. (30) we have extended the lower limit of integration from -R/o to -00 and set p =R in the co efficient of the term (R' + ... )2. The reason for doing this is not simply because R/o is assumed to be large, though we do assume this, but because of the recognition that the "true" function for Mp (which we approximate by Mp ==Ms sechA), must vanish at p== -0 (A == -R/o) as was pointed out above. In fact, it can be seen that the function must vanish at least as fast as p2 = (R + AO)2. De Bonte28 has shown that an appropriate behavior is obtained if we redefine A == (p -R)/o + lnp/R. The resulting corrections to the energy terms are negligible, however, when exp(R/o)>> 1. Taking this inequality (as assumed), we obtain to first order in o/R r dp p('1U)2::::: 2(R/o){[1 + (R' /R)2]}. (31) o Similarly, the exchange and anisotropy terms give C. Kinetic terms We must evaluate i Mh£ a dp T(p, t) == -~ dp V(p, t) at cosU(p, t). Op Y flp Noting that (see Fig. 1) J. Appl. Phys., Vol. 45, No.8, August 1974 3575 a I (0' ) I -cosU = --po' V cosU at plate of P bubble' (33) where the derivatives on the left-and right-hand side are with respect to the fixed (plate) and moving (bubble) coordinate systems, respectively, we obtain a I ...] atcosu I = -sech2 A{[o + p-l(xo sin</> -Yo cos</>W M-1 plate " • • + o-I[R + Xo cos</> + Yo Sin</> + p-l(yo cos</> -Xo sin</»R ]}. (34) Avoiding the singularity at p == 0 as discussed above, the integration is straightforward and yields f dp T(p, t) = (2M .. hjy) t" d</> V( </>, t)[R( </>, t)(Rp ~ 0 + Xo cos</> + Yo sin</»] +0(1j2/R2)+O[exp(-R/o)], (35) where Rp ==R + WI R'(xo sin</> -Yo cos</» is our abbreviated notation for the derivative in the plate coordinates. D. Dissipative terms Utilizing Eq. (15) and noting that (aU)2 2 (a )2 at =csc U at cosU , we obtain as above (36) (37) 1 (OU)2 f2' [R.. . ] dp at = h d</> 2 a (Rp + Xo cos</> + Yo sin</»2 op 0 + ° [(iYl (38) Similarly i (aV)2 fh . dp sin2U at = 2h d<pR(</>, t)o(</>, t)y;, Up 0 (39) where Vp:; V + Wl V'(XO sin</> -Yo cos</» (40) is the time derivative of V in plate coordinates. Com bining the quantities derived above, we obtain finally f drG(r, t) == 2h ir d</>R(</>, t) {[K +A('1V)2]O(</>, t) o + [A/o( <p, t)] [1 + (R'/R)2] + 21TM~o(</>, t) [1 + (R' /R)2]-1 [sinlji -(R' /R) coslji]21 ( 41) where we have put W H = -f dr M· He' The kinetic energy is given by Eq.e (35), and the Rayleigh diSSipation function is f dr J (r, t) = (aMsh/y) fo2. d</> ([R( </>, t)/o( </>, t)) x [Rp( </>. t) + Xo cos</> + Yo sin </>]2 +R(cp, t)O(</>, t)Y;(<p, t)}+O[(o/Rj21. (42) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.63.180.147 On: Sat, 22 Nov 2014 20:36:483576 Cape, Hall and Lehman: Magnetic domain dynamics III. THE WALL-MODEL LIMIT Since the wall model generally omits inertial effects, we must restrict it to slow ("quasistatic") wall ve locities_ We, therefore, assume that the spins at the center of the domain wall, R(cf», are always tangent to the wall, i. e., we put tall</!=R' /R [see Eq. (56b)J. This makes the term Wi vanish. Similarly, the terms in K and A are minimized by 0= {(A/K) [1 + (R' /R)2]y /2 (43) with which the 0 terms combine to give W wall = 4(AK)1 /2 h rr dcf> R( cf» [1 t (R' /R)2Jl/2 =4(AK)1/2S=O"wS, (44) where S is the wall area of the domain and O"w=4(AK)1/2 is the wall-energy density. These terms thus give the wall-energy term of the wall model. 9 The kinetic terms are normally completely absent in the wall model and dissipative effects, as discussed in Sec. I, are included only in the ad hoc fashion embodied in Eq. (1). Note that Eq. (1) for a circular bubble implies a dissipation function f dr ]wM=(aMsh/r) t.r dcf> (R/o)R2 (45) Q which is an obvious approximation to Eq. (42). The quantities W. and W H diverge if evaluated for a domain embedded in an infinite plate. One therefore substracts the energy of the "saturated" (no domain) plate to obtain a finite result as has been done previously for the wall model. To accomplish this, we note that M. can be written M .=M. -M. [1-cosU(p, t)J =M. -M.[l-tanhA(p, t»). (46) Substituting into Eq. (20) and abbreviating Q -={IP -p' 1-1_ [(p -p,)2 + h2J-l/2} we obtain W.=M: 1 dp 1. dp' Q(p, p') Op Op -2M2 J dp (1 -cosU) 1 dp' Q(p, p') & 1'1 OJ, +M2 1 dp 1 dp' (1-cosU)(l- cosU')Q(p, p'), (47) & 0 0' where the first term is clearly W!,at the self-energy of the uniformly magnetized plate. In the remaining terms, since the factor 1 -cosU is zero outside of the domain volume 0, the corresponding regions of integration are reduced from Op and O~ to 0 and 0' . Similarly, the external field energy can be written f dpM.(p)H.(P)=M& fndpH,(P) -Ms 10 dp (1-cosU)H.(P), (48) where the first term on the right-hand side is ~at. e J. Appl. Phys., Vol. 45, No.8, August 1974 3576 The wall-model limit obtains as 0 -0, i. e., when cosU=tanh[(p-R)/o]- ±1 for p~R, respectively, In this limit, since fo dp' Q(p, p') = 2rrN o(p) is the "demagnetization factor" of the volume 0, we find (W. + WH )6-0= -8rrM! 1 dp [Np(p) -N Q(p)] e Q which is the wall-model result obtained previously [Ref. 9, Eq. (9)]. The wall-model limit of the remaining terms in G obtains if we neglect 6(VV)2 compared to 0-1 and (20/acf»2 compared to (2R/acf»2. If we denote the wall-model limit of W. by U M (to agree with the notation used previously9,13,25) it is straightforward to show that W.= U M[1 + O(02/R2)]. (50) To see this, we put l-cosU=2[1-I1(A)]+w (where 11 is the step function) and find W.=UM+2M!hf dtj>J dppw(p,cf»=UM+U l (51) with w(p <R)= -(1 +tanhA) and w(p >R) =tanhA. The integral on the right-hand side, Up is evaluated readily giving U1 =~h.r dcf>02(cf» [2 -exp(-R/o) + ... ] (52) which is smaller than U M by a factor of -o2/R2. We are therefore able to use the simpler form U M in place of W. in the remainder of this discussion. In particular, for circular bubbles, a convenient approxi mate form for U M' given by Callen and Josephs, 12 is quite useful. 25 In summary, we see that we obtain precisely the energy function of the wall model9 if we (i) ignore kinetic terms, (ii) retain terms to order o/R only in the magnetostatic energy, and (iii) assume tan1/!=R' /R, i. e., the magnetization has no normal component at the center of the domain wall. IV. EQUATIONS OF MOTION The Euler equations are derived from Eqs. (41) and (42) as described above from variation of the Lagrangian and the dissipation function. We define _ -1 auM· F u~ (4rrMshR) --aB' (53) 2. ()=(21T)-1 J dcf> ( ), (54) Q and cose -= [1 + (R' /R)?)-1/2 (55) and retain terms to first order only in olR (except when this quantity is multiplied by potentially large quantities such as V'). We are led to the set of equations (56a) (56b) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.63.180.147 On: Sat, 22 Nov 2014 20:36:483577 Cape, Hall and Lehman: Magnetic domain dynamics 3577 ;1T £ dP~~~~= -(a~ (v O~ (R sincp»)) -a(~ (1'~p + xocoscp + Yo sincp) oOcp (R sincp»), ~~ dp a~e =-(o~ (v a~ (RCOSCP»)) + (~(ilp+XoCOSCP+YoSincp) o~ (RCOSCP»), (56c) (56d) and Equations (56a)-(56e) are the equations of motion for a single domain of cross section n [enclosed by R(cp)] and describe, in principle, both translatory and radial motion. The equations are approximate mainly in the respect that z dependence has been ignored and that 6/R is assumed small compared to unity. V. QUALITATIVE FEATURES OF BUBBLE COLLAPSE Equations (56a)-(56e) reduce to those presented previously25.29 for bubble collapse in a uniform field, if we take Rand <P independent of cp (normal circular bubbles) and take the bubble center to be stationary (x, y, VHe = 0). We obtain then two coupled first-order differential equations which govern the time development of Rand 1/J: (1 + a2)~=y[He- F(R, 1/J)] -~41TMsya sin21/J, (57a) (1 + a2)R = -y6a[He -F(R, 1/J)] -~41TMsy6 sin21/J. (57b) Here 1/J defined by Fig. 2, is, for normal circular bubbles, the angle by which the magnetization vector at the domain wall tips away from the wall tangent and F is the wall-model function of Eq. (3), plus a small term proportional to sin21J!.25 In order for Eqs. (57a) and (57b) to have a time independent solution, He must be smaller than the maximum value of F. In that case, the bubble radius and wall magnetization at which the system can rest are determined by the conditions He-F(R, 1/J) = sin21/J = O. In the interval 0 "" 1/J < 21T there are eight such solutions, two at each multiple of ~1T. Only two of these are stable, namely, the larger-radius solutions at ?/!=O and 1/J=1T, corresponding to right-and left-handed bubbles, respectively. These solutions are stable in the sense that a small perturbation in R or 1/J from their stable point values will die away in time. The smaller-radius solutions at 1/J = 0 and 1T and the larger-radius solutions at 1/J = ~1T and ~1T are saddle points. If the values of Rand 1/J for a particular bubble lie in the vicinity of such a point, these values may evolve initially either toward the saddle-point values or away from them, but their further development eventually takes them away from SA, unless the initial values lie on one of the two exceptional trajectories which end upon the saddle point. The remaining two solutions, which lie at 1/J = ~1T and ~7T, are source points: small perturbations from equilibrium grow in all directions. Finally, F turns negative at suffiCiently small R, with the consequence that bubbles of smaller radii will collapse spontaneously. These properties of the motion of the bubble radius R and the wall magnetization angle 1J! can be conveniently summarized by plotting the bounds of all possible trajectories R(1/J) allowed by Eqs. (57a) and (57b), using J. Appl. Phys., Vol. 45, No.8, August 1974 (56e) IR as the radius and 21J! as the azimuth angle [we use 21/J rather than 1J! because the right-hand sides of Eqs. (57a) and (57b) have period 1T in 1/J]. This has been done in Fig. 3 (from Ref. 25) for the particular case a = 0.02, 6 = 5X 10-6 cm, 41TMsY = 5. 6X 109 sec-I, and we have taken the Callen-Josephs12 approximate form for U M' We then obtain F(R, 1/J) = 41TMs{(1 + ~R/h}l -(V2R)[1 + (21TM~/K) sin2(/J]1/2} (58) and we have taken ~ = 2. 5x 10-4 cm, K= 105 ergcm-3, and h= 10-3 cm for Fig. 3. In this figure the usual stable bubble configuration with 1J! = 0 (or 1T) is labelled SI (for sink), and its radius is taken as unity for con venience. Similarly, the saddle pOints (at R = 0.3, 1J!=0 and at R=O. 9, 1J!=t1T) are labelled SA, and the source point at R =0.3, 1/J= ~1T is labelled SO. The solid lines connecting these points are the trajectories in R and 1/J which a bubble must follow to either start or end upon a saddle point; these lines divide the plane into four distinct regions, denoted II-V (I and II are ob viously connected). Because solutions of Eqs. (57a) and (57b) cannot cross, a trajectory beginning in one region must eventually end in that region; for zones II-IV, this """ """ '-... \ \ SPIN DIRECTION AT WALL CENTER \ \ WALL TANGENT ....... \ .p~ '-... " 8 -p -------------+~,~~--~--L-------x FIG. 2. The moving coordinate system (p,1» ("bubble coordi nates") centered at (xo' yO> and the variables of the dynamical theory. The angles V, ¢, and 9 measure, respectively, the spin direction, at the wall center, relative to the positive x axis, the inclination of the wall-center spins from the perpen dicular to the radius vector, and the angle between the wall tangent and the perpendicular to the radius vector. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.63.180.147 On: Sat, 22 Nov 2014 20:36:483578 Cape, Hall and Lehman: Magnetic domain dynamics I FIG. 3. Bounds of the possible R(~) trajectories following termination of the field pulse. Cross-hatched area is the re gion of bubble collapse. Note that the critical collapse radius depends on the initial ~ value which is a high-frequency func tion of the magnitude of the field pulse. endpoint is the stable configuration SI, while for the "whale"-shaped shaded region V, it is the origin, i. e. , trajectories within the zone V lead to bubble collapse. The boundaries of this region of bubble collapse are formed by the four solutions of Eqs. (57a) and (57b) which start from the source point SO and end on the inner saddle point SA. For typical values of the material parameters, these solutions spiral outward from the source point, so that trajectories converging to the stable point SI are closely interwrapped with those collapsing to the origin. Consequently, in the vicinity of SO, zone V is interwrapped with the stable zone, region III, forming a spiral "whale's tail" on the collapse zone. In the usual bubble-collapse experiments, an initially stable bubble is made to contract by increasing the field He beyond the maximum value of F. From Eq. (57a), one sees that if the excess field Hp is small com pared to WMOI/2y (= 27TM 01), l/! will remain near its initial value (0 or 7T) and ~ can be neglected, with the result that R decreases according to Eq. (3). If the excess field is removed before R has decreased below the saddle point radius, which marks the boundary of the collapse region at l/! = 0 and 7T, then the bubble will simply return to its original size. For this range of excess field, there appears to be a well-defined critical radius for bubble collapse, and a straightforward analysis of Eq. (3) confirms that the time required to reach this radius is inversely proportional to the magnitude of the excess field. 12 However, for excess fields of the order of 27TM 01 or larger the situation be comes complicated because l/! no longer can remain near its initial value. Under the influence of a large excess field, l/! will necessarily be large. In fact, from Eqs. (57a) and J. Appl. Phys., Vol. 45, No.8, August 1974 3578 (57b) one can show that i/J will rotate through 27T radians for a decrease in R of only 27T00I. Thus, in a cylindrical coordinate plot such as Fig. 3, the point representing the bubble will spiral inward, cirCling the origin many times before its trajectory crosses into the region of bubble collapse, an intersection which may now occur anywhere along the collapse boundary, the whale of Fig. 3. Furthermore, because this boundary is not Circular, and because the term in sin2l/! in Eq. (57b) imposes an oscillation on R, continued application of the excess field will carry the bubble back out of the collapse re gion, and then into it again, and so on, until the bubble radius has finally decreased far enough below the smallest radius on the boundary that the oscillation in R cannot take it out again. The consequences of this behavior for the usual plot of inverse pulse length against the magnitude of the excess field pulse required for bubble collapse are as follows: At any excess field magnitude greater than 27TM 01 there will be several distinct bands of inverse pulse length corresponding to bubble collapse, separated by bands in which the bubble returns to its original radius, with a final collapse interval extending from some value below the inverse pulse length required to pass inside the smallest critical radius, downward to zero. At very high fields, such that Hp»27TM/0I, the oscillation in R can be neglected and the boundaries of these bands become straight lines on the plot of inverse pulse length versus pulse magnitude. At fields H p between 27TM 01 and 27TM / 01 the oscillations in R will cause the bands to oscillate up and down with a period H given roughly by AH/H"'27T00l/t:.R, where R is the difference between the initial bubble radius and the radius at which the path of the bubble intersects the boundary of the collapse region. In the light of the foregoing qualitative predictions, it appears that the results of Vella-Coleiro et al. , 30 in which a band of reciprocal pulse lengths were observed, may be under stood without invoking hard bubbles, though that possibility is not ruled out. For excess fields smaller than 27TM 01, >jJ no longer rotates beyond 27T and the trajectory of the point representing the bubble becomes nearly radial, with the consequence that it generally intersects the bound ary of the collapse region only once. (For an entry near a source point, several intersections may still occur because of the spiral shape of the boundary in that vicinity. ) Thus, the bands eventually coalesce into a Single interval whose boundary becomes linear at small excess fields. In the small i/J limit, (sin2i/J -2i/J) if a step increment in He is applied at t=O, Eqs. (57a) and (57b) predict an instantaneous increment in ~ at t = 0 damping quickly to zero with the time constant TM "'(1 + 0I2)/2wMOI. In the low-pulse-field bubble-collapse limit (Hp« 27TM 01) the collapse time to is generally much greater than T M' Hence, i/J is sensibly constant during collapse. It can be seen that the approximation ~ = 0 reduces Eqs. (57a) and (57b) to a single equation in k which is precisely the Callen-Josephs equation12 if we identify iJ.w,=yO/Oi. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.63.180.147 On: Sat, 22 Nov 2014 20:36:483579 Cape, Hall and Lehman: Magnetic domain dynamics VI. TRANSLATIONAL MOTION If a gradient is superimposed on the external mag netic field, a translational force will be exerted on the domain. Taking the origin of plate coordinates at the initial center of the bubble, a gradient in the x direction can be represented by dRe H =H +x-, e 0 dx (59) where x=xo + p coscp, Ho is now understood to be con stant and dR/dx may be turned on and off. If dR/dx is sufficiently small, the equations of motion [Eqs. (56a) (56e)], can be linearized by treating the departure from R = Ro' IjJ = 0 as small and ignoring products of small quantities such as (R -Ro)1jJ or IjJdH/dx. The linear equations admit solutions of the form R =Ro + ~ coscp + 1'/ sincp, and ljJ=u coscp + v sincp + w. Defining wM = 47TMsY' wA = 2yA/MsR~ Mw=ya/a, /: ~ [Mwe:;) R -~J1, o we find the steady-state solutions w=O, Ro=O, ~=MwRo(:e) (;J2/:WM(WM+WAt\ 1'/ = (RO/aa)(wA/w M), v=2Mwa :e (WM+WAt1 [1+(;J2 (;u:)] which is generally very small compared to u, • dHe xo= -MwRo fiX' and (60) (61) (62a) (62b) (62d) (62e) (62f) The steady state of ~,1'/, u, v, xo' and Yo is approached approximately in proportion to the factor [1-exp(-tiT)], where T=(l + a2)j(wM+wA)a is a characteristic time for the system. Similarly, when the gradient pulse is turned off, these quantities damp to zero exponentially in the same fashion. An interesting prediction is that during the period of the pulse, the bubble is displaced along the y axis in a direction which may be rep resented by mX VHe, where m is the magnetic moment of the bubble. The physical cause of this very small displacement is not clear except that it is a result of the kinetic properties and not of V(M· H) forces. Significantly, the displacement implies a transverse velocity component during the application of the pulse. J. Appl. Phys., Vol. 45, No.8, August 1974 3579 Possibly this is the explanation of the transverse motion reported earlier. 13.30 For small gradients, the time to required to displace a bubble a reasonable distance-say one or more radii-should substantially exceed the damping time T; the net motion is given very nearly by .uo '" -MwRo(dRe/dx)t o' which is in agreement with Eq, (2). When large pulse gradients are imposed, the non linear behavior of the equations must be taken into account. Qualitatively it is possible to argue that in the high pulse limit, the translation of the bubble occurs almost entirely after the pulse is over; i. e., the motion is more analogous to impact and recoil. This con clusion can be drawn from the following approximate and highly speculative argument. If we assume that, as in the weak-pulse limit, the predominant term in IjJ is u coscp, and that R is again sensibly adiabatic, we obtain the equation of motion for u: (1+a2}it =YR: -aWm~1+(~)I2+e~a)I3+(::)ul (63) where /1, 12, and 13 are dimensionless integrals of the order of unity. In response to a a-function pulse in dH/dx, u rises "instantly" in time to the value uo=yR(dH/dx)t o(1 + a2)"1, hence in general if W AyRo(dH/dx)to[wM(l + a2)]-1» 1, the terms in II to 13 may be dropped and the resulting equation is again linear. This approximation leads to the result (1 + a2)xo = -a2 M",R :~ -W A au ( dH a.) = -M",R-+-u (1+a2) dx a . In response to a a-function pulse, we then obtain for t <i; to (1 2) 2 dHe + a Axo = -a M wRo -d to' < x (64) (65) After the pulse is over, u "unwinds" slowly according to the linear equation (1 + a2)u = -W A au, which governs the motion until (wA/wM)u-1 after which the damping is accelerated (wM» wA). Consequently, the motion in xo' after the pulse, is given approximately by (1 + a2).:ho = -M",Ro dRd e to > x (66) and the total displacement is dRe Axo = -l.I.ufio ax to (67) which is the same as the weak-pulse result. In the strong-pulse limit, however, only a fraction 02/( 1 + a2) of the motion occurs during the pulse, the rest of the motion coming afterward is a result of inertia. If this analysis, with the assumption 1]; "'u coscp (we have also neglected Yo in the strong pulse case), is reasonably accurate, an interesting conclusion is presented. That [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.63.180.147 On: Sat, 22 Nov 2014 20:36:483580 Cape, Hall and Lehman: Magnetic domain dynamics is, that in response to a pulsed gradient, no matter how intense the pulse, the time of the motion is of the order of TA which may be written TA = (M&/2K) (R2/052) (1/ya), and is '" 10-2_10-4 sec for typical bubble materials. This conclusion may in some sense indicate limiting transit times in bubble propagation and, thereby, provide guidance in the selection and design of materials with a view to optimizing data rate in bubble-domain device applications. APPENDIX Consider where K(p) = t dz .r: dz' [(z -Z,)2 + p2]-1 /2. (A2) o 0 In cylindrical coordinates, S may be expressed as the sum of three terms: (A3) (2r 100 Su =.10 dcp 0 dpMI/>(p, cp) a [r 1" a Xacp 0 dcp' 0 dP'MI/>(P',CP')acp,K(P-P'), (A4) (A5) We shall see that the largest contribution to S for a circular bubble with wall thickness 6 «R comes from Spp, and is of the order of o5/R. Rewrite Spp = J dp M p(p, cp) x a~ f dp' {M p(p', cp) + [M ip' , cp') -Mp(p', cp)]} a x ap,K(P -p'); (A6) the first term will be identified as S p' while the re mainder, arising from the bracketed difference, will be called IlS p' For S p' the cp' integration can now be done analytically, using the Fourier trnasform ex pression for K: [00 a 100 Sp= dpMp(p, cp) ap dp' p'M/p', cp) o 0 X [_ 41Th ioo dk (1-[1-e~~(-hk)]) JO(k)J1(kP')} (A7) Making use of the identity31 .( dkJo(kp)Jl(kp')=O, P' <p J. Appl. Phys., Vol. 45, No.8, August 1974 3580 =l/p', P'>p, (A8) one finds that Sp= -41Th J dpM!M! -41T J dpMp(p, cp) x J 00 dp' p'Mp(p', cp) Joo dk [1-exp( -hk)]Jj(kp)J1(kp'). (A9) The first integral above may be evaluated using the wall-model form [see Eq. (26) and the accompanying discussion] Mp =M sech[(p -R)/o5] cos(V -cp), with the result -41Th J dpM~= -81TM2MR [j:r dcp cos2(V -cp) + O(o5/R)], (AIO) the terms of higher order arising from the departure of (20)-1 sech2(x/o) from a Dirac 6 function. The re mainder of S p' because of the two p integrations, is easily shown to be proportional to (0/R)2 in the wall model. However, the k integral in Eq. (A9) possesses a logarithmic singularity at p' = p which, on integration, multiplies (05/R)2 by In(R/o). This is the dominant cor rection; its form can be deduced using the identity31 r dk exp(-hk)J1(kp)Jl(kp') o _( 2 ,)-1/2Q (h2+p2+P'2) -1T PP 1/2 2pp' , (All) where Ql/2 denotes the legender function of the second kind of order~. For h = 0, the asymptotic form of Eq. (All), as p' -p, contains the term (pp' )-lln I p -p' I. The remaining term in Spp, namely, IlSp can be re arranged to read 1 12< f2' 100 100 IlSp=-2 0 dcp 0 dcp' 0 dpp 0 dp' P' xrMip, cp)-Mp(p, cp')] a2 x [Mp(p', cp) -Mp(p', cp')] ap ap' K(p -p'). (A12) In the wall model, each bracketed term is proportional to cos [V( cp) -cp] -cos[V( cp') -cp'], which can be pre sumed to vanish at least linearly in cp -cp' as cp' -cpo A straightforward computation using the expression for K in Eq. (A2) shows that the most singular term in a2K/ap ap', ariSing from the In Ip -p' 1 in K, is h{2R2 sin2[~(cp -cp,)]}-l at p=p' =R. Thus, the angular integration is bounded, while the two p integrations make IlSp proportional to (0/R)2. The expression (A4) for Su may be rearranged in a similar fashion, giving Su = -~ Ia2 < dcp t'dCP 102< dp ;:' dp' [Mp(p, cp) -M I/>(p, cp')] X [MI/>(p', cp)-MI/>(p', cp')] acp~2cp' K(p-p'). (A13) Once again, in the wall model the bracketed terms should vanish linearly in cp -cp' as cp' -cpo Also, the most singular term in a2K/a cp a cp' is proportional to hsin-2[~(cp-cp')]atp=p'=R, soSI/>I/>' justasllSp, will be proportional to (0/R)2. Finally, computation of a2K/ap acp' reveals that this kernel posses no singularity as p' -p. Thus, SPI/> will also be proportional to (O/R)2. This completes the dem- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.63.180.147 On: Sat, 22 Nov 2014 20:36:483581 Cape, Hall and Lehman: Magnetic domain dynamics onstration that for circular bubbles in the wall model S is given indeed to lowest order in (5/R) by Eq. (AIO). IR.C. Sherwood. J.P. Remeika, andH.J. Williams, J. Appl. Phys. 30, 217 (1959). 2C. KOoy and U. Enz, Philips Res. Rep. 15, 7 (1960). 3A. H. Bobeck. Bell Syst. Tech. J. 46, 1901 (1967). 4C. D. Mee, Contemp. Phys. 8, 385 (1967). 5Z. Malek and V. Kambersky, Czech. J. Phys. 21, 416 (1958). 6C. Kittel, Rev. Mod. Phys. 21, 541 (1949). 7J. Kaczer and R. Gemperle, Czech. J. Phys. 10, 505 (1960). 8A.A. Thiele, Bell Syst. Tech. J. 48, 3287 (1969); J. Appl. Phys. 41, 1139 (1970). 9J.A. Cape and G. W. Lehman, J. Appl. Phys. 42, 5732 (1971) . IOJ.A. Cape and G.W. Lehman, Solid State Commun. 8,1303, (1970). lIA.A. Thiele, Bell Syst. Tech. J. 50, 7253 (1971). 12H. Callen and R. M. Josephs, J. Appl. Phys. 42, 1977 (1971). 13J.A. Cape, J. Appl. Phys. 43, 3551 (1972). 14W. F. Brown, Micromagnetics (Interscience, New york, 1963). 15L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935). 16W. Doring, Z. Naturforsch. A3, 374 (1948). 17L.R. Walker (unpublished), described in J.F. Dilion, Jr. in A Treatise on Magnetism, Vol. III, edited by G. T. Rado and H. Suhl (Academic, New York, 1963). 18B.E. Argyle, J.C. Slonczewski, andA.F. Mayadas, AlP Conf. Proc. 175 (1972). J. Appl. Phys., Vol. 45, No.8, August 1974 3581 19E. SchlOmann, Appl. Phys. Lett. 19, 274 (1971); 20, 190 (1972). 20J.C. Slonczewski, Int. J. Magn. 2, 85 (1972);J. Appl. Phys. 44, 1759 (1973). 21Such a configuration has been utilized experimentally by Kurtzig [A.J. Kurtzig, IEEE Trans. Magn. MAG-6, 497 (1970)] and suggested as a "gedanken" configuration by Cape (Ref. 13) to provide a definition of the wall mobility. 22A. H. Bobeck, IEEE Trans. Magn. MAG-6, 445, (1970). 23A. H. Bobeck, I. Danylchuk, J. P. Remeika, L. G. Van Uitert, and E. M. walters, in Proceedings of the Internation al Conference on Ferrites, edited by Y. Hoshino, S. !ida, and M. Sugimoto (University of Tokyo Press, Tokyo, 1971). 24G. W. Lehman (unpublished). 25J.A. Cape, W.F. Hall, andG.W. Lehman, Phys. Rev. Lett. 30. 801 (1973). 26T. 'L. Gilbert, Armour Research Foundation Report No. 11, 1955 (unpublished). 27E. SchlOmann, Appl. Phys. Lett. 21, 227 (1972). 28W.J. De Bonte, J. Appl. Phys. 44, 1793 (1973). 29There are two errors in Eqs. (6)-(8) of Ref. 25. First, for normal circular bubbles, R', 1// = 0, a derivation consistent to first order in 6/R gives 62=A(R+ 21rM; sin2 1jJ) and not Eq. (6) of Ref. 25. Accordingly, 6 = 0 in Eqs. (7) and (8) of Ref. 25. Second, the sin21jJ term of Eq. (8) of Ref. 25 should be omitted as it is already incorporated by definition in F (R). These errors do not appear in Eqs. (10) and (11) of Ref. 25 nor do they change any subsequent results or conclusions of that paper. 30G.p. Vella-Coleiro, F.B. Hagedorn, Y.S. Chen, and S. L. Blank, Appl. Phys. Lett. 22, 324 (1973). 3IG.N. Watson, Bessel Functions, 2nded. (Macmillan New York, 1948), pp. 389-391. ' [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.63.180.147 On: Sat, 22 Nov 2014 20:36:48
1.2837494.pdf	Validity of thermal activation volume estimated by using Wohlfarth’s equation in perpendicular recording hard disk media Masukazu Igarashi , Fumiko Akagi , and Yutaka Sugita Citation: Journal of Applied Physics 103, 07F529 (2008); doi: 10.1063/1.2837494 View online: https://doi.org/10.1063/1.2837494 View Table of Contents: http://aip.scitation.org/toc/jap/103/7 Published by the American Institute of PhysicsValidity of thermal activation volume estimated by using Wohlfarth’s equation in perpendicular recording hard disk media Masukazu Igarashi,1,a/H20850,b/H20850Fumiko Akagi,1,a/H20850,c/H20850and Yutaka Sugita2,d/H20850 1Hitachi, Ltd., Central Research Laboratory, 1-280 Higashi-Koigakubo Kokubunji, Tokyo 185-8601, Japan 2Tohoku Institute of Technology, Taihaku-ku, Sendai 982-8577, Japan /H20849Presented on 6 November 2007; received 11 September 2007; accepted 12 November 2007; published online 5 March 2008 /H20850 The physical volume of thermal switching unit, the activation volume for perpendicular recording media, has been investigated using the Wohlfarth equation based on micromagnetic simulation. Itwas found that the activation volume obtained from the Wohlfarth equation V ac-Wohlfarth exhibits signiﬁcant ﬁeld dependence, which is very different from direct observation of thermal switchingprocess. This shows that V ac-Wohlfarth is invalid for perpendicular recording media. The reason for that was interpreted in terms of the narrow dispersion of the energy barrier for thermal switching,originating from the high degree of easy axes orientation along the direction perpendicular to theﬁlm plane. © 2008 American Institute of Physics ./H20851DOI: 10.1063/1.2837494 /H20852 I. INTRODUCTION With increase in the storage density of hard disk drives, it becomes more critical to design hard disk recording mediakeeping thermal stability. The thermal stability factor K /H9252 /H20849=KuVac/kT, where Ku,Vac, and kTare the uniaxial aniso- tropy energy, the activation volume, and the thermal energy,respectively /H20850is widely used to estimate the validity term of magnetic information. 1Here, the activation volume Vacis deﬁned to be the physical volume of thermally switchingunit. Micromagnetic simulations of thermal switching haveshown that the activation volume obtained by using theWohlfarth equation V ac-Wohlfarth /H20849Ref. 2/H20850in longitudinal re- cording media is valid and depends only slightly on the re-verse ﬁeld. 3On the other hand, in recording media with per- pendicular anisotropy, large change of Vac-Wohlfarth as a function of the reverse ﬁeld has been observedexperimentally. 4However, direct observation of switching units by computer simulation has shown that Vacof recording media does not depend on the reverse ﬁeld signiﬁcantly.5,6 In this study, Vac-Wohlfarth is compared with directly ob- served switching volume based on the micromagnetic simu-lation. A failure of the conventional Wohlfarth equation isinvestigated. II. MODELING Arrays of 250- and 1000-polygonal prism grains with perfect mirror image for soft underlayer were used to simu-late sections of perpendicular recording media. The averagegrain size /H20855D/H20856of 7.1 nm with the thickness of magnetic layer t magof 12 nm /H20849the average grain volume Voof 480 nm3/H20850was used. The standard deviation of the grain size /H9268D//H20855D/H20856was set to be 20% in a log-normal dispersion by growth controlof Voronoi lattice. The easy axis of each grain was perpen- dicular to the ﬁlm with an angular dispersion of 3.0° in aGaussian dispersion. A medium with three-dimensional/H208493D /H20850-random easy axis orientation was also investigated. The value of the uniaxial anisotropy energy K uwas 2.8 Merg /cm3in a normal dispersions of /H1100615% or /H1100645%. The saturation magnetization of each grain was830 emu /cm 3. The intergrain exchange areal energy density wwas changed from 0 to 1.0 erg /cm2. The time evolution of the magnetizations of the grains was calculated by solvingthe Langevin equation, 7,8based on Landau–Lifshitz–Gilbert equation as shown below /H208491+/H92512/H20850dM dt=−/H9253/H20849M/H11003H/H11032/H20850,H/H11032=Heff+/H9251M/H11003Heff M. /H208491/H20850 Here, Heffis the effective ﬁeld consisting of ﬁve terms: the applied ﬁeld Hext, the uniaxial anisotropy ﬁeld Ha, the mag- netostatic ﬁeld Hd, the exchange ﬁeld from a neighboring grain- i/H20841Hexc− i/H20841/H20849=wSi/MsV, where SiandVare boundary area to the neighboring grain- iand the volume of the grain dis- cussed, respectively /H20850. The thermal excitation ﬁeld Hthermal was calculated under thermally accelerated condition /H20849the ac- celeration temperature Taccof 500–1500 K /H20850.9,10The gyro- magnetic ratio /H9253of 1.93 /H11003107//H20849Oe s /H20850/H20851corresponding to the gvalue of 2.19 /H20849Ref. 11/H20850/H20852and the Gilbert damping constant /H9251of 0.05 were used. From time-dependent magnetization under constant re- verse ﬁelds, Vac-Wohlfarth was calculated using the Wohlfarth equation, Vac-wohlfarth =kT MsHf,Hf=/H20873S /H9273irr/H20874=/H20879dH dln/H20849t/H20850/H20879 Mirr, /H208492/H20850 where Msand Hfare the saturation magnetization and the ﬂuctuation ﬁeld, respectively. S,/H9273irr,t, and Mirrare the vis- cosity coefﬁcient, the irreversible susceptibility, the time, andthe irreversible /H20849remanence /H20850magnetization, respectively. 2,12a/H20850Tel.:/H1100181-42-323- 1111. FAX: /H1100181-42-327-7844. b/H20850Electronic mail: mauskazu.igarashi.qu@hitachi.com. c/H20850Electronic mail: fumiko.akagi@hitachi.com. d/H20850Tel.: /H1100181-22-305-3226. FAX: /H1100181-22-305-3202. Electronic mail: ysugita@tohtech.ac.jp.JOURNAL OF APPLIED PHYSICS 103, 07F529 /H208492008 /H20850 0021-8979/2008/103 /H208497/H20850/07F529/3/$23.00 © 2008 American Institute of Physics 103 , 07F529-1The directly observed switching volume /H20849Vac/H20850was obtained from the difference of the magnetization distributions in very short time of around 1 ns after a long time had elapsed fromwhen the dc ﬁeld reversed /H20849DM method /H20850. 3,5,6Vacis almost constant in a wide elapse time region. III. RESULTS AND DISCUSSIONS The DM method is explained brieﬂy in Fig. 1showing magnetization dispersion of demagnetized state by dc reverseﬁeld and switched grains in media with wof 0.4 erg /cm 2. The dark cells in the ﬁgure on the left hand side are thegrains whose magnetizations have switched. The dark cellsin the ﬁgure on the right hand side show the switched grainsin 1 ns /H20849which corresponds to the elapse time of 10 ms at T accof 1000 K /H20850, approaching the magnetization dispersion in the demagnetized state. It is seen that almost all the switchedgrains are single and isolated. To determine the value of V ac, averaging the data using 12 frames was used. Figure 2shows average magnetization of grain at ther- mal switching as a function of time for a medium with wof 0.4 erg /cm2. The average switching time tswis deﬁned as the time between /H1100690% of the saturation magnetization Ms. The value of tswfor this medium is 0.13 ns and is independent of the applied ﬁeld, the magnetization, and the /H20849acceleration /H20850 temperature Tacc.Figure 3shows tswas a function of Taccfor media with w of 0 and 1.0 erg /cm2.tswdoes not depend on Taccsigniﬁ- cantly. tswfor the medium with wof 0 is 0.13 ns, similar to that for wof 0.4 erg /cm2, while tswforwof 1.0 erg /cm2is 0.17 s, a little higher and has larger deviation. This is be-cause sequential switching of the neighboring grains in-creases t swin media with larger intergrain exchange cou- pling. Those results show that once the dynamic switchingprocess is activated, the acceleration temperature does notinﬂuence the switching time. 1 ns is sufﬁcient time to ob-serve the sequential switching of the neighboring grains be-cause 1 ns is several times larger than t sw. This is the reason why in the DM method we observe grains switched during1 ns. Thus, the difference of the magnetization dispersions in1 ns can give us the thermally switching unit. Figure 4shows V acas a function of the reverse ﬁeld obtained from the DM method with Taccof 1000 K. Vacdoes not depend signiﬁcantly on the reverse ﬁeld irrespective ofthe values of w. This result is similar to that for longitudinal recording media. 3 Figure 5shows Vac-Wohlfarth as a function of the reverse ﬁeld for various media with different wobtained from Eq FIG. 1. Magnetization distribution of demagnetized state and the switched grains for media with wof 0.4 erg /cm2,/H20849250-polygonal prism grains /H20850. FIG. 2. Magnetization of grains at thermal switching as a function of time /H20849w=0.4 erg /cm2/H20850. The average switching time tswis deﬁned here as the time between /H1100690% of the saturation magnetization Ms. FIG. 3. /H20849Color online /H20850tswas function of Taccfor media with wof 0 and 1.0 erg /cm2. FIG. 4. Vacas a function of the reverse ﬁeld obtained from the DM method.07F529-2 Igarashi, Akagi, and Sugita J. Appl. Phys. 103 , 07F529 /H208492008 /H20850/H208492/H20850. For smaller w,Vac-Wohlfarth does not depend on the re- verse ﬁeld strongly. However, the values are a little higherthan the correct values of V acobtained from the DM method. For larger w,Vac-Wohlfarth has a large peak with increasing the reverse ﬁeld. Those results mean that Wohlfarth equation is invalid. A possible reason for this discrepancy is as follows. In perpen-dicular recording media, the energy barrier for thermalswitching has a narrow dispersion because the easy axes arealmost aligned. For smaller w, the dispersion of the energy barrier is likely to be a little larger due to the different de-magnetizing ﬁelds, while in longitudinal media, the energybarrier has a wide dispersion because the easy axes are ran-domly oriented in the ﬁlm plane. Figure 6shows V ac-Wohlfarth as a function of the reverse ﬁeld for media with wof 1.0 erg /cm2and larger /H9004Ku/Kuof /H1100645% along with /H1100615%, and a 3D-random medium. Vac-Wohlfarth for/H9004Ku/Kuof/H1100645% decreases drastically with small peak and closes to the values of Vacfrom the DM method. The activation volume as a function of Hfor the medium with /H9004Ku/Kuof 45% is similar to that for 15%, as shown in Fig. 4.Vac-Wohlfarth for the 3D-random medium does not depend on the reverse ﬁeld at all and the values are thesame as those from the DM ethod. V ac-Wohlfarth is easily af- fected by energy barrier dispersion. Those results are reason-able, since Wohlfarth 2and Street–Woolley–Smith12equa- tions are based on a ﬂat dispersion of energy barrier. IV. CONCLUSIONS The activation volume for perpendicular recording me- dia was investigated using micromagnetic simulations, andthe following results were obtained:/H208491/H20850The activation volume obtained from the Wohlfarth equation V ac-Wohlfarth exhibits signiﬁcant ﬁeld depen- dence, while the correct value obtained using the DMmethod does not. Thus, V ac-Wohlfarth is invalid for perpen- dicular recording media. /H208492/H20850The reasons for the above results are explained as fol- lows. The Wohlfarth2and Street–Woolley–Smith12equa- tions are based on a ﬂat dispersion of energy barrier. Inperpendicular recording media, the energy barrier forthermal switching has a narrow dispersion because theeasy axes are well aligned. /H208493/H20850The validity of V ac-Wohlfarth for longitudinal recording media is explained in terms of the wide energy barrierdispersion, which originated from random easy-axes-orientation in the ﬁlm plane. 1Y. Hosoe, I. Tamai, K. Tanahashi, T. Yamamoto, T. Kanbe, and Y. Yajima, IEEE Trans. Magn. 33, 3028 /H208491997 /H20850. 2E. P. Wohlfarth, J. Phys. F: Met. Phys. 14, L155 /H208491984 /H20850. 3M. Igarashi, F. Akagi, and Y. Sugita, IEEE Trans. Magn. 37, 1386 /H208492001 /H20850. 4R. Sbiaa, M. Mochida, Y. Itoh, and T. Suzuki, IEEE Trans. Magn. 36, 2279 /H208492000 /H20850. 5M. Igarashi, F. Akagi, and Y. Sugita, IEEE Trans. Magn. 39, 1897 /H208492003 /H20850. 6M. Igarashi, F. Akagi, and Y. Sugita, IEEE Trans. Magn. 42, 2393 /H208492006 /H20850. 7R. W. Chantrell, J. D. Hannay, M. Wongsam, and A. Lyberatos, IEEE Trans. Magn. 34,3 4 9 /H208491998 /H20850. 8H. N. Bertram and Q. Peng, IEEE Trans. Magn. 34, 1543 /H208491998 /H20850. 9J. Xue and R. H. Victora, Appl. Phys. Lett. 77, 3432 /H208492000 /H20850. 10M. Igarashi, M. Hara, Y. Suzuki, A. Nakamura, and Y. Sugita, IEEE Trans. Magn. 39, 2303 /H208492003 /H20850. 11M. Igarashi, T. Kambe, K. Yoshida, Y. Hosoe, and Y. Sugita, J. Appl. Phys. 85, 4720 /H208491999 /H20850. 12R. Street, J. C. Woolley, and P. B. Smith, Proc. Phys. Soc. London, Sect. B65, 679 /H208491952 /H20850. FIG. 5. Vac-Wohlfarth as a function of the reverse ﬁeld obtained from Eq. /H208492/H20850. Here, wandVoare intergrain exchange areal energy density and grain vol- ume, respectively /H20849/H9004Ku/Ku=/H1100615% /H20850. FIG. 6. Vac-Wohlfarth as a function of the reverse ﬁeld for media with /H9004Ku/Ku of/H1100645% along with /H1100615%, and a 3D-random medium /H20849w=1.0 erg /cm2/H20850.07F529-3 Igarashi, Akagi, and Sugita J. Appl. Phys. 103 , 07F529 /H208492008 /H20850
1.5144691.pdf	J. Appl. Phys. 127, 183905 (2020); https://doi.org/10.1063/1.5144691 127, 183905 © 2020 Author(s).Three terminal nano-oscillator based on domain wall pinning by track defect and anisotropy control Cite as: J. Appl. Phys. 127, 183905 (2020); https://doi.org/10.1063/1.5144691 Submitted: 09 January 2020 . Accepted: 17 April 2020 . Published Online: 13 May 2020 Oscar O. Toro , Sidiney G. Alves , Vagson L. Carvalho-Santos , and Clodoaldo I. 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Carvalho-Santos,1,a) and Clodoaldo I. L. de Araújo1 AFFILIATIONS 1Departamento de Física, Universidade Federal de Viçosa, Viçosa, 36570-900 Minas Gerais, Brazil 2Departamento de Estatística, Física e Matemática, CAP, Universidade Federal de São João del Rei, 36420-000 Ouro Branco, Minas Gerais, Brazil a)Author to whom correspondence should be addressed: vagson.santos@ufv.br ABSTRACT The proper understanding of the dynamical properties of magnetization collective modes is a cornerstone for future applications in spin- tronic devices based on the domain wall (DW) motion. In this work, through micromagnetic simulations and analytical calculations, we study the rotation of a DW pinned by a T-shaped defect on an anisotropic magnetic nanostripe. We show that the competition between the torques produced by the magnetostatic field generated by the T-shaped defect and the applied electric current makes the DW stop at a spe-cific position along the track, and start to turn around the in-plane direction with a specific rotation frequency depending on anisotropyand current density. It is also shown that the distance between the DW position and the T-shaped structure position depends on the anisot- ropy constant of the nanostripe. Finally, it is proposed as an experimental setting considering that the DW rotation mode can be used to induce the rotation of magnetization of a magnetic nanodisc by a magnetic tunnel junction device. We have then shown that this experi-mental arrangement can be considered as a three-terminal nano-oscillator. Published under license by AIP Publishing. https://doi.org/10.1063/1.5144691 I. INTRODUCTION The competition among different magnetic interactions can yield different magnetization configurations in a ferromagnetic nanoparticle. Among such configurations, we can highlight thedomain walls (DWs), which consist of collective modes of magneti-zation that can be controlled by magnetic fields and currents. These collective modes can be included in technological devices based on the prominent concepts of magnonics and spintronics. Inthis context, the knowledge on the generation, manipulation, anddetection of DWs is a cornerstone of research studies in magne-tism. Indeed, the proper control of the DW dynamics is crucial formaking possible the design of “racetrack ”memory devices. 1,2 Nano-oscillators are another up-and-coming concept for applica- tions regarding the DW dynamical properties. Nano-oscillators aredevices generally based on a magnetic tunnel junction (MTJ), 3 developed in nanometric stacks having two main features: the lowresistivity of the insulator and the magnetization of one of theferromagnetic layers pointing slightly perpendicular to the stack. This second feature can be achieved by using materials with some degree of out-of-plane anisotropy. 4Due to the low resistance of the stack, the current density flowing through it does not present highpower consumption. In this case, the current is polarized by thein-plane orientation of the first ferromagnetic material and exert a torque on the second misaligned layer. The spin-transfer torque (STT) 5,6is then responsible for the oscillation of the second ferro- magnetic layer, which can be read in the tunnel magnetoresistance(TMR) response. From the available fabrication techniques, nano-oscillators can reach tiny sizes 7in such a way that they have received much atten- tion for possible applications in several and different contexts innanotechnology. Indeed, nano-oscillators are potential candidatesto be used in frequency signal generation 8and modulation,9micro- wave generators for magnonic based devices,10and more recently in neuromorphic, which utilizes in-phase or out-of-phase oscilla- tions to mimetic brain network interactions.11Nevertheless, due toJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 183905 (2020); doi: 10.1063/1.5144691 127, 183905-1 Published under license by AIP Publishing.high current needed in spin-transfer torque technology, these devices operate in a regime very near the insulator threshold voltage, decreasing its durability. Therefore, the development ofnew technologies allowing low voltage drop in the barrier shouldbe developed to circumvent such limitations. Among the several and different settings that have been consid- ered to compose nano-oscillator devices, one can highlight the concept of three-terminal devices based on spin –orbit torque (SOT). In those structures, heavy metal contacts are used just below the firstferromagnetic electrode in the tunnel junction. A high current flowsthrough the metal, and the interaction between electronic spins, angular momentum, and angular orbital momentum generates a spin current polarization. 12The polarized spins impose an STT in the neighbor ferromagnetic contact, whose magnetization is alignedslightly perpendicular to the stack plane, providing a magnetizationoscillation, needed for STT technology nano-oscillators. Regarding this concept, an experimental realization of nano-oscillators combin- ing STT and SOT has been reported in a three-terminal device withsupport of spin splitting in tantalum. 13 Another handy magnetic system to be considered as a candi- date in applications in three-terminal devices consists of a DW dis- placing along a “racetrack, ”2which has the potential for application in magnetic random access memory14and in magnonics.15The proper control of DW pinning is essential for its application inpractical devices like racetrack memories 1or three-terminal MRAM,14for instance. Several approaches have been proposed to control the DW pinning, such as the insertion of a triangular notchin the track border, 16DW attraction and repulsion by a square notch and anti-notch,17,18curvature-induced effective magnetic interactions,19,20and the presence of magnetic defects changing locally the track magnetic properties.21,22Nevertheless, the realiza- tion of DW-based nano-oscillators demands not only the pinningof the DW along the nanostripe, but also the control of the DWrotation frequency. In this context, it was shown that a magneticanisotropy step, 24and the competition between the actions of dc current and magnetic field25yield a DW pinning at a fixed position along a nanostripe, and the current density control the rotation fre-quency. Additionally, a DW based three-terminal nano-oscillatorwas proposed with working frequencies of some few gigahertz,driven by non-uniform spin currents. 23 Based on the above and on the fact that modulation in a cylin- drical magnetic nanowire can produce a magnetic field that pins aDW, 26we propose the analysis of the DW dynamics along a mag- netic nanostripe with a T-shaped defect having an out-of-plane anisotropy. Through micromagnetic simulation and analytical cal- culations, it is shown that due to the competition between thecurrent density and an effective magnetostatic field (generated bythe track defect), the domain wall is pinned in a specific coordinatealong the track. Additionally, it is shown that the DW phase pre- sents a rotation around the in-plane direction. The specific DW pinning position and rotation frequency depend on both: thecurrent density and anisotropy of the track. From the obtainedresults, we propose the use of this set in a three-terminal nano-oscillator based on DW pinning by a defect in a magnetic nano- stripe. Finally, we estimate the proportional magnetoresistive response of the proposed three-terminal nano-oscillator as a func-tion of the anisotropy and the current density.This work is divided as follows: In Sec. II, we present the con- sidered system and the theoretical model. Section IIIpresents the results and discussions. In Sec. IV, we present our conclusions and prospects. II. THEORETICAL MODEL The analyzed system, depicted in Fig. 1(a) , consists of a mag- netic tunnel junction (MTJ) composed of (i) a thin magnetic nano- stripe having perpendicular anisotropy and a T-shaped defect; (ii) a thin insulator separator; and (iii) a magnetic nanodot presenting anin-plane single domain magnetization, which can be obtained bydesigning a specific shape or coupling with an antiferromagneticlayer. The nano-oscillator MTJ pillar could be developed by a con- ventional top-down fabrication technique, while the thicker trans- verse region would be developed afterward by the bottom-up(lift-off) technique. The nano-oscillator operation is based on the dynamic proper- ties of a DW displacing along the T-shape nanotrack. In this context, our main focus is on describing the dynamics of local mag- netization Minside a domain wall (DW). To reach our objectives, we parametrize the DW in a spherical coordinate system lying on aCartesian basis, that is, M¼M S(s i nθcosf,s i n θsinf,c o s θ). Under this framework, the DW profile can be represented by θ(y)¼2a r c t a n ey/C0y0 Δ/C16/C17 , (1) where we have considered that the center of the DW is located at y0, where θ¼π=2. In the above equation, Δis the domain wall width, whose value depends on the anisotropy constant ( K), exchange stiff- ness ( A), and the magnetostatic contribution, here characterized by the demagnetizing terms of the demagnetizing tensor of a magneticrectangular body ( N x,Ny,a n d Nz),27,28that is, Δ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ A Kþμ0M2 S 2N0s , (2) where we have used the definition in which the DW width is based on the slope of the magnetization angle,29,30andN0;N0(f) ¼(Nxcos2fþNysin2f). The magnetization dynamics is deter- mined by the Landau –Lifshitz –Gilbert equation (LLGe)31,32in the presence of spin-torque terms,33,34written as @M @t¼γHeff/C2Mþα MSM/C2@M @t/C0u@M @yþβ MSM/C2@M @y, (3) where γis the gyromagnetic ratio, MSis the saturation magnetiza- tion, αis the Gilbert damping coefficient, βis the phenomenological non-adiabatic spin-transfer parameter, and Heffis the effective field that the DW experiences when it displaces along the magnetic nano- track. Additionally, u¼gJeμBP=2eMShas the dimension of velocity and depends on the electrical current Je,gis the Landé factor, μBis the Bohr magneton, eis the electron charge, and Pis the polarization factor of the electric current. The effective field is composed of the external magnetic field, and the effective fields generated by magne- tocrystalline anisotropy, exchange, and dipolar interactions.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 183905 (2020); doi: 10.1063/1.5144691 127, 183905-2 Published under license by AIP Publishing.The DW equilibrium position and rotation frequency have been obtained from micromagnetic simulations and analytical cal-culations. The micromagnetic simulations were performed withGPU-accelerated micromagnetic simulator MuMax 3,37which per- forms iterations for transitions between spin configurations and energy minimization based on the LLGe. The analytical calcula-tions have been developed following the ideas of Mougin et al. , 35,36 in which the different torques acting on the DW can be written in terms of a spherical basis ( ^ρ,^θ,^f). Therefore, from the general kinetic momentum theorem, we can determine the torque compo- nents ΓθandΓfin terms of their associated rotation velocities in the following way: _θ¼/C0γ MSΓθand _f¼/C0γ MSΓf: (4) where the torques are given by Γθ¼μ0M2 SN2þαMS γ_fþMSu γΔ/C20/C21 sinθ, (5) Γf¼/C0 MSHþβMSu γΔþcosθ(μ0M2 SN1þK)/C20/C21 sinθ/C0αMS γ_θ, (6) whereN1¼N0/C0Nz, andN2;N2(f)¼(Ny/C0Nx)sinfcosf. The analyzed system is then divided into two parts. The first one is the track in which the DW propagates under the action of the current density. The second part is a thicker region, here calledT-shaped structure (TS). The track dimensions used in our simula-tions are 100 nm /C220 nm /C22 nm, while TS dimensions are 10 nm /C250 nm /C28 nm. The nanostripe was divided into cubic cells of 2 nm /C22n m/C22 nm, lower than the exchange length of theconsidered material. For the track, we have considered CoPt param- eters as saturation magnetization M CoPt S¼5/C2105Am/C01, exchange stiffness ACoPt ex¼1:5/C210/C011Jm/C01, and the Gilbert damping α¼0:3. The adopted track dimensions allow us to obtain the demagnetizing factors of the DW region by assuming that it lies in a rectangular prism,27,28giving Nx/C250:0875, Ny/C250:149, and Nz/C250:763. Due to state of the art in nanofabrication, the chosen device dimensions aim to investigate smaller feature sizes possiblefor practical nano-oscillators. The MTJ is simulated with a separation of 2 nm above the track, representing the insulator thickness between the track andreference layer. The reference layer presents a single domain mag-netization state along the in-plane direction. Under spin-transfertorque, generated by spin-polarized current applied in the extremi- ties track contacts, the DW can displace throughout it [ Fig. 1(b) ]. Due to the competition between the torques produced by the elec-tric current and the magnetostatic field generated by the TS, theDW is pinned in an equilibrium position along the track. 26Based on this, we propose that by keeping currents below the critical value, the described competition induces a stationary DW dynam-ics, in which the DW is at equilibrium near to TS, but rotatesaround the in-plane direction, transforming periodically from Néelto Bloch DW [see Fig. 1(c) ]. Under this assumption, an MTJ put just above the pinning region could measure the rotational frequency. III. RESULTS AND DISCUSSIONS A. Static Starting from a random magnetization configuration, we have performed micromagnetic simulations to obtain the magnetization ground state in the proposed structure. We have considered that materials with different anisotropies compose the track region and FIG. 1. (a) The proposed design to the three-terminal nano-oscillator based on DW pinning by a thickerregion forming a T-shape geometry. (b)The DW motion through the track with magnetization rotation in the xy-plane. (c) The magnetization oscillation beforeand after DW pinning.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 183905 (2020); doi: 10.1063/1.5144691 127, 183905-3 Published under license by AIP Publishing.the TS region. In this way, we have varied the perpendicular anisot- ropy of the track from K1¼2/C2105Jm/C03toK1¼5/C2105Jm/C03, while the anisotropy of the TS was kept constant, given byK 2¼5/C2105Jm/C03.F o r K1/C212/C2105Jm/C03, a Néel DW, separat- ing regions in which the magnetization points along upward anddownward directions [see Fig. 2(a) ], has been obtained. It can be observed that the increase in the anisotropy yields a reduction of the distance between the DW center and TS. Indeed forK 1¼2/C2105Jm/C03, the DW distance from the TS region isdDW/difference57 nm, while for K1¼5/C2105Jm/C03, this distance is reduced to dDW38 nm. The fitting of the results by an exponential decay equation showed in the inset of Fig. 2(b) reveals that for K1,1:23/C2105Jm/C03, the ground state magnetization configura- tion of the nanotrack would consist of a single domain pointingalong the out-of-plane direction and the minimum DW distance achieved for higher K 1would be dDW¼37 nm. B. Dynamics We now focus our analysis on the DW dynamics under the action of a spin-polarized current. From adopting the previously described anisotropy values, we have investigated the range of cur- rents able to move the DW in the track by STT(J[[6/C210 7,6/C2108]A=cm2). One can observe that for J/C202/C2108A=cm2, the DW is pinned in a specific position along the track, near to the TS, and its phase rotates around the in-plane direction, performing a non-harmonic oscillation, as depicted inFig. 1(b) . On the other hand, when J.6/C210 8A=cm2, spin waves start to be pumped throughout the thicker region, similarly what asdemonstrated by Michele Voto and collaborators. 15The analysis of the generation and propagation of these spin waves is not the focus of this work. Therefore, we will give attention to the rotation of thepinned DW as a function of the anisotropy and current density.The main results are presented in Fig. 3(a) , in which one can see that the DW rotation frequency depends on both the anisotropy of the track and current density. From the Fourier transform of the obtained results [see Fig. 3(b) ], it is possible to observe that by tuning the applied electric current and anisotropy, a broad range offrequencies (up to 30 GHz) can be obtained. Figure 3(c) presents the dependence of the frequency on the applied current for differ- ent anisotropy values. In this case, it can be noticed an increase of the DW precession frequency as a function of the current densityand anisotropy. Indeed, the frequency behavior as a function ofcurrent density is very similar for all the considered anisotropy values. From the analysis of the inset of Fig. 3 , one can notice that all curves collapse quite perfectly, with better results for higheranisotropies. This very good collapse indicates that the followingscaling Ansatz, F/differencef(j)=κ σ u (7) can be used to describe the DW frequency as a function of the current density. Here, κu¼Ku(μ0M2 S)/C01is the dimensionless anisotropy constant, the exponent σ/C250:35 is the scaling exponent, and f(j) is a function of the dimensionless current density j¼JS(eμ0MSγ)/C01, where Sis the area of the cross section of the nanotrack. The above-described results can be understood from assuming that the TS structure acts as a modulation in the nanotrack,26 which can generate an effective dipolar field, HT, pointing along thez-axis direction, competing with the torque produced by a current density, J. Therefore, from Eqs. (4),(5), and (6), the veloc- ity terms for the DW center ( θ¼π=2) are given by _θ¼/C0γαHT 1þα2/C0μ0MSγN2 1þα2/C0u(1þαβ) Δ(1þα2)(8) FIG. 2. (a) Ground state obtained from the simulation showing different DW dis- tances dDWand sizes, depending on the anisotropy of the track material. (b) DW distance as a function of track anisotropy, evidencing that, below K1¼1:23/C2105J=m3, the DW would be outside the track, which means the presence of an out-of-plane single domain magnetization.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 183905 (2020); doi: 10.1063/1.5144691 127, 183905-4 Published under license by AIP Publishing.and _f¼γHT/C0μ0MSγαN2 1þα2þu(β/C0α) Δ(1þα2): (9) Assuming that _θ¼0 when the DW is pinned, we can determine the magnetic field that produces the pinning in the DW, evaluated as HTpin¼/C0μ0MSγN2þu Δ(1þαβ) γα: (10) Due to the dependence of N2onf, the pinning field depends on the DW phase, presenting its maximum and minimum values when f¼π=4(f¼5π=4) and f¼3π=4(f¼7π=4), respec- tively. Nevertheless, from considering that hsin 2fi¼0, the aver- aged value of HTpindoes not depend on f. In this case, from the analysis of Fig. 4 , one can observe that the averaged pinning field increases with both current density and anisotropy constant. These results are in good agreement with that obtained in the analysis ofa DW displacing under the action of electric current and magneticfield. Indeed, it was observed that for a current density of J¼1/C210 8Ac m/C02, the pinning field is in the order of 10/difference100 mT.25The rotation velocity when the DW is pinned can be obtained from the substitution of Eq. (10) in(9), resulting in _fc¼/C0μ0MSγN2 α/C0u Δα: (11) Some main features can be highlighted at this point. The velocity (and consequently the frequency) of the DW phase preces-sion does not depend on the non-adiabatic spin-transfer parameter β. Additionally, because the DW width depends on the anisotropy, Eq.(11) reveals that the rotation velocity is a function of both current density and magnetocrystalline anisotropy (see Fig. 5 ). These results are in good agreement with that obtained in micro-magnetic simulations. Nevertheless, from the micromagnetic simu- lations, one can observe a small deviation from the linear behavior of rotation frequency as a function of the current density. Thisdeviation is caused by the nonlinear effects in the presence of spinwaves and the variation in DW width when the DW rotates aroundthe in-plane direction, which induces small deviations in the previ- ously obtained frequencies. Aiming at obtaining a complete description of the influence of the anisotropy on the DW frequency, we will determine the solu-tions of Eq. (11) numerically by considering the dependence of the DW width on its phase. Therefore, assuming that the pinning field is approximately constant, in Fig. 6 , we present a comparison of the FIG. 3. (a) Variation of the in-plane magnetization dynamics in the pinning position of the DW as a function of different anisotropies and applied currents. ( b) Fourier trans- formation of the magnetization dynamical behavior. A summary of the rotational frequency as a function of the applied current is presented in (c).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 183905 (2020); doi: 10.1063/1.5144691 127, 183905-5 Published under license by AIP Publishing.behavior of the DW phase for the cases where the DW presents a fixed (black line) and variable (red-dashed line) width during itsrotation. It can be noticed that variations of the DW width during its rotation can induce non-linear effects that generate differences in the DW rotation frequency. The effects of the DW width in thephase rotation are more evidenced for smaller values of anisotropyand current density. Indeed, from the Fourier analysis presented inFig. 3(b) , it is difficult to identify the DW rotation frequencies for smaller values of anisotropy. The differences in the DW frequencies for fixed and variable widths diminishes when anisotropy andcurrent densities increase, evidencing that the appearance of spinwaves is the cause of the deviation from the linear behavior of the rotation frequency as a function of the current density, observed in Fig. 3(c) .C. Output From the measurement of the magnetization properties pre- sented above, an MTJ should be used as a nano-oscillator deviceapplication. Under this assumption, we have performed micromag-netic simulations of a ferromagnetic nanodisc of 10 nm diameter and 2 nm thickness, separated from the nanostripe by a spacer of 2 nm in order to simulate the insulator, which would act as atunnel barrier in a real device. A magnetic nanodisc positioned5 nm before the TS can measure the DW rotation. The expected tunnel magnetic conductance (TMG) is then calculated by the scalar product of magnetization, for each adjacent simulationcell in the nanostripe and disc facing the separation region. 38In Fig. 7(a) , we present an example of such a coherent result for the calculated TMG in a nanotrack with an anisotropy of K1¼2/C2105J=m3and a current of J¼2/C2108A=cm2.I nFig. 7(b) , we present the zoom view of a specific interval of time, in which eachpoint of the TMG curve is highlighted and attached to images repre-senting the scalar product results presented in Fig. 7(c) . The dependence of the TMG as a function of anisotropy and current density is presented in Fig. 7(d) , which depicts the frequen- cies of signals that would be measured by a lock-in amplifier in theexperimental characterization of such a kind of device. The evolu-tion of TMG shows that the nano-oscillator sensitivity depends onthe anisotropy and current density. The higher values of TMG are obtained for a low anisotropy and current density. The decrease in the TMG as a function of anisotropy and current is related to theDW size decrease, which yields a misalignment with the referencelayer in the stack position. By increasing the current density, the DW starts to be pressed to the TS, and then the magnetization rotation effectively read by the MTJ lowers. Once lower anisotropy FIG. 5. Averaged DW rotation frequency as a function of the current density for different values of K1. FIG. 6. DW phase as a function of time. Black lines represent the DW rotation considering that the DW does not change its width during the rotation. Red-dashed lines show the DW phase in the case in which the DW width is a function of the phase. FIG. 4. Averaged value for the pinning field as a function of the current density for different anisotropy constants.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 183905 (2020); doi: 10.1063/1.5144691 127, 183905-6 Published under license by AIP Publishing.values lead to bigger DW widths, the misalignment does not affect the signal much. Even with the losses caused by the positioning ofMTJ, all the sensitivity measured is reasonable for the applicationin nano-oscillators. In the low-frequency regime, reached forJ,1/C210 8A=cm2, the TMG signal is approximately constant. The TMG signal is low but still possible to be measured in a real device. D. Influence of the temperature on the DW rotation frequency Investigations of nanotrack heating by current pulses have been performed, and a temperature of T¼780 K was estimated for permalloy nanotracks, under the current densities similar to those utilized in this paper, and longer pulses of /difference4 ns.39In this context, the high DW rotation frequency measured in this work allows theutilization of lower current pulse /difference1 ns. Additionally, the thick cobalt utilized in the track is expected to have high T c. These prop- erties prevent any damage or information loss in the proposeddevice by Joule heating. Therefore, aiming at extending our observations for the dependence of DW pinning and rotation behavior under external current at higher temperatures, we have performed thermal depen- dent simulations for a nanotrack with an anisotropy ofK 1¼5/C2105Jm/C03, under the applied current density of J¼6/C2108A=cm2,a t T¼300 K and T¼700 K. For the itera- tions, we have used the implemented Mumax3 second-order Heun ’s solver with a fixed time step of 1 /C210/C015s.38The main results are depicted in Fig. 8 , which evidences that the DW pinning and rotation are still observed, but present an increase in the signalnoise and phase shift when compared with the previously obtained FIG. 8. (a) Magnetization evolution in time for temperatures 0 K, 300 K, and 700 K for the track with K5¼5/C2105J=m3under J¼6/C2108A=cm2. (b) Operating frequency shift in function of temperature. FIG. 7. (a) Calculated TMG obtained from simulations performed with track anisot- ropy of K1¼2/C2105J=m3. (b) Zoom of the calculated TMG under the action of a current density of J¼2/C2108A=cm2. In (c), we present the evolution of TMG measured by the MTJ stack for different track anisotropies from K1¼ 2/C2105J=m3(hexagons) to K1¼5/C2105J=m3(circles) under different currents.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 183905 (2020); doi: 10.1063/1.5144691 127, 183905-7 Published under license by AIP Publishing.results [ Fig. 8(a) ]. The Fourier transform of the magnetization pre- sented in Fig. 8(b) shows a systematic decrease in frequency as a function of the temperature, suggesting a possible deviation in thefrequency range in practical devices. IV. CONCLUSIONS In summary, we have analyzed the domain wall dynamics along a magnetic nanostripe having a T-shaped geometrical defect andmagnetic anisotropy. We showed that the domain wall is pinned in an equilibrium position along the nanostripe due to the action of an effective magnetic field generated by the T-shaped structure. In thepinning position, the domain wall phase rotates around the in-planedirection with a well-defined frequency. The pinning field and theobtained frequencies are functions of the anisotropy of the stripe and applied current density. Analytical results show that the dependence of the rotation frequency on the domain wall width can be neglectedfor higher values of anisotropy and current density. Based on theobtained results, we propose a three-terminal nano-oscillator. It isshown that the DW width variation as a function of current can affect the TMG signal and that the practical device frequency can decrease as a function of temperature. Nevertheless, it remains highenough for practical applications in a nano-oscillator covering abroad range of frequencies. ACKNOWLEDGMENTS This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de N ıvel Superior - Brasil (CAPES) — Finance Code 001. The authors would like to thank the Brazilianagencies CNPq and FAPEMIG. The authors are also grateful toA.P. Espejo and F. Tejo for fruitful discussions on the theoreticalmodel. REFERENCES 1S. Parkin and S.-H. Yang, Nat. Nano 10, 195 (2015). 2S. S. Parkin, M. Hayashi, and L. Thomas, Science 320(5873), 190 –194 (2008). 3J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey, Phys. Rev. Lett. 74, 3273 (1995). 4S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. L. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nat. Mater. 9, 721 (2010). 5J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 6L. Berger, Phys. Rev. B 54, 9353 (1996). 7N. Perrissin, G. Grégoire, S. Lequeux, L. Tillie, N. Strelkov, S. Auffret, L. D. Buda-Prejbeanu, R. C. Sousa, L. Vila, and B. Dieny, J. Phys. D: Appl. Phys. 52, 234001 (2019). 8E. Bankowski, T. Meitzler, R. S. Khymyn, V. S. Tiberkevich, A. N. Slavin, and H. X. Tang, Appl. Phys. Lett. 107, 122409 (2015).9M. Quinsat, F. Garcia-Sanchez, A. S. Jenkins, V. S. Tiberkevich, A. N. Slavin, L. D. Buda-Prejbeanu, A. Zeltser, J. A. Katine, B. Dieny, and M.-C. Cyrille, Appl. Phys. Lett. 105, 152401 (2014). 10A. Slavin and V. Tiberkevich, Phys. Rev. Lett. 95, 237201 (2005). 11J. Torrejon, M. Riou, F. A. Araujo, S. Tsunegi, G. Khalsa, D. Querlioz, P. Bortolotti, V. Cros, K. Yakushiji, and A. Fukushima, Nature 547, 428 (2017). 12A. Manchon and S. Zhang, Phys. Rev. B 79, 094422 (2009). 13M. Tarequzzaman, T. Böhnert, M. Decker, J. D. Costa, J. Borme, B. Lacoste, E. Paz, A. S. Jenkins, S. Serrano-Guisan, and C. H. Back, Commun. Phys. 2,2 0 (2019). 14C. I. L. de Araújo, J. C. S. Gomes, D. Toscano, E. L. M. Paixão, P. Z. Coura, F. Sato, D. V. P. Massote, and S. A. Leonel, Appl. Phys. Lett. 114, 212403 (2019). 15M. Voto, L. Lopez-Diaz, and E. Martinez, Sci. Rep. 7(1), 13559 (2017). 16D. Petit et al. ,Phys. Rev. B 79(21), 214405 (2009). 17M. C. Sekhar et al. ,J. Phys. D: Appl. Phys. 44(23), 235002 (2011). 18H. Y. Yuan and X. R. Wang, Phys. Rev. B 89(5), 054423 (2014). 19K. V. Yershov, V. P. Kravchuk, D. D. Sheka, and Y. Gaididei, Phys. Rev. B 92, 104412 (2015). 20R. Moreno, V. L. Carvalho-Santos, A. P. Espejo, D. Laroze, O. Chubykalo-Fesenko, and D. Altbir, Phys. Rev. B 96, 184401 (2017). 21V. A. Ferreira et al. ,J. Appl. Phys. 114(1), 013907 (2013). 22D. Toscano et al. ,J. Magn. Magn. Mater. 419,3 7–42 (2016). 23S. Sharma, B. Muralidharan, and A. Tulapurkar, Sci. Rep. 5, 14647 (2015). 24J. H. Franken, R. Lavrijsen, J. T. Kohlhepp, H. J. M. Swagten, and B. Koopmans, Appl. Phys. Lett. 98, 102512 (2011). 25R. Hiramatsu, K.-J. Kim, T. Tanigushi, T. Tono, T. Moriyama, S. Fukami, M. Yamanuochi, H. Ohno, Y. Nakatani, and T. Ono, Appl. Phys. Express 8, 023003 (2015). 26A. P. Espejo, F. Tejo, N. Vidal-Silva, and J. Escrig, Sci. Rep. 7, 4736 (2017). 27A. Aharoni, J. Appl. Phys. 83, 3432 (1998). 28A. Smith, K. K. Nielsen, D. V. Christensen, C. R. H. Bahl, R. Bjørk, and J. Hattel, J. Appl. Phys. 107, 103910 (2010). 29A. Hubert and R. Schäfer, Magnetic Domains: The Analysis of Magnetic Microstructures (Springer-Verlag, 1998). 30A. P. Chen, J. Gonzalez, and K. Y. Guslienko, Mater. Res. Express 2, 126103 (2015). 31L. Landau and E. Lifshitz, Ukr. J. Phys. 53, 14 (2008). 32T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 33A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 (2005). 34S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). 35A. Mougin, M. Cormier, J. P. Adam, P. J. Metaxas, and J. Ferré, Europhys. Lett. 78, 57007 (2007). 36M. Yan, A. Kákay, S. Gliga, and R. Hertel, Phys. Rev. Lett. 104, 057201 (2010). 37A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, AIP Adv. 4, 107133 (2014). 38C. I. L. de Araújo, S. G. Alves, L. D. Buda-Prejbeanu, and B. Dieny, Phys. Rev. Appl. 6, 024015 (2016). 39J. Leliaert et al. ,AIP Adv. 7(12), 125010 (2017).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 183905 (2020); doi: 10.1063/1.5144691 127, 183905-8 Published under license by AIP Publishing.
1.1555377.pdf	Effect of pole tip anisotropy on the recording performance of a high density perpendicular head Mohammed S. Patwari, Sharat Batra, and R. H. Victora Citation: Journal of Applied Physics 93, 6543 (2003); doi: 10.1063/1.1555377 View online: http://dx.doi.org/10.1063/1.1555377 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/93/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Micromagnetic simulation of various pole-tip design perpendicular magnetic recording heads J. Appl. Phys. 105, 07B711 (2009); 10.1063/1.3077213 Stitched pole-tip design with enhanced head field for perpendicular recording J. Appl. Phys. 93, 6540 (2003); 10.1063/1.1557935 Track edge effects in tilted and conventional perpendicular recording J. Appl. Phys. 93, 7840 (2003); 10.1063/1.1557819 Micromagnetics of perpendicular write heads with extremely small pole tip dimensions J. Appl. Phys. 91, 6833 (2002); 10.1063/1.1452662 Micromagnetic simulation of an ultrasmall single-pole perpendicular write head J. Appl. Phys. 87, 6636 (2000); 10.1063/1.372795 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 139.78.24.113 On: Sun, 21 Dec 2014 05:03:21Effect of pole tip anisotropy on the recording performance of a high density perpendicular head Mohammed S. Patwaria) Seagate Research, Waterfront Place, Pittsburgh, Pennsylvania 15222 and Department of Electrical and Computer Engineering, University of Minnesota,Minneapolis, Minnesota 55455-0154 Sharat Batra Seagate Research, Waterfront Place, Pittsburgh, Pennsylvania 15222 R. H. Victora Department of Electrical and Computer Engineering, University of Minnesota,Minneapolis, Minnesota 55455-0154 ~Presented on 12 November 2002 ! Aperpendicular recording head for an areal density of 1 Tbit per square inches has been developed using a self-consistent three-dimensional ~3D!micromagnetic model. The soft underlayer and the recording layer have been included in the model. The head consists of a probe type tip protrudingfrom a collar. The tip has saturation magnetization ( M s) of 24 kG while the collar has lower Msof 10 kG. The magnitude and orientation of anisotropy ﬁeld ( Hk) in the tip is varied to obtain the best recording performance. A perpendicularly oriented Hkin the tip reduces ﬂux spreading, thereby enhancing the recording ﬁeld in the writing track while reducing the offtrack ﬁeld. However,simulations show that the head’s performance suffers in terms of high remanent ﬁeld and slowerfrequency response. Simulations show that a lower remanent ﬁeld can be achieved by applying acdemag pulse to the tip. A detailed comparison has been made between two cases of perpendicularH kof 1 kOe and longitudinal Hkof 10 Oe in the tip of the head. Results show a clear tradeoff in terms of recording ﬁeld and frequency response. Finally, simulations show that the designed head iscapable of recording areal density of 1 TBPSI on a recording layer of M sof 700 emu/cc, thickness of 20 nm, Hkof 18 kOe and average grain diameter of 6 nm. © 2003 American Institute of Physics. @DOI: 10.1063/1.1555377 # For the last decade or so, the areal density in magnetic disk storage has been increasing at a rapid rate ~60% annual growth rate !. For current longitudinal recording media, physical limitations such as the superparamagnetic limitthreaten to stymie this growth rate. 1To avoid this problem other recording schemes such as perpendicular recordinghave been proposed. 2To overcome superparamagnetism, while maintaining a high growth rate in areal density, thetrend in technology is to use thin ﬁlm media with small grainsize distribution and high anisotropy ( K u) materials but the scaling down of the media grains is limited by the write ﬁeldcapabilities of the recording head. The ﬁeld produced by thewrite pole is limited by the saturation magnetization(4 pMs). This work is concerned with the design of a per- pendicular single-pole type head with double layer media for an areal density of 1 Tbit/in.2 A self-consistent 3D micromagnetic model has been used to analyze the new head.3The model includes both magnetically soft and hard materials. The recording headpoles and the soft underlayer ~SUL!have been discretized into cubic cells of dimension 10 nm 310 nm 310 nm. The recording layer is represented by a grain conﬁguration ofplanar Voronoi cells. 4The medium parameters such as aver-age grain diameter of 6 nm with a standard deviation of about 30%, media thickness of 20 nm, and media magneti-zation of 400–700 emu/cc are chosen for an areal density of1 Tbit per square inches. The grains are magnetically decou-pled and the easy axis has a dispersion of 5° with respect tothe normal. The time for the media instability is calculatedusing the following formula: 5 a!Electronic mail: patwari@ece.umn.edu FIG. 1. ~a!Design of the novel head ~b!cross section of lower part and ~c! upper part.JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 10 15 MAY 2003 6543 0021-8979/2003/93(10)/6543/3/$20.00 © 2003 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 139.78.24.113 On: Sun, 21 Dec 2014 05:03:21DE5KuVF12H HkG2 5kBTln~2f0t!, ~1! where,Kuis the anisotropy constant of the material, kBis the Boltzmann constant, f0~attempt frequency !5109Hz and tis the time scale for the magnetic reversal to occur. Now, forten years of storage time, K uV/kBT>41. By putting Hk 5Hpeak~peak ﬁeld !and calculating the average offtrack ﬁeld, tcan be computed. The proposed head consists of a probe type tip protrud- ing from a collar.6The tip has saturation magnetization ( Ms) of 24 kG while the collar has lower Msof 10 kG.The tip has a thickness of 150 and 50 nm in the downtrack andcrosstrack directions, respectively ~see Fig. 1 !. In the earlier design, 6the anisotropy ﬁeld ( Hk) of the tip was oriented vertical. Although, the perpendicularly oriented and highvalue ofH k~1000 Oe !was able to reduce the ﬂux spreading from the narrow tip, it could suffer from slower frequencyresponse and high remanent ﬁeld.Also, the effect of the me-dia on the recording ﬁelds was not included. In this article,we attempt to address these pertinent issues and vary thedirection of H kto optimize the recording performance. In order to expedite the simulations, the recording layer was omitted for the following analysis.The distance betweenbottom of the pole and SULis set to 25 nm.With 5 nm of airbearing surface and carbon coating, 20 nm of media can bedeposited. SUL is 50 nm thick. Figure 2 indicates the per-pendicular ﬁeld proﬁles of three cases: perpendicular H kof 1 and 4 kOe as well as longitudinal Hkof 10 Oe both along the downtrack and crosstrack directions. For perpendicular Hkof 1 kOe, a peak ﬁeld of 11.2 kOe was generated with peakﬁeld gradients of 180 and 230 Oe/nm along the downtrackand crosstrack directions, respectively. When perpendicularH kis 4 kOe, the peak ﬁeld further improves to about 12 kOe with the corresponding peak gradients of 183 and 253 Oe/nm. However, for longitudinal H kof 10 Oe, the peak ﬁeld is 10.7 kOe with ﬁeld gradients of 175 Oe/nm along the down-track and 210 Oe/nm along the crosstrack directions. Notethe asymmetry of the ﬁeld proﬁle in the crosstrack direction,especially, when the pole tip anisotropy is in the horizontaldirection. This is due to the radically directed anisotropyﬁeld~20 Oe !of soft underlayer. Such bending of ﬂux is less pronounced for the perpendicular H kcases. From the contour plot ~see Fig. 3 !the ﬁeld along the center line of the next adjacent track, averaging over a dis-tance of about 0.2 mm, is ;3.3 kOe. Therefore, based on thermal relaxation calculations, the recorded bits in the adja-cent tracks would be erased in about 0.4 and 0.1 s for per-pendicular and longitudinal H kcases, respectively. Consid- ering that the head is moving at a linear velocity of 100 m/sover the same track again and again, the media in the neigh-boring tracks would become thermally unstable after about200 million pass lines for perpendicular H kof 1 kOe and 45 million pass lines for longitudinal Hkcase. For perpendicular Hkof 4 kOe, it takes about 1.2 s for the head to erase the FIG. 2. Perpendicular ﬁeld pattern ~a!along downtrack direction and ~b! along crosstrack direction. FIG. 3. Contour plot of perpendicular ﬁeld at the recording plane. FIG. 4. Perpendicular ﬁeld proﬁle along downtrack direction. FIG. 5. Pulse applied to the pole tip.6544 J. Appl. Phys., Vol. 93, No. 10, Part s2&3 ,1 5M a y 2003 Patwari, Batra, and Victora [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 139.78.24.113 On: Sun, 21 Dec 2014 05:03:21recorded bits in the next adjacent track, which corresponds to about 550 million pass lines. Micromagnetically it was found7that the recording layer has nonunity permeability. Therefore, from the reluctanceviewpoint, 6magnetically the head would see a thinner media than it is physically. Our simulations collaborate this conclu-sion. Simulations were run under different bias conditions ofthe recording layer ~ac erased, positive and negative dc erased !. The bias condition of the recording layer does not affect the recording ﬁeld much, hence the permeability of therecording media. Figure 4 shows the intrack ﬁeld proﬁlewhen a recording layer of 20 nm thick, H kof 18 kOe and magnetization of 700 emu/cc is included. In this case, theanisotropy ﬁeld of the tip is 10 Oe in the horizontal direction.According to Ref. 7, this corresponds to a relative permeabil-ity of about 1.4 for the recording layer. Figure 4 illustrates the effect of the recording layer. Pres- ence of the recording layer is found to reorient the magneticﬂuxes in the tip in the perpendicular direction, with a conse-quent increase in the recording ﬁeld. For this particular case,the peak ﬁeld is 13.1 kOe, about 22% more than the previouscase where the media was replaced by air of relative perme-ability of 1.0. The effect of the recording layer is found todepend on the magnetization and anisotropy ﬁeld of the me-dia, consistent with other calculations. 7Similar results are found for perpendicular Hkof the pole tip. In the calculations of remanent ﬁeld, the recording layer is not included and the distance between the SULand bottomof the pole tip is set to 15 nm. When the applied ﬁeld isturned off instantaneously, the peak remanent ﬁelds are 5.8,7, and 9 kOe for longitudinal H kof 10 Oe, perpendicular Hk of 1 and 4 kOe, respectively. Micromagnetic simulations show that if an ac demag pulse such as the one in Fig. 5, isapplied to the tip, the peak remanent ﬁeld reduces to 2.7, 3.5,and 2.4 kOe, respectively, for perpendicular H kof 1 and 4 kOe as well as longitudinal Hkof 10 Oe. This particular pulse has an initial full negative ﬁeld of width 1 ns followedby shorter pulses of width 0.5 ns with amplitude decreasingwith a time constant of 2.5 ns. The frequency responses of the heads are determined by switching on the applied ﬁeld instantaneously and then re-solving the recording ﬁelds at the trailing edge as a function of time. The initial conditions are the ones in the remanentstates at the end of the pulses shown in Fig. 5. We havechosen a starting point ( t50) where the ﬁeld at the trailing edge is small at the end of ac demag pulses. The dampingconstant ~ a!is set to 0.1 in the Landau–Lifshitz–Gilbert equation ~LLG!.3As indicated in Fig. 6, it takes about 0.9 ns for the recording ﬁeld to reach about 90% of the peak ﬁeldwhen the H kis in the longitudinal direction. However, for the perpendicularly directed Hk~1 kOe !of the pole tip, the temporal response is a bit slower, especially in the initialtimes. It takes about 1.4 ns to reach about 90% of the peakﬁeld. However, for higher perpendicular H kof 4 kOe, the temporal response is even slower ~about 2.0 ns !. Simulations show that a high perpendicular Hkreduces the ﬂux spreading from the narrow tip; therefore, it enhancesthe recording ﬁeld. However, such a head suffers from higherremanent ﬁeld and slower temporal response compared tothose when H kof the pole tip is in the longitudinal direction. The simulations show that remanent ﬁelds can be reduced byapplying an ac demag pulse such as shown in Fig. 5. Thedesigned head has a little head induced media erasure prob-lem. Figure 7 illustrates that the designed head with longitu-dinalH kof 10 Oe in tip can write on a recording layer of Ms of 700 emu/cc, thickness of 20 nm, Hkof 18 kOe and aver- age grain diameter of 6 nm. The authors wish to acknowledge J. Hannay for useful discussions and J. Xue for providing the Voronoi media. 1S. H. Charap, P. L. Lu, and Y. He, IEEE Trans. Magn. 33,9 7 8 ~1997!. 2R. Wood, IEEE Trans. Magn. 36,3 6~2000!. 3R. H. Victora and M. Khan, IEEE Trans. Magn. 38, 181 ~2002!. 4J. Xue and R. H. Victora, J. Appl. Phys. 87, 6361 ~2002!. 5B. D. Cullity, Introduction to Magnetic Materials ~Addison-Wesley, Read- ing, MA, 1972 !, pp. 413–414. 6R. H. Victora, J. Xue, and M. Patwari, IEEE Trans. Magn. 38,1 8 8 6 ~2002!. 7K. Senanan and R. H. Victora, Appl. Phys. Lett. 81,3 8 2 2 ~2002!. FIG. 6. Temporal response of the head. FIG. 7. Recording a bit on a recording layer of Msof 700 emu/cc and Hkof 18 kOe.6545 J. Appl. Phys., Vol. 93, No. 10, Part s2&3 ,1 5M a y 2003 Patwari, Batra, and Victora [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 139.78.24.113 On: Sun, 21 Dec 2014 05:03:21
1.5081665.pdf	J. Chem. Phys. 150, 174105 (2019); https://doi.org/10.1063/1.5081665 150, 174105 © 2019 Author(s).Algebraic-diagrammatic construction scheme for the polarization propagator including ground-state coupled-cluster amplitudes. II. Static polarizabilities Cite as: J. Chem. Phys. 150, 174105 (2019); https://doi.org/10.1063/1.5081665 Submitted: 15 November 2018 . Accepted: 07 April 2019 . Published Online: 01 May 2019 Manuel Hodecker , Dirk R. Rehn , Patrick Norman , and Andreas Dreuw ARTICLES YOU MAY BE INTERESTED IN Algebraic-diagrammatic construction scheme for the polarization propagator including ground-state coupled-cluster amplitudes. I. Excitation energies The Journal of Chemical Physics 150, 174104 (2019); https://doi.org/10.1063/1.5081663 Analytical gradient for the domain-based local pair natural orbital second order Møller- Plesset perturbation theory method (DLPNO-MP2) The Journal of Chemical Physics 150, 164102 (2019); https://doi.org/10.1063/1.5086544 van der Waals interactions in DFT using Wannier functions without empirical parameters The Journal of Chemical Physics 150, 164109 (2019); https://doi.org/10.1063/1.5093125The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Algebraic-diagrammatic construction scheme for the polarization propagator including ground-state coupled-cluster amplitudes. II. Static polarizabilities Cite as: J. Chem. Phys. 150, 174105 (2019); doi: 10.1063/1.5081665 Submitted: 15 November 2018 •Accepted: 7 April 2019 • Published Online: 1 May 2019 Manuel Hodecker,1 Dirk R. Rehn,1,2 Patrick Norman,2 and Andreas Dreuw1,a) AFFILIATIONS 1Interdisciplinary Center for Scientiﬁc Computing (IWR), Ruprecht–Karls University Heidelberg, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany 2Department of Theoretical Chemistry and Biology, KTH Royal Institute of Technology, Roslagstullsbacken 15, S-10691 Stockholm, Sweden a)Electronic mail: dreuw@uni-heidelberg.de ABSTRACT The modiﬁcation of the algebraic-diagrammatic construction (ADC) scheme for the polarization propagator using ground-state coupled- cluster (CC) instead of Møller–Plesset (MP) amplitudes, referred to as CC-ADC, is extended to the calculation of molecular properties, in particular, dipole polarizabilities. Furthermore, in addition to CC with double excitations (CCD), CC with single and double excitations (CCSD) amplitudes can be used, also in the second-order transition moments of the ADC(3/2) method. In the second-order CC-ADC(2) variants, the MP correlation coefﬁcients occurring in ADC are replaced by either CCD or CCSD amplitudes, while in the F/CC-ADC(2) and F/CC-ADC(3/2) variants, they are replaced only in the second-order modiﬁed transition moments. These newly implemented variants are used to calculate the static dipole polarizability of several small- to medium-sized molecules, and the results are compared to the ones obtained by full conﬁguration interaction or experiment. It is shown that the results are consistently improved by the use of CC amplitudes, in particular, for aromatic systems such as benzene or pyridine, which have proven to be difﬁcult cases for standard ADC approaches. In this case, the second-order CC-ADC(2) and F/CC-ADC(2) variants yield signiﬁcantly better results than the standard third-order ADC(3/2) method, at a computational cost amounting to only about 1% of the latter. Published under license by AIP Publishing. https://doi.org/10.1063/1.5081665 I. INTRODUCTION The algebraic-diagrammatic construction (ADC) scheme for the polarization propagator1–4has become a versatile and reli- able tool for the calculation of excitation energies and transition moments1,5–10and has also been applied successfully to static and dynamic polarizabilities,11,12X-ray absorption spectroscopy,13–15 two-photon absorption,16and C6dispersion coefﬁcients,12partic- ularly exploiting the formalism of the intermediate state represen- tation (ISR).3,11In a recent work on static polarizabilities and C6 dispersion coefﬁcients,12aromatic systems such as benzene have proven to be a difﬁcult case for standard ADC approaches, yieldingrather poor results compared to other theoretical approaches or experiment. We extended the previous implementation of second- order ADC with ground-state coupled-cluster (CC) amplitudes17in a development version of the Q-C HEM program package18to the cal- culation of molecular properties and tested its performance on static polarizabilities of several small- to medium-sized molecules. This approach has been inspired by similar works on the related second- order polarization propagator approximation (SOPPA) method by Geertsen, Oddershede, and Sauer.19,20Furthermore, a variant of the implementation relevant for molecular properties has been made by replacing the amplitudes in the transition moment vectors only, but not in the ADC secular matrix itself. This variant has also been J. Chem. Phys. 150, 174105 (2019); doi: 10.1063/1.5081665 150, 174105-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp implemented for the ADC(3/2) method in which the eigenvectors (and response vectors) of the third-order ADC matrix are used to calculate properties with second-order dipole matrices. The cur- rent implementation allows for the use of CC with double excita- tions (CCD) as an underlying coupled-cluster model as well as CC with single and double excitations (CCSD), where the singles ampli- tudes replace a part of the second-order density-matrix correction as described in Sec. II. Experimentally, static polarizabilities can, for instance, be obtained by considering the relative dielectric permittivity or the refractive index.21Here, we would like to refer to the comprehensive work of Hohm22in which experimental data for 174 molecules are compiled. Alternatively, static polarizabilities and other properties such as inelastic scattering cross sections of charged particles, Lamb shifts, or dipole-dipole dispersion coefﬁcients can be estimated using the so-called dipole oscillator strength distribution (DOSD), which is constructed using various pieces of experimental information such as photoabsorption spectra, refractivity, and electron scattering as well as constraints from quantum mechanics.12,23,24 As ﬁrst example of the performance of the new CC-ADC vari- ants on molecular properties, static dipole polarizabilities of sev- eral small- to medium-sized atomic and molecular systems are reinvestigated. In general, care has to be taken when comparing with experiment, in particular, due to vibrational or environmen- tal effects. For example, the compilation of Hohm22often includes estimates of vibrational contributions to the static polarizability, but such effects are not considered in the present computational study.25,26DOSD estimates, on the other hand, often include zero- point vibrational effects, and a previous study on methane reported an increase in its static polarizability by about 5% when including zero-point vibrational averaging (ZPVA).27While, in the static limit, pure vibrational contributions can be of the same order of magni- tude as the electronic contributions for some molecules, ZPVA has been observed to change polarizabilities, in general, by only a few percent.25,28 II. THEORETICAL METHODOLOGY AND IMPLEMENTATION The underlying theory and the ADC formalism for calculat- ing polarizabilities has been discussed in detail elsewhere.11,12Here, only a brief outline of the basic equations and principles for the calculation of dipole polarizabilities within the intermediate state representation shall be given. Apart from the original derivation of the ADC equations with the propagator approach,17an alternative exists via the so-called intermediate state representation (ISR).3,4,29,30The ISR not only gives direct access to excited states and transition properties but also offers a straightforward way to transform expressions from time-dependent response theory into closed-form matrix expres- sions.11,16The components of the frequency-dependent molecu- lar dipole polarizability αAB(ω) (with A,B∈{x,y,z}) are given as αAB(ω)=−/uni27E8Ψ0/divides.alt0ˆµA(/uni0335hω−ˆH+E0)−1ˆµB/divides.alt0Ψ0/uni27E9 +/uni27E8Ψ0/divides.alt0ˆµB(/uni0335hω+ˆH−E0)−1ˆµA/divides.alt0Ψ0/uni27E9, (1)with the electric dipole operator ˆµ=∑ pqµpqˆa† pˆaq. The exact sum- over-states expression is obtained by inserting the resolution of the identity of exact states, 1=∑ n/divides.alt0Ψn/uni27E9/uni27E8Ψn/divides.alt0.11If instead the resolution of the identity of intermediate states, 1=/divides.alt0Ψ0/uni27E9/uni27E8Ψ0/divides.alt0+∑ I/divides.alt0˜ΨI/uni27E9/uni27E8˜ΨI/divides.alt0is inserted, one arrives at the ADC formulation of the polarizability.12 For a static perturbation ( ω= 0), it is given by αAB(0)=F† AM−1FB+F† BM−1FA, (2) where we introduced vectors of modiﬁed transition moments Fwith elements FI=/uni27E8˜ΨI/divides.alt0ˆµ/divides.alt0Ψ0/uni27E9=/summation.disp pqµpq/uni27E8˜ΨI/divides.alt0ˆa† pˆaq/divides.alt0Ψ0/uni27E9=/summation.disp pqµpqfI pq (3) and used the deﬁnition of the modiﬁed transition amplitudes, fI pq =/uni27E8˜ΨI/divides.alt0ˆa† pˆaq/divides.alt0Ψ0/uni27E9. In order to obtain ADC expressions, the interme- diate states are constructed as described in the literature3,30and the exact ground-state wave function and energy are replaced by the Møller–Plesset (MP) perturbation series expansions31 /divides.alt0Ψ0/uni27E9=/divides.alt0Ψ(0) 0/uni27E9+/divides.alt0Ψ(1) 0/uni27E9+/divides.alt0Ψ(2) 0/uni27E9+: : :, (4) E0=E(0) 0+E(1) 0+E(2) 0+: : :. (5) Algebraic expressions are obtained by using the MP- partitioning of the molecular Hamiltonian and by collecting terms for the ADC matrix up to a given order n. When both the secular matrix and the transition moments are described consistently up to a certain order, this is then referred to as ADC( n). The second-order scheme ADC(2) formally depends on the MP wave-function and energy correction up to second order and describes single excitations correct in second order of perturbation theory. The ADC(3) scheme depends on the MP energy up to third order and describes single excitations consistent in third order and double excitations consistent in ﬁrst order of perturbation theory. However, for both ADC(2) and ADC(3), the excitation space is lim- ited to single and double excitations, i.e., the ADC matrix Mis of the same size as the conﬁguration interaction singles and dou- bles (CISD) matrix. Currently, the modiﬁed transition amplitudes are only available up to second order. Combining the second-order modiﬁed transition amplitudes with the third-order ADC matrix yields the so-called ADC(3/2) model.16The ﬁrst-order MP doubles amplitudes which are deﬁned as tab ij=/uni27E8ab/divides.alt0/divides.alt0ij/uni27E9 εa+εb−εi−εj, (6) where/uni27E8ab\|\|ij/uni27E9is an antisymmetrized two-electron integral and the εpare HF orbital energies, occur for the ﬁrst time in the second- order contribution to the p-h/p-hblock of the ADC matrix.1 They have already been replaced here for the calculation of exci- tation energies by CC doubles amplitudes,17which are calcu- lated in an iterative manner according to the CC amplitude equations32for the doubles /uni27E8Φab ij/divides.alt0e−ˆTˆHeˆT/divides.alt0Φ0/uni27E9=0, (7) J. Chem. Phys. 150, 174105 (2019); doi: 10.1063/1.5081665 150, 174105-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where \| Φ0/uni27E9is the Hartree–Fock reference determinant, ˆTis the cluster operator that is either approximated as ˆT=ˆT2for CCD orˆT=ˆT1+ˆT2for CCSD, and /divides.alt0Φab ij/uni27E9is a doubly excited determinant. The MP amplitudes also occur in the ﬁrst- and second-order contribution to the modiﬁed transition amplitudes fI pq,1where they were replaced by CCD or CCSD doubles amplitudes as well. Fur- thermore, in a similar spirit to the work of Sauer,20thep-hpart of the second-order one-particle density matrix correction10 ρ(2) ia=−1 2(εa−εi)/uni23A1/uni23A2/uni23A2/uni23A2/uni23A2/uni23A3/summation.disp jbctbc ij/uni27E8ja/divides.alt0/divides.alt0bc/uni27E9+/summation.disp jkb/uni27E8jk/divides.alt0/divides.alt0ib/uni27E9tab jk/uni23A4/uni23A5/uni23A5/uni23A5/uni23A5/uni23A6(8) was replaced by the corresponding CCSD singles amplitudes. Since Eq. (8) corresponds precisely to the second-order contribution of ˆT1, i.e., the lowest order where the singles occur in the MP wave- function expansion, CCSD was considered to an equal extent as CCD here, in contrast to Paper I on excitation energies.17These singles amplitudes are not replaced when CCD is chosen as the coupled-cluster model, but ρ(2) iais calculated instead with the CCD T2amplitudes. We would like to mention that the CC-ADC approach pre- sented here is still size consistent (size intensive), since, on the one hand, in the ISR, the ground state is completely decoupled from the excited conﬁgurations, and, on the other hand, as described before,17 the form of the ADC equations is still the same in the CC-ADC variants, which means that local and nonlocal excitations are exactly decoupled as well. The CCD and CCSD amplitudes were combined with ADC(2) to yield the variants termed CCD-ADC(2) and CCSD-ADC(2). Fur- thermore, in order to check for the importance of the amplitudes in different parts of the calculation, more variants of ADC(2) as well as ADC(3/2) have been implemented, in which the ampli- tudes are replaced in the modiﬁed transition moments F, but not in the ADC matrix M. These variants are then referred to as F/CC- ADC(2) and F/CC-ADC(3/2), where CC stands for either CCD or CCSD. III. RESULTS AND DISCUSSION In the following, static dipole polarizabilities of a series of small and medium-sized atomic and molecular systems are calculated using different ADC and CC-ADC variants and the results are com- pared to full conﬁguration interaction (FCI), CC3, or experimental values. In a previous study,11it was shown that double-zeta basis sets are clearly insufﬁcient for the calculation of polarizabilities at the wave-function correlated level. Furthermore, one set of diffuse functions is crucial, whereas adding further sets of diffuse functions seemed to be of minor importance at the triple-zeta level. Thus, a basis set like aug-cc-pVTZ represents a good compromise between basis-set size and accuracy.11Since the purpose of this study is to compare different CC-ADC variants with other methods, in partic- ular, standard ADC, no attempt was made to optimize the employed one-particle basis set. Instead, the basis sets of previous studies were employed for comparability. Most of the geometries were taken from the literature as well.11A. Comparison with FCI 1. The case of Li− As a ﬁrst step, we reinvestigate the case of the lithium anion, Li−, which has been a prominent test case for the calculation of dipole polarizabilities with many correlated methods.33–37Sauer chose to investigate this anion ﬁrst as an “ideal test case” for the SOPPA variant referred to as SOPPA(CCSD),20where he replaced MP by CCSD amplitudes, based on earlier works by Geertsen et al.19,38Thus, it was chosen as the ﬁrst test case for the CC-ADC approaches using the same uncontracted (16 s12p4d) Gaussian one- electron basis set.20 The values for the static dipole polarizability calculated with different ADC- and SOPPA-based methods compared to FCI are shown in Table I. A graphical representation of the relative error deﬁned asα(X)−α(FCI) α(FCI), where Xis the corresponding method, is depicted in Fig. 1. As can be seen, both standard second-order methods, ADC(2) and SOPPA, show only a small improvement compared to the ﬁrst-order random-phase approximation (RPA) which has a relative error of about 50% (corresponding to 400 a.u.). They still overestimate the static polarizability signiﬁcantly by more than 30% (about 250 a.u.). The use of coupled-cluster ampli- tudes within these methods lowers the value of the polarizability in all cases, but the magnitude of the effect varies strongly for the different variants. While SOPPA(CCSD) yields better results than Geertsen’s coupled-cluster polarization propagator approximation (CCSDPPA) variant,20this also holds true for the ADC(2) variant with CCD, but not for the one with CCSD amplitudes. In the lat- ter case, the polarizability is underestimated by more than 40% or 350 a.u. With CCD amplitudes, the underestimation is less than 25% (200 a.u.). A further improvement can be observed for the vari- ants in which the amplitudes are only substituted in the modiﬁed transition moments F. While for the F/CCSD-ADC(2) the error is still−30% (about 240 a.u.), the best result of all compared methods could be obtained with F/CCD-ADC(2), where the underestima- tion of 6% (50 a.u.) is even smaller than for SOPPA(CCSD) with 8% (65 a.u.). It can already be seen in this system that the ampli- tudes in the Fvectors play a larger role than the ones in the sec- ular matrix, since the change in going from standard ADC(2) to F/CCD-ADC(2) is already almost 300 a.u., and when the amplitudes TABLE I . Static dipole polarizability (in a.u.) of Li−calculated with different methods. Method α RPAa1198.39 SOPPAa1061.70 CCSDPPAa620.80 SOPPA(CCSD)a732.60 ADC(2) 1039.17 CCD-ADC(2) 601.66 F/CCD-ADC(2) 747.59 CCSD-ADC(2) 448.38 F/CCSD-ADC(2) 558.30 FCIa797.77 aTaken from Ref. 20. J. Chem. Phys. 150, 174105 (2019); doi: 10.1063/1.5081665 150, 174105-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . The relative error of the static dipole polarizability αfor Li−of results presented in Table I with respect to FCI. are additionally substituted in the secular matrix in CCD-ADC(2), the polarizability decreases by another 145 a.u. For CCSD ampli- tudes, this trend is even more pronounced: the difference between ADC(2) and F/CCSD-ADC(2) amounts to 480 a.u., and between F/CCSD-ADC(2) and “full” CCSD-ADC(2) only 110 a.u. However, the results obtained with the different methods do not appear to be very systematic, and especially, the best result obtained with the F/CCD-ADC(2) variant seems rather fortuitous. Since the lithium anion is a system with a diffuse charge cloud that is easily polarizable, it is understandable that the computed polar- izability is very sensitive to small changes in the parameters. This makes it, however, questionable whether the Li−ion is really an ideal test case and whether the observed improvements were obtained for the right reasons and not fortuitously. Furthermore, Li−is isoelec- tronic to the beryllium atom which, in turn, is known to be a strongly correlated system, and therefore, perturbation theories at low order and even single-reference coupled-cluster approaches may not be appropriate such that in this case a real multireference treatment would be needed. In order to further investigate the CC-ADC methods and deduce some general trends when using different t-amplitudes within ADC, additional calculations on more standard chemical sys- tems have been carried out and analyzed as shall be discussed in the following. 2. Neon and hydrogen ﬂuoride We turn our attention to two more small systems, namely, neon and hydrogen ﬂuoride. The static dipole polarizabilities of Ne and HF have been calculated with various ADC methods, and the results are compared to FCI. The basis sets used here are only of double- zeta quality, but since the reference FCI values were calculated in the same one-particle basis, the deviations from FCI stem solely from the approximations in the respective ADC method. Table II shows the static dipole polarizability of the Ne atom calculated with the d-aug-cc-pVDZ basis set,40,41and the relative error is depictedTABLE II . Static dipole polarizability (in a.u.) of Ne (d-aug-cc-pVDZ basis set) and HF (aug-cc-pVDZ basis set) obtained with different variants of the ADC scheme compared to FCI. NeHF Method α α xxαzz ¯α ADC(2) 2.83 4.55 6.71 5.27 CCD-ADC(2) 2.78 4.43 6.47 5.11 F/CCD-ADC(2) 2.78 4.43 6.48 5.11 CCSD-ADC(2) 2.83 4.53 6.58 5.21 F/CCSD-ADC(2) 2.83 4.53 6.59 5.22 ADC(3/2) 2.70 4.29 6.32 4.97 F/CCD-ADC(3/2) 2.65 4.19 6.12 4.84 F/CCSD-ADC(3/2) 2.70 4.28 6.21 4.93 FCIa2.67 4.29 6.21 4.93 aTaken from Refs.11 and 39. in Fig. 2. The deviation of the standard ADC(2) result from FCI of 6% (0.16 a.u.) is improved by 0.05 a.u. when using CCD ampli- tudes such that the deviation is only 4% or 0.11 a.u. When CCSD doubles amplitudes are employed, the polarizability increases again to the same value as standard ADC(2) and hence no improvement is observed. We can see, however, that the results for both CCD- ADC(2) and F/CCD-ADC(2) as well as for CCSD-ADC(2) and F/CCSD-ADC(2) are the same, underlining the greater importance of the amplitudes in the modiﬁed transition moments Fcompared to the ones in the secular matrix Mfor the calculation of the polariz- ability. The same trend as for ADC(2) is observed for the third-order variants, where standard ADC(3/2) slightly overestimates the static polarizability by 1.0% compared to FCI. The use of CCD amplitudes within the second-order modiﬁed transition moments Flowers the obtained value and improves it slightly with a relative error of −0.7%, whereas with F/CCSD-ADC(3/2), the same value as for standard ADC(3/2) is obtained. The dipole polarizability of hydrogen ﬂuoride was calculated with the aug-cc-pVDZ basis set,42and the results can also be found in Table II and Fig. 2. Again, the results for the CC-ADC and F/CC- ADC variants are almost identical. Focusing ﬁrst on the isotropic polarizability of HF ¯α=1 3(αxx+αyy+αzz), withαxx=αyyfor sym- metry reasons, standard ADC(2) overestimates its value by 6.9% or 0.34 a.u. As before, the use of CC amplitudes in ADC lowers the static polarizability and thus improves its value compared to stan- dard ADC. CCD amplitudes again yield a better result in ADC(2) than CCSD ones, with the error of the former being only 3.7% (0.18 a.u.) compared to about 5.8% (0.28 a.u.) of the latter. So again, when CCSD amplitudes are employed, the polarizability is raised compared to CCD ones, making the result more similar to stan- dard ADC(2). A similar trend is observed for the ADC(3/2) method. Here, however, F/CCD-ADC(3/2) underestimates the polarizability by 1.9% or 0.09 a.u. due to the already very good result of standard ADC(3/2), having an error of only 0.8% or 0.04 a.u. The F/CCSD- ADC(3/2) method again raises the value of the polarizability to some extent compared to F/CCD-ADC(3/2) and is in this case in almost perfect agreement (relative error <0.1%) with the FCI result of 4.93 a.u. for the isotropic polarizability. J. Chem. Phys. 150, 174105 (2019); doi: 10.1063/1.5081665 150, 174105-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Relative error of the isotropic polarizability ¯αfor Ne and HF of results presented in Table II with respect to FCI. Having a look at the individual values of the polarizability ten- sor, all ADC(2) variants describe the components of the polarizabil- ity perpendicular to the molecular axis (that is, αxxandαyy) better than the component parallel to the axis, αzz. The relative improve- ment when using CCD amplitudes, however, is larger for the parallel zcomponent than for the perpendicular ones. A similar observation holds for the ADC(3/2) method. Here, however, the standard ver- sion is already in agreement with FCI for the diagonal xandycom- ponents of the polarizability, whereas the error of the zcomponent amounts to 0.11 a.u. When using CCSD amplitudes in the Fvectors, the perpendicular components remain virtually unchanged, whereas the parallel zcomponent is lowered to be in perfect agreement with the FCI value as well. B. Comparison with experiment In the following, we will evaluate the accuracy of the CC-ADC methods for molecular systems of increasing size and with larger basis sets and compare the obtained results to the ones obtained inexperiments, often by means of the dipole oscillator strength dis- tribution (DOSD).24Since no FCI results are available for these systems, the results of the third-order approximate coupled clus- ter (CC3) method43were taken as a theoretical reference when they were available. Additionally, the polarizability anisotropy deﬁned as ∆α=/radical.alt4 (αxx−αyy)2+(αyy−αzz)2+(αzz−αxx)2 2(9) is compared. Previous studies have shown that ADC(2) yields, in general, rather large discrepancies in the anisotropies due to a poor reproduction of longitudinal polarizability components.11,12 1. Water and carbon monoxide Let us start with the investigation of the water molecule, using the rather large d-aug-cc-pVTZ basis set41in order to allow for a proper comparison of theory and experiment.11The results obtained for H 2O are shown in Table III, and the relative error with respect to TABLE III . Static dipole polarizability (in a.u.) of H 2O and CO calculated with different ADC variants (d-aug-cc-pVTZ basis set) compared to CC3 and experiment. H2O CO Method αxxαyyαzz ¯α ∆α α xxαzz ¯α ∆α ADC(2) 9.79 10.41 10.17 10.13 0.54 11.88 17.32 13.70 5.43 CCD-ADC(2) 9.48 9.97 9.83 9.76 0.44 11.45 16.92 13.27 5.47 F/CCD-ADC(2) 9.48 9.97 9.85 9.77 0.45 11.47 17.07 13.34 5.61 CCSD-ADC(2) 9.81 10.11 10.05 9.99 0.28 11.51 17.14 13.38 5.63 F/CCSD-ADC(2) 9.81 10.12 10.06 10.00 0.28 11.55 17.27 13.46 5.72 ADC(3/2) 9.30 10.09 9.71 9.70 0.69 12.07 16.35 13.50 4.29 F/CCD-ADC(3/2) 9.03 9.70 9.43 9.39 0.58 11.68 16.28 13.21 4.59 F/CCSD-ADC(3/2) 9.33 9.82 9.63 9.59 0.43 11.78 16.45 13.33 4.67 CC3a9.38 9.96 9.61 9.65 0.51 11.95 15.57 13.16 3.62 Experimenta9.83 0.67 13.08 3.59 aTaken from Refs. 11 and 44–48. J. Chem. Phys. 150, 174105 (2019); doi: 10.1063/1.5081665 150, 174105-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp CC3 is depicted in Fig. 3. Compared to CC3, the standard ADC(2) variant overestimates the polarizability by almost 5%. This can be signiﬁcantly improved to almost 1% by using CCD amplitudes, inde- pendent of whether they are used everywhere or only in the Fvec- tors. When using CCSD amplitudes, the results are with a relative error of about 3.5% worse, but still better than for the standard ADC(2) variant. ADC(3/2), however, yields a result very similar to CC3, having a relative error of only 0.5%. The trend of using CCD or CCSD amplitudes within ADC(3/2) is the same as for the pure second-order method. Here, however, this means a deterioration in the case of CCD amplitudes, since the polarizability is underes- timated by about 2.7%. F/CCSD-ADC(3/2) has roughly the same relative error compared to CC3 as the standard variant, just with the opposite sign. When taking the experimental value as reference, which was obtained using refractive index data,45,47similar trends are observed. ADC(2) overestimates the polarizability by 3% or 0.3 a.u., and the use of CC amplitudes again lowers the obtained values, thus gen- erally improving the results. As for Ne and HF, CCD amplitudes yield better results than CCSD ones and the difference between the CC- and F/CC-ADC variants is negligible. CCSD-ADC(2), however, still overestimates the static polarizability by about 1.6% (0.16 a.u.), whereas the variants with CCD amplitudes now underestimate its value by 0.06 a.u. Overall, (F/)CCD-ADC(2) yields the best results of all compared methods with a relative error of only about −0.65%. In fact, the result with CCD-ADC(2) agrees even better with exper- iment than the CC3 one, which for the previously studied systems yielded results almost identical to FCI, but here underestimates the polarizability by 1.8% (0.18 a.u.) compared to experiment.11,39A signiﬁcant difference to previous results is observed for the third- order ADC scheme. The effect of the CC amplitudes of lowering the values is still the same, but since standard ADC(3/2) already underestimates the polarizability compared to experiment by 1.3% (0.13 a.u., thus being still more accurate than CC3); in this case, the results deviate stronger when using CCD or CCSD amplitudes within the second-order Fvectors. Deviations from experiment of −0.44 and −0.24 a.u. corresponding to relative errors of −4.5% and−2.4% were obtained for F/CCD-ADC(3/2) and F/CCSD-ADC(3/2), respectively. Having a look at the polarizability anisotropy ∆α, standard ADC(2) yields the best result of 0.54 a.u. with respect to CC3 or experiment compared to all other second-order methods. CCD amplitudes lower this value only by 0.1 a.u., but with CCSD amplitudes, the result is with 0.28 a.u. the worst of all. Standard ADC(3/2) yields the best result of all with respect to experiment, even better than CC3. Taking CC3 as a reference, on the other hand, the ADC(3/2) value can be slightly improved by using CC amplitudes. Another molecular system under investigation here is carbon monoxide, which was also calculated using the d-aug-cc-pVTZ basis set. As can be seen from the results for the isotropic polarizabil- ity shown in Table III and the relative error with respect to CC3 depicted in Fig. 3, standard ADC(2) overestimates its value signif- icantly by 4.1% or 0.34 a.u. The use of CCD amplitudes in both the Fvectors and the secular matrix Mof ADC(2) lowers this error sig- niﬁcantly to 0.11 a.u., yielding again the best result of all ADC(2) variants compared to CC3 with a relative error of only about 0.9%. With CCSD amplitudes, the deviation is 1.7% (0.22 a.u.), which is still less than half as large as for standard ADC(2). The difference between the CC-ADC(2) and F/CC-ADC(2) variants is for CO larger than for Ne or HF, but the trend is the same as for Li−: employ- ing CC amplitudes only in the modiﬁed transition moments has the largest inﬂuence and lowers the value of the dipole polarizability sig- niﬁcantly, with F/CCD-ADC(2) and F/CCSD-ADC(2) resulting in a relative error of about 1.3% and 2.3%, respectively, while the addi- tional substitution in the secular matrix Mhas the same effect, but to a smaller extent. Going to the third-order description in the secular matrix only yields a small improvement compared to pure second- order; the error of standard ADC(3/2) still amounts to 2.6% or 0.34 a.u. Replacing the MP amplitudes in the second-order transition moment vectors by CC ones gives an improvement for both CCD and CCSD doubles amplitudes. In this case, however, the variant with CCD amplitudes yields better result than that with CCSD ones. While F/CCSD-ADC(3/2) still deviates from experiment by 1.3% FIG. 3 . Relative error of the isotropic polarizability ¯αof H 2O and CO of results presented in Table III with respect to CC3. J. Chem. Phys. 150, 174105 (2019); doi: 10.1063/1.5081665 150, 174105-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (0.17 a.u.), F/CCD-ADC(3/2) yields the best result of all presented ADC variants with a deviation of only 0.05 a.u., corresponding to a relative error of about 0.4%. It is also remarkable at this point that all “hybrid” CC-ADC variants, even the pure second-order ones, yield better results than the (third-order) standard ADC(3/2) method. For example, the relative error of F/CCD-ADC(3/2) is only one third of the standard ADC(3/2) one, and the relative error of CCD-ADC(2) is about half as large as the one of standard ADC(3/2) and only one third of the standard ADC(2) one. All observed trends and results hold as well when taking experiment11,46as a reference for the isotropic polarizability, just that the absolute deviation is 0.08 a.u. larger for all ADC variants. A different picture is observed for the individual components of the polarizability tensor. For the two components perpendicular to the molecular axis, αxxandαyy, the standard ADC approaches with MP amplitudes have a smaller deviation from the CC3 results than the ones with CC amplitudes, the order of magnitude of the deviation for the former being about 0.1 a.u., whereas for the latter, it is up to 0.5 a.u. However, for the component along the molec- ular axis,αzz, the largest difference can be observed between the pure second-order ADC variants and the ADC(3/2) ones. The third- order description of the secular matrix Msigniﬁcantly improves the description of αzzby about 1.0 a.u. for the standard ADC approaches. The inﬂuence of the chosen amplitudes in the Fvectors on the ADC(3/2) results is rather negligible. At the ADC(2) level, this inﬂuence is somewhat larger, and the largest improvement is again obtained with CCD amplitudes replacing the MP ones every- where, with the error of CCD-ADC(2) being 0.4 a.u. smaller than the one of the standard ADC(2) variant. These differences, of course, explain the changes in the polarizability anisotropy. While all ADC variants overestimate its value compared to experiment11,48or also CC3,44the use of CC amplitudes within ADC generally raises ∆α, thus worsening the results. For CCSD amplitudes, the effect is more pronounced than for CCD ones. 2. Aromatic systems Finally, we turn our attention now to some larger chemical sys- tems: aromatic and heteroaromatic compounds. Due to the lack ofCC3 or similar values in the literature for these systems, they are compared to experimental values only. The prototype of aromatic systems is, of course, the benzene molecule, which is considered as a ﬁrst example using the Sadlej-pVTZ basis set.52Experimental values in the literature were obtained by applying ultraviolet Stark spectroscopy49or through a series of experimental and theoretical data using the DOSD technique.50For standard ADC methods, the benzene molecule has proven to be a difﬁcult case,12which can be seen in the results shown in Table IV and Fig. 4 (left). Compared to the DOSD value, standard ADC(2) overestimates the static polar- izability signiﬁcantly by 5.14 a.u., corresponding to a relative error of 7.6%. Expanding the secular matrix Mto third order in standard ADC(3/2) improves the result only slightly and still overestimates ¯αnotably by 6.1% or absolutely by 4.13 a.u. Using CC amplitudes within ADC again improves the values for the polarizability sig- niﬁcantly by lowering the computed values. Here, the difference between CCD and CCSD amplitudes is replacing the MP ones either only in the Fvectors or both in Fand the secular matrix Mis rather negligible, with the difference between the two correspond- ing CC-ADC(2) and F/CC-ADC(2) variants being ≤0.1%. Using CC amplitudes within ADC(2) in the modiﬁed transition moment vec- tors only yields a deviation from experiment of about 3.2% (2.2 a.u.), whereas the error is about 2.9% (less than 2.0 a.u.) when the amplitudes are replaced everywhere in CC-ADC(2). A signiﬁcant improvement is also observed when using CC amplitudes in the F vectors of the ADC(3/2) variant, with the deviation from experiment being merely about 2.2% (1.5 a.u.), thus yielding the best results for all compared ADC variants. Hence, the improvement obtained when using CC amplitudes within ADC for the calculation of the static polarizability lies in the order of 63%, which is the most signif- icant one of all systems compared so far. Again, all CC-ADC vari- ants show a substantial improvement over the standard ones with the relative error of CC-ADC(2) methods being only about half as large as the one for standard ADC(3/2). A possible explanation for the better performance of the CC-ADC variants compared to the standard ADC ones is the better description of excitation energies, especially for the lowest ones, as shown in Paper I.17Yet, the transi- tion moments seem to be a more important factor. They are, how- ever, hard to compare with the literature or especially experiment. TABLE IV . Static dipole polarizability (in a.u.) of benzene, pyridine, and naphthalene calculated with different ADC variants (Sadlej-pVTZ basis set) compared to DOSD values. Benzene Pyridine Naphthalene Method αxxαzz ¯α ∆α α xxαyyαzz ¯α ∆α α xxαyyαzz ¯α ∆α ADC(2) 86.32 46.14 72.93 40.18 82.64 42.21 78.49 67.78 38.53 182.3 134.4 69.2 128.6 98.3 CCD-ADC(2) 81.68 45.99 69.78 35.69 78.41 42.09 74.67 65.05 34.60 172.8 128.5 69.4 123.6 89.8 F/CCD-ADC(2) 81.90 46.08 69.96 35.82 78.51 42.14 74.88 65.18 34.69 171.0 128.7 69.6 123.1 88.2 CCSD-ADC(2) 81.79 45.57 69.72 36.22 78.70 41.83 75.20 65.24 35.26 172.9 129.0 68.9 123.6 90.4 F/CCSD-ADC(2) 82.14 45.75 70.01 36.39 78.96 41.97 75.54 65.49 35.41 171.6 129.5 69.3 123.4 89.1 ADC(3/2) 84.89 45.97 71.92 38.92 80.91 41.95 76.59 66.48 36.99 178.1 130.7 68.6 125.8 95.2 F/CCD-ADC(3/2) 80.91 46.08 69.30 34.82 77.27 42.07 73.46 64.27 33.46 168.3 126.0 69.3 121.2 86.1 F/CCSD-ADC(3/2) 81.12 45.75 69.33 35.37 77.67 41.87 74.05 64.53 34.13 168.8 126.6 68.9 121.4 86.8 Experimenta67.79 31.5 62.88 117.4 86.8 aTaken from Refs. 24 and 49–51. J. Chem. Phys. 150, 174105 (2019); doi: 10.1063/1.5081665 150, 174105-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Relative error of the isotropic polarizability ¯αfor benzene, pyridine, and naphthalene of results presented in Table IV with respect to DOSD values. In Ref. 9, only oscillator strengths were compared, but those also depend linearly on the excitation energy. Not only the isotropic polarizability but also its anisotropy is improved signiﬁcantly compared to the experimental value49when using CC amplitudes in ADC. While it does not seem to play a signiﬁcant role whether they are employed both in the secular matrix and the modiﬁed transition moments, CCD amplitudes again yield slightly better results than the corresponding versions with CCSD amplitudes. Other experimental results give the polarizabil- ity anisotropy of benzene as 35.02 a.u.,51,53which is in almost per- fect agreement with CCD-ADC(2) or F/CCSD-ADC(3/2) results, for instance. Another system closely related to benzene is the six-membered heteroaromatic compound pyridine, the geometry of which has been optimized using the Gaussian 09 program package54at the MP2/cc- pVTZ level of theory. For the calculation of the static polarizability again the Sadlej-pVTZ basis set was used, the results are shown next to the ones for benzene in Table IV and the relative errors are depicted in Fig. 4. The experimental value of its isotropic polariz- ability was obtained using the DOSD method.24However, no value for the individual components or its anisotropy could be found in the literature. The deviation of the standard ADC(2) method from the DOSD value is with 7.8% or 4.9 a.u., very similar to the one for the benzene molecule, while the deviation of the standard ADC(3/2) variant is with 5.7% (3.6 a.u.) slightly smaller (0.5 a.u. in absolute numbers) for pyridine than for benzene. However, a clear improve- ment is observed again for all ADC variants when using CC instead of MP amplitudes. The difference between the individual variants is slightly larger in this case than for benzene, though all variants are still very similar. The best result for the pure second-order ADC method is again obtained when CCD amplitudes are used through- out, i.e., CCD-ADC(2). Here, the error amounts to 3.46% (2.17 a.u.), as compared to 3.76% (2.36 a.u.) when CCSD amplitudes are used, or 3.65% and 4.15% corresponding to 2.30 and 2.61 a.u. when CCD or CCSD amplitudes are used in the Fvectors only, respectively. This corresponds to an improvement of up to 55% compared tothe relative error of the standard ADC(2) method. Another signiﬁ- cant improvement is observed when F/CCD-ADC(3/2) is employed. With a deviation from experiment of 2.21% (1.39 a.u.), the F/CCD- ADC(3/2) variant again yields the best result, which corresponds to an improvement of 61% as compared to the standard ADC(3/2) vari- ant. The F/CCSD-ADC(3/2) variant yields a comparable result with a relative error of 2.62%. Again, the results obtained with all hybrid CC-ADC variants show a signiﬁcant improvement over the standard ones, even CC-ADC(2) over standard ADC(3/2), at a lower overall computational cost. The results for the last and largest system discussed here, the naphthalene molecule, are summarized in Table IV and Fig. 4, as well calculated with the Sadlej-pVTZ basis set. As noted by Mille- ﬁori and Alparone,51experimental results of the polarizability and its anisotropy were obtained from the Cotton–Mouton effect,55molar Kerr constants, and refractions,56,57as well as from laser Stark spec- troscopy.58,59Concerning the isotropic polarizability, the standard ADC(2) variant has an even larger deviation from experiment than for benzene and pyridine, the relative overestimation amounting to 9.6%, its absolute error being 11.22 a.u. As previously, signiﬁcant improvement is obtained when CC amplitudes are used. For CCD- ADC(2), CCSD-ADC(2), and F/CCSD-ADC(2), the relative error lies between 5.1% and 5.3%, with the absolute error between 6.0 and 6.2 a.u. In this case, the F/CCD-ADC(2) variant again stands some- what out, having the smallest error of all compared methods with 4.9% or 5.72 a.u. Thus, the improvement obtained when using CC amplitudes is up to almost 50% compared to the standard ADC(2) variant. The standard third-order ADC(3/2) method again shows no signiﬁcant improvement compared to standard ADC(2) and has an error of 7.2% corresponding to 8.4 a.u. The use of CC amplitudes within the second-order Fvectors improves notably upon this value, yielding the best result of all compared methods with 3.2% corre- sponding to 3.8 a.u. As for the aromatic systems studied before, all CC-ADC variants yield better results compared to experiment than the standard ones, especially CC-ADC(2) yields better results than standard ADC(3/2) while the computational cost remains J. Chem. Phys. 150, 174105 (2019); doi: 10.1063/1.5081665 150, 174105-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp signiﬁcantly lower. On the other hand, an improvement in the rela- tive error of more than 50% is obtained when going from standard ADC(3/2) to F/CCD-ADC(3/2) at a computational cost increase that only amounts to about 1% in this case. Again, a possible explanation for the improved description of the polarizability is the improve- ment in excitation energies.17Even more pronounced than for ben- zene, signiﬁcantly improved results for the polarizability anisotropy ∆αcompared to experiment are obtained when using CC amplitudes within ADC, especially in the ADC(3/2) scheme where the F/CCSD- ADC(3/2) variant is in perfect agreement with the experimental value. Two more related aromatic systems, quinoline and isoquino- line, have been calculated as well (see the supplementary material for the results), and the results show the same trends and improve- ments for the CC-ADC methods, underlining the consistency of the improvement for this class of molecules. IV. SUMMARY In this work, the existing implementation of the algebraic- diagrammatic construction scheme for the polarization propagator with coupled-cluster amplitudes17has been extended to molecu- lar properties and in this special case tested for dipole polarizabil- ities recently implemented for standard ADC using the damped response formalism.12Furthermore, in addition to CCD, CCSD amplitudes can be used as well, also in the second-order tran- sition moments of the ADC(3/2) method. This new approach is inspired by similar works done on the SOPPA method by Geert- sen, Oddershede, and Sauer.19,20In the new CC-ADC(2) variants, the Møller–Plesset correlation coefﬁcients that occur in ADC are replaced by either CCD or CCSD amplitudes; in the F/CC-ADC(2) and F/CC-ADC(3/2) variants, they are replaced only in the second- order modiﬁed transition moments F, but not in the secular matrix M. In order to test the performance of the new CC-ADC variants, the static dipole polarizabilities of several small- to medium-sized chemical systems have been calculated and compared to FCI, CC3, DOSD, or experimental reference values. As a ﬁrst test case, the Li−ion was chosen since it served previously as a reference.20In our opinion, however, this is not a good test case since the results are very sensitive with respect to the amplitudes employed in the calculation, and hence, the values vary very strongly and unsys- tematically. Although the result obtained with the F/CCD-ADC(2) variant is very close to FCI, this seems to be rather fortuitous than systematic and hence does not allow for many general conclusions regarding the use of CC amplitudes within ADC, except that the polarizability becomes smaller when using CC amplitudes. For the ten-electron systems neon and hydrogen ﬂuoride, the standard ADC methods show a relatively large deviation from FCI that could be improved when employing CCD amplitudes. Since, however, the third-order ADC(3/2) scheme already provided very good results with relative errors ≤1%, no signiﬁcant improvement was obtained with CC amplitudes in the Fvectors. A slightly different picture is obtained when experimental values are used as reference. While for the water molecule notable improvements, especially with CCD amplitudes, could be observed for the second-order ADC method, an increased deviation is observed for ADC(3/2) because the stan- dard variant already underestimates the static polarizability by about 1%, and the use of CC amplitudes in the Fvectors generally lowersits absolute value even more. For carbon monoxide and, in par- ticular, the aromatic systems benzene, pyridine, and naphthalene, which have proven to be very problematic cases for standard ADC,12 very consistent improvements for all CC-ADC variants compared to the standard schemes are obtained. The CCD-ADC(2) results, for instance, even exhibit a notably smaller relative error than the considerably more expensive ADC(3/2) method. For benzene, the relative errors of both the CC-ADC(2) and F/CC-ADC(3/2) variants amounted only to about 35%–50% compared to the one of standard ADC(3/2). Due to the less favorable scaling of CCD/CCSD compared to MP2, the CC-ADC(2) variants are, of course, computationally some- what more demanding than standard ADC(2), but still signiﬁcantly cheaper than the standard third-order ADC(3/2) or equation-of- motion (EOM)-CC methods. At this point, it seems appropriate to consider some computational efﬁciency aspects of the different stan- dard ADC, CC-ADC, and standard (EOM-)CC approaches in terms of their formal scaling with system size a bit more in detail. Both MP2 and ADC(2) scale as O(N5)(the latter in an iterative man- ner, however), whereas ADC(3) and both (EOM-)CCSD and CCD scale as O(N6), where Nis the number of basis functions. The price that has thus to be paid for the improvement of the results for the static polarizability with CC-ADC(2) is the O(N6)iterative ground- state calculation with CCD or CCSD instead of just the single O(N5) MP2 one. The successive excited-state calculation, however, scales more favorably for ADC(2) than for ADC(3) or CCSD. Thus, while the ground-state calculation has become one order of magnitude more expensive compared to MP2, the excited-state calculation still scales as O(N5)and the results obtained with the CC-ADC(2) vari- ants are notably better than the ones for standard ADC(3/2). In this way, one obtains very good results at an overall lower cost than stan- dard third-order ADC or CCSD methods which are sometimes even comparable to the very accurate iterative CC3 method that, however, scales very unfavorably as O(N7). As an example, in the ADC(2) and CC-ADC(2) computations of the aromatic systems, the central pro- cessing unit (CPU) time needed for the ADC (and CC) calculations amounts to only about 1% compared to ADC(3/2). On the other hand, the additional time needed for the CC calculation in F/CC- ADC(3/2) also amounts to only about 1% of the total time, and the improvement in the results is remarkable. We thus believe that especially the CC-ADC(2) variants will become useful and versatile alternatives to standard ADC in the calculation of molecular properties such as polarizabilities since it combines a reliable iterated CC ground state and retains the advan- tageous features of ADC with its Hermitian eigenvalue problem and low computational cost. SUPPLEMENTARY MATERIAL See supplementary material for geometries of all considered molecules as well as additional results for the static polarizabilities of the quinoline and isoquinoline molecules. ACKNOWLEDGMENTS M.H. acknowledges ﬁnancial support from the Heidelberg Graduate School “Mathematical and Computational Methods for the Sciences” (GSC 220) and many helpful discussions with Adrian J. Chem. Phys. 150, 174105 (2019); doi: 10.1063/1.5081665 150, 174105-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp L. Dempwolff. 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1.2713373.pdf	Spin transfer oscillators emitting microwave in zero applied magnetic field T. Devolder, A. Meftah, K. Ito, J. A. Katine, P. Crozat, and C. Chappert Citation: Journal of Applied Physics 101, 063916 (2007); doi: 10.1063/1.2713373 View online: http://dx.doi.org/10.1063/1.2713373 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Perfect and robust phase-locking of a spin transfer vortex nano-oscillator to an external microwave source Appl. Phys. Lett. 104, 022408 (2014); 10.1063/1.4862326 Zero field high frequency oscillations in dual free layer spin torque oscillators Appl. Phys. Lett. 103, 232407 (2013); 10.1063/1.4838655 Injection locking at zero field in two free layer spin-valves Appl. Phys. Lett. 102, 102413 (2013); 10.1063/1.4795597 Nonuniformity of a planar polarizer for spin-transfer-induced vortex oscillations at zero field Appl. Phys. Lett. 96, 212507 (2010); 10.1063/1.3441405 Influence of the Oersted field in the dynamics of spin-transfer microwave oscillators J. Appl. Phys. 101, 09C108 (2007); 10.1063/1.2712946 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.97.178.73 On: Tue, 02 Dec 2014 15:39:29Spin transfer oscillators emitting microwave in zero applied magnetic ﬁeld T . Devoldera/H20850and A. Meftah Institut d’Electronique Fondamentale, CNRS UMR 8622, Université Paris Sud, Bâtiment 220, 91405 Orsay, France K. Ito Hitachi Cambridge Laboratory, Hitachi Europe, Ltd., Cavendish Laboratory, Madingley Road,Cambridge CB3 0HE, United Kingdom J. A. Katine Hitachi GST, San Jose Research Center, 650 Harry Road, San Jose, California 95120 P . Crozat and C. Chappert Institut d’Electronique Fondamentale, CNRS UMR 8622, Université Paris Sud, Bâtiment 220,91405 Orsay, France /H20849Received 7 November 2006; accepted 21 January 2007; published online 29 March 2007 /H20850 Using pillar-shaped spin valves with the magnetization of the reference layer being pinned perpendicularly to the easy axis of the free layer, we show that spin-transfer-induced microwaveemission can be obtained at exactly zero applied magnetic ﬁeld and in its vicinity. The frequencytunability /H20849typically 150 MHz/mA /H20850, the spectral purity /H20849quality factor up to 280 /H20850, and the power /H20849up to 5 nV/Hz 1/2/H20850of the emission compares well with other spin-transfer oscillators based on spin-valve nanopillars. This ability to get satisfactory microwave emission without needing bulkymagnetic ﬁeld sources may arise from a small nonvanishing ﬁeld-like term in the current-inducedtorque. It may be of interest for the design of submicron microwave sources. © 2007 American Institute of Physics ./H20851DOI: 10.1063/1.2713373 /H20852 I. INTRODUCTION A spin-polarized current ﬂowing into a ferromagnet can transfer spin angular momentum to the magnetization,thereby causing a so-called spin-transfer torque /H20849STT /H20850acting on the magnetization. 1In spin valves, STT can be used to switch the magnetization of the free layer.2When combined with an applied ﬁeld typically greater than the anisotropyﬁeld of the free layer, a sufﬁciently high STT can set the freelayer magnetization into a stationary precessional motion. 3 Since the electrical resistance depends on the magnetizationof the free layer, this precessional motion generates a micro-wave voltage across the spin valve. Under optimized condi-tions, this emission can have a high spectral purity, togetherwith a large tunability by the current and the magnetic ﬁeld,making it an interesting system to design compact micro-wave sources. 4However, several breakthrough are needed before this concept can be efﬁciently used in applications;one of them is to get rid of the need for magnetic ﬁeld sourcethat would be detrimental to both the fabrication cost of theoscillator and its compactness. Following the early experimental demonstrations of mi- crowave emission without applied ﬁeld, 5,6several ideas have been proposed. For instance Diény et al. have proposed a spin transfer oscillator /H20849STO /H20850with a magnetic stack7com- prising /H20849i/H20850a polarizer with its magnetization perpendicular to the multilayer plane and /H20849ii/H20850a free layer and a readout layer both with collinear in-plane magnetizations. Despite promis-ing predicted characteristics, the experimental demonstrationof such a device is still lacking. Another route to get zeroﬁeld microwave emission has been suggested by Barna śet al. 8Using transport calculations, they have conjectured that in some speciﬁc asymmetric nanopillars comprising two na-nomagnets, the STT can destabilize both the parallel and theantiparallel conﬁgurations. As a result, stationary preces-sional modes were predicted to occur at zero magnetic ﬁeld. 9 The experimental proof of concept has been recently done10 but the technological potential of this concept still needs tobe assessed. In this article, we report another route to obtain micro- wave emission in zero applied ﬁeld in STO. We start fromthe consensual idea that in spin valve systems, radio fre-quency /H20849rf/H20850emission requires two ingredients: a ﬁeld and a STT that favor different magnetization orientations and com-pete. While in most devices the ﬁeld is simply an externalﬁeld, Tularpurkar et al. 11have shown that at least in some systems when the spin polarization pis perpendicular to the magnetization mof the free layer, the current density Jin- duces a torque that comprises not only a Slonczewski term,i.e.,m/H11003/H20849m/H11003p/H20850but also a signiﬁcant build-in ﬁeld-like term, i.e., m/H11003p. We propose that this term be used to re- place or complement the external ﬁeld and show an experi-mental situation where this condition is likely realized. Weuse a nanopillar etched from a spin valve where the syntheticantiferromagnet reference layer is pinned in the sampleplane, at an orientation p yperpendicular to the easy axis /H20849x/H20850 of the free layer. All layers are magnetized in-plane. With this speciﬁc stack, microwave emission is obtained experi-mentally at zero applied ﬁeld and for both current polarities.This ability to get satisfactory microwave emission without a/H20850Electronic mail: thibaut.devolder@ief.u-psud.frJOURNAL OF APPLIED PHYSICS 101, 063916 /H208492007 /H20850 0021-8979/2007/101 /H208496/H20850/063916/5/$23.00 © 2007 American Institute of Physics 101 , 063916-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.97.178.73 On: Tue, 02 Dec 2014 15:39:29needing bulky magnetic ﬁeld sources may be of interest for the design of submicron STO. II. MODEL Phenomenologically, we can decompose any torque act- ing on the magnetization on the base of two orthogonal vec-tors. We will thus write the equation describing the magne-tization dynamics in our samples, by incorporating both aSlonczewski torque and a ﬁeld-like term, i.e., dm dt=⌊/H20873/H9262B tMs/H20874/H20873/H9016J /H20841e/H20841/H20874/H20849py/H11003m+/H9252py/H20850 +/H92530Heff−/H9251dm dt⌋/H11003m, /H208491/H20850 where /H92530=221 kHz mA−1is the gyromagnetic ratio, /H9262Bis the Bohr magneton, and eis the electron charge. The symbol pyis a unit vector describing the magnetization of the refer- ence layer. All through this article, it is assumed static along/H20849y/H20850./H9016is a dimensionless effective spin polarization. It is assumed constant throughout this article, with no angular dependence. We use the so-called sine approximation of thespin torque term. Jis the electronic current, which we will hereafter write J yto recall the fact that it should transport a spin polarization that is directed along /H20849y/H20850./H9007effis the effec- tive ﬁeld of the free layer, including shape anisotropy and applied ﬁeld. /H9251is the Gilbert damping factor. The symbol t stands for the free layer thickness. MSis its magnetization. All along this article, we write /H20849z/H20850as the growth direction and /H20849xy/H20850as the plane of the layers, /H20849x/H20850being the easy axis of the free layer. The dimensionless parameter /H9252describes the relative strength of the ﬁeld-like torque compared to theSlonczewski-like torque. There is some controversy in theliterature about the strength of /H9252, which should depend on the thickness tof the free layer and on the decay length of the transverse component of the spin accumulation inside thefree layer. When this decay length is considered as zero,there no ﬁeld-like torque. Stiles et al. 12and Waintal et al.13 support this view and consider /H9252to be negligible in most metallic systems of interest. When the decay length is ﬁnite,the alignment of the spin of the conduction electron towardthe background magnetization of the free layer is not com-plete. As a result, the exchange ﬁeld that these conductionelectrons apply on the background magnetization does notvectorially sum to zero, and the current creates both aSlonczewski-like spin torque, i.e., mÃ/H20849mÃp/H20850, and a ﬁeld- like spin torque, i.e., /H9252mÃp. Gmitra et al. consider for in- stance /H9252to be negative and typically less than 0.1 /H20849Ref. 14/H20850 while Shpiro et al.15have predicted that the effective ﬁeld term can be as large as the spin-torque term provided that thefree layer thicknesses be in the range of 2 nm. For Shpiro et al., the ratio /H9252is near maximum when py/H11036mand its sign is negative in the writing convention of Eq. /H208491/H20850. In the standard conﬁguration3where the reference layer magnetization px, the applied ﬁeld Hx, and the free layer easy axis /H20849x/H20850are all collinear, the presence of a ﬁnite /H9252term would not alter signiﬁcantly the overall behavior. The phasediagram in the /H20853Hx,Jx/H20854plane comprises stable and bistable states, together with dynamical precession modes; the fron-tiers between the area of existence of these modes are onlyslightly distorted by the presence of a ﬁnite /H9252. Subtle extrac- tion procedures16are needed to deduce /H9252from experimental data. Dedicated experiments16on a free layer of 3 nm of Co concluded that /H9252=−0.2. Conversely, in the experimental situation that we have chosen, i.e., with the spin polarization pyperpendicular to the easy axis of the free layer, the nature of the expectedstability diagram /H20853H x,Iy/H20854isqualitatively changed by the presence of a ﬁnite /H9252, provided that /H9252is negative. Let us recall the situation when /H9252=0, as calculated by Morise et al. /H20849see Fig. 2 of Ref. 17/H20850. The stability diagram comprises ﬁve zones, all of them having one or two staticstable magnetization states. When /H9252=0, the magnetizations are predicted17to be directed according to the dominant torque: there is a ﬁeld-dominated zone where mx/H11015sgn /H20849Hx/H20850 /H20849we will refer to these zones as Hand − H/H20850, a bistable, ﬁeld- history dependent zone /H20849zone HH /H20850, and a current-dominated zone where my/H11015sgn /H20849Iy/H20850/H20849zones Iand − I/H20850. When /H9252=0, there is no deﬁnite I−Hborder, since the magnetization rotates continuously when we vary IyandHxfrom one region to the other. Note that when /H9252=0 no precession mode ever occur in the whole /H20853Hx,Jy/H20854plane. We have calculated that this is also the case when /H9252is chosen positive /H20849not shown /H20850. In contrast, the stability diagram comprises precession states when /H9252/H110210, as exempliﬁed for /H9252=−0.05 in Fig. 1. The boundary between the regions IandHdivides itself into two stationary precession zones. The narrowest one, labeled IPP,comprises modes that recall the in-plane precession modes ofthe standard conﬁguration, with a shell-like trajectory of themagnetization vector. 3The amplitude of precession de- creases continuously to zero as approaching the Hand − H regions. At zero applied ﬁeld, these IPP states precess aroundthe /H20849y/H20850axis. The precession axis rotates gradually from /H20849y/H20850to /H20849x/H20850as the ﬁeld H xis increased /H20849not shown /H20850. The IPP modes redshift when increasing the current, until they suddenly transform into out-of-plane /H20849OPP /H20850precession modes with ap- proximately twice lower frequency, and then blueshift. Atmuch larger currents, the magnetization ﬁnally freezes in theIregion. Note that the part of the OPP region near H x=0 FIG. 1. /H20849Color online /H20850Calculated stability diagram of a thin uniaxial mac- rospin submitted to both an easy axis ﬁeld and a current carrying a spinpolarization along the hard axis. Calculation parameters are H k/MS=0.01, /H9251=0.02, /H92620MS=0.85 T, /H9016=0.37, and /H9252=−0.05.063916-2 Devolder et al. J. Appl. Phys. 101 , 063916 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.97.178.73 On: Tue, 02 Dec 2014 15:39:29/H20849dotted bold line /H20850can sustain two different precession modes with different precession frequencies. The appearance of onemode or the other depends on the initial conditions. Notealso that at zero applied ﬁeld, a value of /H9252as small as /H110020.02 /H20849not shown /H20850is sufﬁcient to ensure the existence of precession modes in a very wide interval of current. In summary, setting the spin polarization orthogonal to the easy axis of the free layer is a method to obtain steady-state precessions in zero applied ﬁeld, provided that thecurrent-induced torque comprises a ﬁnite negative ﬁeld-liketerm /H9252and that the macrospin approximation is valid. Mi- crowave emission should be then possible in the four quad-rants of the /H20853H x,Jy/H20854. We have thus examined this situation experimentally, in order to try and get rf emission in zeroapplied ﬁeld. III. EXPERIMENTS Our experimental conﬁguration consist of a pinned layer magnetization along pyand an applied ﬁeld Hxalong the easy axis of the free layer. We use spin valves of compositionPtMn 17.5/CoFe 1.8/Ru 0.8/CoFe 2/Cu 3.5/CoFe 1/NiFe 1.8 /H20849thick- ness in nanometers /H20850. The stack is etched into a pillar-shaped elongated hexagon of size 50 /H11003100 nm2. The stacks are identical to those used in Refs. 18and19, except that the magnetization of the reference layer CoFe 2, hereafter quoted aspyis along /H20849y/H20850, i.e., perpendicular to the shape-induced easy axis /H20849x/H20850of the free layer CoFe 1/NiFe 1.8/H20851see inset, Fig. 2/H20849a/H20850/H20852. The shape of the resistance versus ﬁeld loops is con- sistent with this perpendicularity.20Stoner astroid measure- ments /H20849not shown /H20850indicate that the anisotropy ﬁeld of the free layer was /H92620Hk=30 mT, with back and forth coercivitiesbeing/H1100219 and 18 mT. The hard axis loop were off-centered by 4 mT along /H20849y/H20850, indicating the presence of stray ﬁeld radiated by the synthetic antiferromagnet layers magnetized along /H20849±y/H20850. We have performed resistance versus current R/H20849Iy/H20850loops in a ﬁeld Hxapplied along the easy axis /H20849x/H20850of the free layer /H20851Fig.2/H20849a/H20850/H20852. All R/H20849Iy/H20850loops consists of three portions of pa- rabola separated by reversible steps, subtracting an extra re- sistance at high negative and positive currents. The steps arefor instance indicated by the two arrows in Fig. 2/H20849a/H20850.N o hysteresis was ever measured in R/H20849I y/H20850loops. Within our ex- perimental accuracy, the step positions were independent of the applied ﬁeld in our studied ﬁeld interval of −37 /H11021/H92620Hx /H1102114 mT. Steps in R/H20849I/H20850curves are often the signature of micro- wave emission observed with an insufﬁcient measurement bandwidth. To conﬁrm this point, we have measured the fre-quency spectrum of the voltage noise at constant appliedcurrent and ﬁeld, meshed within the /H20853H x,Iy/H20854plane. The setup is similar to that of Ref. 5. Row data have been trans- lated into absolute power spectral densities /H20849PSD /H20850after cali- bration of the frequency-dependent gain of our ampliﬁcationchain. The essential features of the experimental microwave emission are reported in Figs. 3and4. They include some features that recall the macrospin modeling and some addi-tional features. The main agreements with the macrospinmodeling are: /H20849i/H20850the possibility to emit at zero applied ﬁeld for currents higher than ±3 mA /H20849±7.8/H1100310 7A/cm2/H20850forboth current polarities and /H20849ii/H20850the possibility to emit in the four quadrants of the /H20853Hx,Iy/H20854plane, with threshold current that does not vary much with the applied ﬁeld. These two pointsare in qualitative agreement with the predicted stability dia-gram /H20849Fig.1/H20850for negative ﬁeld-like term /H9252. However, the experimental microwave spectra have many additional more complex features that were not pre-dicted in the above crude macrospin model. We review themain differences later. Figure 4reports the noise power integrated in the fre- quency interval between 5 and 10.5 GHz, where most of thenoticeable features in the experimental spectra appeared. Thehighest radiated power is found in the /H20853H x/H110220,Iy/H110220/H20854quad- rant, and the corresponding noise PSD goes up to5n V / H z 1/2. In many portions within the /H20853Hx,Iy/H20854plane, the voltage noise power spectrum is multiply peaked /H20851see Figs. 3/H20849b/H20850and 4/H20852, revealing that several distinct precession modes can either coexist or can blink alternately during the measurement time.This recalls the modeling near H x=0, when two distinct OPP modes of differing frequencies could occur, except that inexperiments the coexistence of several modes in not re-stricted to the vicinity of H x=0. We could follow the experi- mental modes by continuity criteria within the /H20853Hx,Iy/H20854plane, and we have thus labeled them accordingly. Our naming con-vention is depicted in Fig. 3. The modes named /H110011t o/H110015 were detected in positive current. The modes named –1 to –3were detected in negative current. The portions within the/H20853H x,Iy/H20854plane where these modes exist are reported in the experimental stability diagram Fig. 4/H20849b/H20850. FIG. 2. /H20849Color online /H20850/H20849a/H20850Current-induced hysteresis loop. The arrows in- dicate reversible changes in the differential resistance. Inset: sketch of ourgeometry. /H20849b/H20850Representative spectra of the microwave voltage emitted by our nanopillars. The spectra are vertically offset for clarity. The labels recallthe naming of the different modes.063916-3 Devolder et al. J. Appl. Phys. 101 , 063916 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.97.178.73 On: Tue, 02 Dec 2014 15:39:29All modes exhibit a redshift with growing absolute cur- rent, except mode –1 /H20851Fig. 2/H20849b/H20850/H20852that exhibits a blueshift. This could indicate than we almost only excite in-plane pre-cession modes, recalling what has been formerly observed inthe standard conﬁguration of collinear easy axis and refer-ence layer magnetization. 3,19 Among our redshift modes, the one labeled –2 is the sole that can have a linear redshift /H20849/H11002150 MHz/mA /H20850for negative applied ﬁeld greater than /H1100225 mT. The other modes have a vanishing df/dIyslope when they start growing from the noise ﬂow; the frequencies of modes /H110011t o/H110015 are then 6, 9.5, 8, 8.2, and 7.6 GHz. When the current is increased, those redshifting modes have a ﬁnite, steadily growing df/dIywith a nonlinear redshift /H20849see Fig. 2/H20850. The linewidths are scattered from mode to mode. The linewidth is generally in the range of 200 MHz, except formodes 5 and mode –2. The mode –2 has the smallest line-width, which goes go down to 25 MHz /H20849quality factor Q =280 /H20850for an emission centered at 6.958 GHz requiring a command of /H1100229 mT and /H110027.4 mA. In its range of exis- tence, the mode 5 has a constant linewidth of 65 MHz, whichappears for instance with an emission centered at 7.620 MHzfor/H1100215 mT and 4.5 mA. These small linewidths compare well with those previously reported in spin-valve nanopillars. The frequent coexistence of several modes, the quasiab- sence of blueshifting modes and the slight current asymme-try of the magnetic behaviors in positive and negative cur-rents could not be predicted by our crude modeling. Weshould thus reexamine the simplifying assumptions of ourmodel. The main assumptions are the macrospin approxima-tion, the trivial angular dependence of the Slonczewskitorque, the uniform current density, and the static magnetiza-tion of the synthetic antiferromagnet layers. Among theseassumptions, choosing a more elaborate angular dependence of the Slonczewski torque is the only way to induce someasymmetry between the magnetic behaviors in positive andnegative currents. Obtaining coexistence of several preces-sion modes would require to lift the macrospin approxima-tion to obtain the nonuniform magnetization eigen oscillationmodes and to allow mode hopping through ﬁnite temperatureﬂuctuations. Lifting the macrospin approximation also re-quires to take into account the current nonuniformities thatwere washed out by our macrospin approximation. Thiscomplicated task is far beyond the scope our article, whose FIG. 3. /H20849Color online /H20850Field and current dependence of the spectra of the microwave voltage emitted by ournanopillars. The labels recall the naming of the differentmodes. The color scales with the logarithm of thepower spectral density.063916-4 Devolder et al. J. Appl. Phys. 101 , 063916 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.97.178.73 On: Tue, 02 Dec 2014 15:39:29main result is the observation of microwave emission in zero applied magnetic ﬁeld and the conjecture that this phenom-enon is rendered possible by the presence of ﬁnite nonvan-ishing ﬁeld-like spin torque /H9252. IV. CONCLUSIONS In summary, we have developed a spin-transfer oscillator that works both with and without applied magnetic ﬁeld. Thekey point is to use a nanopillar etched from a spin-valvewhere the synthetic antiferromagnet that polarizes the currentis pinned in the sample plane, at an orientation perpendicularto the easy axis /H20849x/H20850of the free layer. Macrospin modeling indicates that if the spin torque includes a ﬁnite ﬁeld-like term, steady state precessional states should exist even zeroapplied magnetic ﬁeld. Without this perpendicularity condi-tion between free layer easy axis and reference layer magne- tization, previous work has indicated that rf emission re-quired ﬁelds of the order of a few times the anisotropy ﬁeld. The tunability, the emission frequencies and linewidths ofour present spin transfer oscillator compare well with otherspin transfer oscillators based on nanopillars. However, themicrowave emission spectra are much richer than anticipatedfrom macrospin modeling, with often coexistence of twomodes. Their understanding would require extensive micro-magnetic modeling that are beyond the scope of this article.Finally, we stress that this ability to get satisfactory micro-wave emission without needing bulky magnetic ﬁeld sourcescould be of great interest for the design of submicron micro-wave sources. 1J. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850; US Patent No. 5,695,864 /H208491997 /H20850. 2Y. Acremann et al. , Phys. Rev. Lett. 96, 217202 /H208492006 /H20850. 3S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoe- lkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850425,3 8 0 /H208492003 /H20850. 4W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. 92, 027201 /H208492004 /H20850. 5T. Devolder, P. Crozat, C. Chappert, J. Miltat, A. Tulapurkar, Y. Suzuki, and K. Yagami, Phys. Rev. B 71, 184401 /H208492005 /H20850. 6Q. Mistral et al. , Mater. Sci. Eng., C 126, 267 /H208492006 /H20850. 7K. J. Lee, O. Redon, and B. Dieny, Appl. Phys. Lett. 86, 022505 /H208492005 /H20850. 8J. Barna ś, A. Fert, M. Gmitra, I. Weymann, and V. K. Dugaev, Phys. Rev. B72, 024426 /H208492005 /H20850. 9M. Gmitra and J. Barnas, Phys. Rev. Lett. 96, 207205 /H208492006 /H20850. 10O. Boulle, J. Grollier, V. Cros, C. Deranlot, F. Petroff, A. Fert, and G. Faini, International Workshop on Spin Transfer, Nancy, Oct. 3, 2006. 11A. A. Tulapurkar et al. , Nature /H20849London /H20850438, 339 /H208492005 /H20850. 12M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407 /H208492002 /H20850. 13X. Waintal, E.B. Myers, P.W. Brouwer, and D.C. Ralph, Phys. Rev. B 62, 12317 /H208492000 /H20850. 14M. Gmitra and J. Barnas, Phys. Rev. Lett. 96, 207205 /H208492006 /H20850. 15A. Shpiro, P. M. Levy, and S. Zhang, Phys. Rev. B 67, 104430 /H208492003 /H20850. 16M. A. Zimmler, B. Özyilmaz, W. Chen, A. D. Kent, J. Z. Sun, M. J. Rooks, and R. H. Koch, Phys. Rev. B 70, 184438 /H208492004 /H20850. 17H. Morise and S. Nakamura, Phys. Rev. B 71, 014439 /H208492005 /H20850. 18T. Devolder, P. Crozat, J.-V. Kim, C. Chappert, K. Ito, J. A. Katine, and M. J. Carey, Appl. Phys. Lett. 88, 152502 /H208492006 /H20850. 19Q. Mistral, J.-V. Kim, T. Devolder, P. Crozat, C. Chappert, J. A. Katine, M. J. Carey, and K. Ito, Appl. Phys. Lett. 88, 192507 /H208492006 /H20850. 20For instance, the remanent resistance was almost ﬁeld-history indepen- dent, with a value compatible with 0.5 /H11003/H20849RP+RAP/H20850, the resistance RPand RAPof the parallel and antiparallel states having been measured previously in other samples where pwas along /H20849x/H20850. FIG. 4. /H20849Color online /H20850/H20849a/H20850Total emitted power versus easy axis ﬁeld and applied current /H20849a. u. /H20850./H20849b/H20850Experimental stability diagram and zones of ex- istence /H20849or coexistence /H20850of the identiﬁed precession modes.063916-5 Devolder et al. J. Appl. Phys. 101 , 063916 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.97.178.73 On: Tue, 02 Dec 2014 15:39:29
1.1578063.pdf	Laser cooling and trapping visualized E. J. D. Vredenbregt and K. A. H. van Leeuwen Citation: American Journal of Physics 71, 760 (2003); doi: 10.1119/1.1578063 View online: http://dx.doi.org/10.1119/1.1578063 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/71/8?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Tapered optical fibers as tools for probing magneto-optical trap characteristics Rev. Sci. Instrum. 80, 053102 (2009); 10.1063/1.3117201 An electrostatic glass actuator for ultrahigh vacuum: A rotating light trap for continuous beams of laser-cooled atoms Rev. Sci. Instrum. 78, 103109 (2007); 10.1063/1.2800777 Laser cooling and trapping, Bose Einstein Condensation, the case of metastable helium AIP Conf. Proc. 748, 230 (2005); 10.1063/1.1896494 A simple laser cooling and trapping apparatus for undergraduate laboratories Am. J. Phys. 70, 965 (2002); 10.1119/1.1477435 One-dimensional laser cooling of an atomic beam in a sealed vapor cell Am. J. Phys. 70, 71 (2002); 10.1119/1.1419098 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 165.123.34.86 On: Wed, 08 Oct 2014 11:05:44Laser cooling and trapping visualized E. J. D. Vredenbregta)and K. A. H. van Leeuwen Physics Department, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands ~Received 13 March 2002; accepted 1 April 2003 ! Laser cooling and trapping have become widely used in the atomic physics laboratory.Acomputer program is presented that simulates some of the most important techniques employed, includingatomic beam collimation, Zeeman slowing, funneling, and magneto-optical trapping. Its applicationranges from experiment design to illustration of course material. © 2003 American Association of Physics Teachers. @DOI: 10.1119/1.1578063 # I. INTRODUCTION Two recent Nobel prizes have celebrated advances in the ﬁeld of Atomic, Molecular, and Optical Physics. The 1997prize recognized the importance of the development of lasercooling techniques. 1This technique paved the way for the work leading to the 2001 prize which was given for achiev-ing Bose–Einstein Condensation 2,3in atomic gases, because this achievement relied heavily on the newly developed abil-ity to trap and cool alkali-metal atoms with lasers. By now,laser cooling has become part of every modern atomic phys-ics laboratory and of many Modern Physics and advancedAtomic Physics courses. Some undergraduate laboratorycourses already demonstrate the magneto-optical atom trap. 4 Because of these developments, a tool to illustrate laser cooling techniques for educational purposes as well as tosimulate or design experiments can be quite useful. In ourownlaboratorywehavehadsomesuccessinboththeseareaswith a computer code that we have gradually developed overa number of years. Originally meant to aid in designing andoptimizing experimental apparatus for producing high inten-sity, cold atomic beams 5–7and atom traps,8we have also found it useful as a means of increasing students’ under-standing of lecture course material by building problem setsaround it. It is the purpose of this paper to make the programavailable to a wider audience. To this end we very brieﬂy review laser cooling in Sec. II and the theory behind the radiation pressure force in Sec. III.The program itself is discussed in Sec. IV. Then we illustrateits use both for design ~Sec.V !and for instructional purposes ~Sec. VI !. Finally, we explain how the code can be obtained. II. LASER COOLING TECHNIQUES Cooling and trapping of atoms 1,9with light ~usually sup- plied by lasers !invariably depends on the change in linear momentum an atom undergoes by the absorption and subse-quent emission of a photon. Each photon carries momentum \k5h/lwherekis the photon’s wave number and lthe corresponding wavelength. Through the absorption-emission process, the atom exchanges momentum and energy with thelight ﬁeld, thereby changing its own velocity. When the netresult of a repeated application of this process is such that the rms velocity vrmsof a collection of atoms is reduced, the atoms are said to have been ‘‘cooled’’by the light. They are then assigned a temperature T5mvrms2/kB, wheremis the atomic mass and kBis Boltzmann’s constant. ~It is under-stood that this is not a true temperature in the thermody- namic sense of the word since thermal equilibrium is notimplied. !The temperatures that can be achieved with laser cooling range from 1 mK to ’1 nK, depending on the de- tails of the technique used. These cooling techniques can be roughly divided into three classes. 1All of these require the atom to return to its original electronic energy level after an absorption-emissioncycle has been completed, so that repeated cycling is pos-sible. Many atoms, including all alkali-metal atoms andmetastable rare gas atoms, possess such closed transitions.The most common technique goes by the name of Dopplercooling 10because it relies on the dependence of the rate of absorption of photons on the Doppler shift experienced by amoving atom. Doppler cooling can be shown to lead to a lower limit on the temperature given by T D5\G/2kB, where Gis the linewidth of the atomic transition involved. Cooling below this limit ~sub-Doppler cooling11!is possible when the light ﬁeld contains polarization gradients and the atom’slower energy state has a multi-level substructure, which oc-curs when its total electronic angular momentum quantum numberfis greater than zero. As with Doppler cooling, po- larization gradient cooling relies on spontaneous emission. As a result, the temperature achievable in this way is limitedby the recoil resulting from the emission of a single photon, T5\ 2k2/mkB. Even deeper cooling is possible with so- called sub-recoil techniques such as velocity-selective coher- ent population trapping,12,13which are not limited by spon- taneous emission. The simulation program discussed in this paper treats only Doppler cooling. Because sub-Doppler cooling leads tolower temperatures, it is often used as the ﬁnal stage in lasercooling experiments. However, it is fair to say that Dopplercooling is the true workhorse of the atomic physics labora-tory, being the main technique used for atomic beam colli-mation, slowing, and focusing, as well as atom trapping. Infact, sub-Doppler cooling is normally only effective for at-oms that have already been cooled to the Doppler limit.Therefore, for much of the application of laser cooling to thedesign of experiments it is sufﬁcient to consider only Dop-pler cooling. From an instructional point of view, Dopplercooling can be understood from the concepts of rate equa-tions and Brownian motion while understanding sub-Dopplercooling involves a rather advanced theoretical frameworkwhich may be beyond the level of an undergraduate physicscourse. 760 760 Am. J. Phys. 71~8!, August 2003 http://ojps.aip.org/ajp/ © 2003 American Association of Physics Teachers This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 165.123.34.86 On: Wed, 08 Oct 2014 11:05:44III. RADIATION PRESSURE Doppler cooling is mediated by radiation pressure. A sta- tionary atom with an f50 ground state and an f51 excited state, placed in a laser beam, scatters photons at a net rate1 R5sG 2~s1L!~1! wheres5I/Isis the saturation parameter, given by the ratio of the laser intensity Ito the saturation intensity Isof the transition. The Lorentzian factor L511(2dl/G)2accounts for the reduction in scattering rate when the laser’s frequency vlis detuned from the transition frequency vaof the atom (dl5vl2va). While the momenta of absorbed photons have a common direction kˆ, spontaneously emitted photons are directed randomly. As a result, absorption-emission cycles lead to a net transfer of momentum to the atom resulting in aforce F5R\k. ~2! When the atom is moving with velocity vor located in a region of nonzero magnetic ﬁeld B, Eq. ~2!holds as well if dlis replaced by d5dl2k"v1vzwhere vzis the Zeeman shift of the atomic transition. Cooling occurs if an atom is irradiated from two opposite directions 6kˆwith identical laser beams of negative detun- ing. To ﬁrst order, the forces of the two beams may be added and the resultant can be shown1to have an approximate lin- ear behavior F52gvv’, with v’5kˆ"vthe transverse veloc- ity and gvthe damping constant. This leads to a cooling rate dEk/dt52gvvrms2withEk5(1 2mvrms2) the rms kinetic en- ergy. In addition, due to the statistical nature of absorption and the random character of spontaneous emission, an atomexecutes Brownian motion in velocity space, characterized by a diffusion constant D5md E k/dt. The temperature limit is found by equating the diffusional heating with the rate of cooling, from which T5D/gv. In addition to cooling, trapping occurs if the atom is irra- diated from opposite directions with laser light of opposite circular @s(1)ands(2)] polarizations in a region of inho- mogeneous magnetic ﬁeld. For a ﬁxed velocity, the net force may now be shown1to have an approximate linear depen- denceF52gsx~wherexis the transverse spatial coordi- nate!if the ﬁeld strength has a linear dependence on xand its direction is assumed to coincide with the propagation direc- tion of the laser beams. In this case gsis a spring constant. For atoms with f.0 in the ground state, the approximate linear dependence of the force holds as well. However, the situation is complicated by optical pumping effects betweenthe ground state sublevels, which leads to dynamical varia-tions of the damping and spring constants when the velocityor spatial coordinates change. These are due to the variationof the transition strength Gwith the ground and excited state magnetic quantum numbers. For a speciﬁc ( f,m)!(f 8,m8) transition, Gis proportional to the squared Clebsch-Gordan coefﬁcient u(fm1euf8m8)u2where the quantum number e 50,61 denotes the polarization of the light.1IV. DESCRIPTION OF THE PROGRAM The program presented here does not make explicit use of expressions for the damping, spring, and diffusion coefﬁ-cients. Instead, it simply keeps track of all velocity changesdue to absorption and emission of photons. 14Whether a pho- ton is absorbed from any particular laser beam during a cer-tain time interval is determined by a Monte Carlo methodbased on the rate of absorption @Eq.~1!#from each laser beam for the speciﬁc position and velocity of the atom at thestart of the interval. After absorption, Monte Carlo methodsare also used to treat optical pumping and to choose thedirection in which the emitted photon departs. A simulation is built up from a number of laser cooling stages, each of which has its own set of laser and magneticﬁelds.Atoms are followed on their trajectories through thesestages. At each point on the trajectory, the rate of absorption of photons for each laser ﬁeld ~indexi!, is calculated for the ground-state sublevel that the atom is in at this point.Atime intervaldtis then chosen such that the probability dP ifor any photon to be absorbed in this interval is small. For each laser ﬁeld, a random pick from a Poisson distribution with averagedPinow decides how many photons Niare actually absorbed, which usually leads to Ni50, but there is a small chance that Ni51 for one speciﬁc laser ﬁeld, i. In that case, the atom’s velocity is changed by the recoil due to this event and it is now considered to be in the excited state. The prob-ability that this excited atom will emit a photon with a par-ticular polarization is proportional to the appropriate squaredClebsh-Gordan coefﬁcient.Arandom pick, weighted accord-ingly, decides which type of photon is emitted, and thus towhat ground-state sublevel the atom returns. An appropriaterecoil, with direction picked randomly from an isotropicemission distribution is then added to the atom’s velocity,and the atom is returned to the ground state. Independent ofwhether an absorption-emission cycle has occurred or not,the atom is now propagated along its trajectory assuming acceleration-free motion during dt. At the new point, the evaluation of absorption rates starts again, and so on, until the end of all stages is reached. The description afforded by the program is limited to two- dimensional situations, i.e., it treats transverse motion with asingle coordinate. In this case it is in practice always possibleto choose the quantization axes of laser and magnetic ﬁeldssuch that a rate equation description remains valid, as is fur-ther explained in a manual for the program. 14True three- dimensional calculations would need to account for the co-herence between magnetic sublevels, requiring a densitymatrix treatment that would make the program substantiallymore complex. In our experience, the lack of a full three-dimensional treatment is not a serious limitation to the pro-gram’s applicability. The program allows the user to deﬁne a number of con- secutive laser cooling stages with adjustable length, each ofwhich has its own laser beam conﬁguration and magneticﬁeld distribution. Several laser beams may be conﬁgured foreach stage. The trajectories of a number of atoms thattraverse these stages may then be calculated and visualizedon the computer screen. These visualizations are two-dimensional graphical representations of the transverse posi-tion, the transverse and the axial velocity of the atoms versustheir axial position. Colors are used to discern between themagnetic sublevels at each point. Binned, transverse position 761 761 Am. J. Phys., Vol. 71, No. 8, August 2003 E. J. D. Vredenbregt and K. A. H. van Leeuwen This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 165.123.34.86 On: Wed, 08 Oct 2014 11:05:44and velocity proﬁles at the end of each stage are also avail- able as well as a phase-space density plot of the transversecoordinates. In addition, numerical values of all position andvelocity variables at the end of each stage are available asnumerical output for further evaluation. For single atom tra-jectories, the entire trajectory can be output in numericalform. Setting up the input to the simulations is controlled by dialog boxes and the input can be saved to a conﬁgurationﬁle for later reuse or modiﬁcation. The number of atomictrajectories to be calculated may be varied as well as thetime-step so that the user can increase the statistical and nu-merical accuracy of the results. Various types of atoms arepre-programmed and new ones can be quickly added. Thedistribution of initial transverse positions and velocities ofthe atoms can be evenly spaced or chosen randomly and theirwidths adjusted. To simulate the distribution of axial veloci-ties for various sources, this can be chosen to be a Gaussianof variable width centered around an adjustable average. On-line help and tooltips are also built in. V. APPLICATION TO EXPERIMENTS While it is quite possible to make reasonable analytical estimates of the effects of laser cooling, experiment designand veriﬁcation often need to go beyond the approximationsthat analytical treatments can provide, e.g., to describe theeffects of optical pumping in multi-level atoms. In such casesnumerical simulations can provide a closer approximation.Here we brieﬂy discuss simulations of magneto-optic com-pression of an atomic beam. Laser cooling of an atomic beam often involves slowing down the beam with the Zeeman technique to bring its aver-age velocity down to the capture range of a trap or to narrowits longitudinal velocity distribution. This always leads to alarge increase in beam divergence, not only because the lon-gitudinal component of the velocity is reduced but also be-cause it requires tens of thousands of photons to be scattered,giving substantial diffusion in the transverse direction. Tocounteract the increased divergence with its correspondingreduction in atomic density, a funnel for atoms can be used tomold the beam back to a pencil shape. Such funnels or com-pressors have indeed been experimentally demonstrated. 15 They are basically two-dimensional versions of the magneto-optical trap, where the atoms are both forced back to thebeam axis as well as transversely cooled during its traversal,while on the other hand the axial motion of the atoms re-mains nearly unchanged. The basic conﬁguration of a compressor is a two- dimensional quadrupole ﬁeld with transverse components B x5G(z)xandBy5G(z)yin combination with four circu- larly polarized laser beams, alternately s(1)ands(2)in character, as illustrated in Fig. 1. The ﬁeld gradient G(z)i s an increasing function of the axial coordinate designed to maximize the spatial capture range rcat the beginning of the device (z50) while at the same time increasing the spatial conﬁnement toward its end ( z5L). Here,rcis given by the radius at which the magnetic ﬁeld tunes the atom into reso- nance with the laser due to the Zeeman shift, and thus in- versely proportional to G(0); the spatial conﬁnement is characterized by the derivative of the radiative force with respect to the radial coordinate, which is proportional to G(L). Optimizing such a device involves a trade-off be- tween the beam velocity for which it remains effective, itslength ~limited by practical requirements such as the avail- able laser power !, its spatial capture range, the laser param- eters and the ﬁnal beam diameter and divergence achieved. In general, a near-linear dependence G(z)’gzwith a large value for gis proﬁtable. While an analytic limit to the value ofgfollows from the maximum possible value of the scat- tering force, in practice gmust be chosen lower to allow efﬁcient optical pumping between magnetic substates at the crucial point where the action of the compressor changesfrom acceleration toward the axis to cooling of the transversevelocity component. In addition, experimental realizations of G(z) in general have a more complicated shape than the simple linear form and in particular do not conform to G(0)50. For these and other reasons, simulations of atomic trajectories are very helpful in dimensioning the parameters of the device and in characterizing its effectiveness, and wehave used our code extensively for this purpose. 5,7 Figure 2 shows a calculation of atomic trajectories done with the program presented here for a device that is close tothe one we actually implemented in our atomic beam apparatus. 7Here we used metastable neon atoms with f52 andvi5100 m/s, appropriately polarized laser ﬁelds charac- terized by d522Gands51 in a device with L50.1 m. The magnetic ﬁeld gradient used smoothly increases from G(0)50.055 T/m to G(L)50.55 T/m and is input from a data ﬁle that contains magnetic ﬁeld measurements done in the actual device. Under these conditions the trajectorieshave a clear funnel shape in real space, while in velocityspace we observe ﬁrst a region of acceleration toward theaxis followed by transverse cooling. The ﬁnal diameter anddivergence of the atomic beam can be obtained from thenumerical output of the program. VI. INSTRUCTIONAL USE In our department we teach an advanced undergraduate course on laser cooling and trapping, based on material fromthe recent book by Metcalf andVan der Straten. 1While some Fig. 1. Schematic view of a magneto-optical compressor.Atoms traveling in thezdirection traverse a two-dimensional quadrupole magnetic ﬁeld pro- vided by four magnets. Laser beams of opposite circular polarization irra-diate the atomic beam from the transverse directions causing a dampedmotion of the atoms toward the beam axis. Positioning the magnets near the end of the device creates a magnetic ﬁeld gradient that increases with z. 762 762 Am. J. Phys., Vol. 71, No. 8, August 2003 E. J. D. Vredenbregt and K. A. H. van Leeuwen This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 165.123.34.86 On: Wed, 08 Oct 2014 11:05:44of this material may be too extensive or too advanced for a generalatomicphysicscourse,itillustratesthebasicsoflasercooling very well through a quite accessible treatment ofDoppler cooling theory. This involves such basic principlesas the linear and angular momentum of photons, excitationline shapes, Zeeman sublevels and Zeeman shift, rate of ab-sorption and emission, and Brownian motion.We have foundit useful to illustrate these with the program presented in thispaper. We create problem sets that require students to applythe theory treated in the lectures in order to ﬁnd appropriateinput for the program, and to check the results of the theoryagainst self-executed simulations. In this way we try to in-crease involvement with the material by giving students akind of hands-on experience with the material. In this re-spect, we are much helped by the wide-spread use of laptopcomputers by undergraduate students at Eindhoven Univer-sity of Technology. The problems treated can easily have different levels of difﬁculty depending on how far the course has advanced.Anearly use of the program in our course is to set up a simula-tion from which the dependence of the radiation pressureforce on velocity can be derived. Through basic questions,students develop appropriate input ~such as the velocity range to consider and the interval length !to enter in the dialog boxes of the program and run the simulation. Thenumerical output of the program can then be used with anyspreadsheet program to produce the resulting velocity changeas a function of initial velocity, which can be compared tothe simple analytical expression @Eq.~2!#in both shape and magnitude, and to ﬁnd the velocity damping coefﬁcient. Fig-ure 3 illustrates the result. Variation of the parameters is thenused to gain additional insight. Later problems build on thisone by, e.g., ﬁrst changing the input to allow for a numericalcalculation of the velocity diffusion for initially stationaryatoms to be compared to theory, and then allowing the simu- lation to run on until a stationary vrmsis found. This value isthen compared to that derived from the damping and diffu- sion constants. The inﬂuence of the polarization of light on excitation of atoms becomes obvious when f.0-atoms are introduced.Acondensed example of a problem set developed along these lines is available from the EPAPS-depository.17 Slowing of a ~typical !beam of atoms with an average axial velocity of 500 m/s and a rms spread of 50 m/s with theZeeman technique 1is the subject of a more advanced prob- lem. Here we ask students to ﬁnd input parameters for theprogram such as polarization of the light and the requiredmagnetic ﬁeld, as found from the Doppler shift of atomsentering the device where the deceleration is effected ~the Zeeman ‘‘slower’’ !. The answers are checked by running the Fig. 2. Transverse position @~a!#and transverse velocity @~b!#as a function of axial coordinate during compression of an atomic beam of f52 atoms. In position space, a clear funnel shape is observed while in velocity space acceleration toward the axis is followed by cooling. Vertical scales are ~a!20.02 ,x(m),0.02~b!250,v’(m/s) ,50 while the horizontal scale is 0 ,z(m),0.1. Colors denote the magnetic quantum number mwith 2f<m<f. The plots are screen shots of the graphical output of the program. Parameters are given in the text. Fig. 3. Change in transverse velocity vs initial velocity. The points arenumerical calculations while the line represents the result of Eqs. ~1!and ~2!. Parameters were d52G,s51, interaction time 6 ms,\k/m 516.1 mm/s and G5(2p)5.8 MHz ~mimicking metastable Ar !. The plot was made with a spreadsheet program using the numerical output of theprogram. 763 763 Am. J. Phys., Vol. 71, No. 8, August 2003 E. J. D. Vredenbregt and K. A. H. van Leeuwen This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 165.123.34.86 On: Wed, 08 Oct 2014 11:05:44simulation. We ask for explanations of the behavior seen in the axial motion ~both deceleration and cooling !and in the transverse motion ~diffusional heating due to spontaneous emission !of the atoms, and for the inﬂuence of the laser detuning ~which allows atoms to exit the slower with adjust- able velocities !. Other considerations are the modiﬁcations necessary to switch from a conventional slower, with maxi-mum ﬁeld at the entrance and zero ﬁeld at the exit, to onethat has ﬁelds of equal magnitude but opposite direction onboth sides, a solution that is popular in experimental labora-tories. Figure 4 shows a typical output of the program underthese conditions. A further problem treats the character of motion in a magneto-optical trap. A realistic ﬁeld gradient is developedby considering the spatial capture range of the trap, whichfollows from a comparison of the Zeeman shift and laserdetuning. Students are asked to investigate and then explainthe inﬂuence of the polarization of the light on the trapping. Depending on the laser and magnetic ﬁeld parameters, vari-ous types of damped harmonic oscillator motion are ob-served in the graphical output of the program. Their fre-quency and damping rate are easily accessible theoretically.The equilibrium position distribution is calculated and com- pared to what is to be expected from the equilibrium tem-perature and the theoretical spring constant. VII. CONCLUSION We have discussed the application of a graphically ori- ented laser cooling simulation program to illustrate andclarify course material as well as to the design of experi-ments. It is our hope that others may ﬁnd it useful as well.We therefore offer the complete code for download from ourwebsite 16as well as the EPAPS-depository17in two forms, a C11source version that should be amenable to extension, and a ready-to-run executable for the Windows platform. Some examples have been included in the downloadable ﬁlesso as to aid in using the code; in addition we welcome in-quiries for clariﬁcation and comments. ACKNOWLEDGMENTS We are pleased to acknowledge contributions from many students and colleagues to the ideas behind this program as Fig. 4. Slowing of metastable neon atoms @\k/m530 mm/s, G5(2p)8.2 MHz] in a midﬁeld-zero Zeeman slower. The magnetic ﬁeld has the optimum B 52B0/21B0A(12z/L) shape with B0553 mT and L52m. Laser parameters are d5263Gands52. Under these conditions the atoms leave the slower with vi’90m/s. The ﬁgures show ~a!transverse position, ~b!transverse velocity, ~c!axial velocity vs axial position. The plots are screen shots of the graphical output of the program. Vertical scale: ~a!20.01<x(m)<0.01, ~b!210<v’(m/s) <10,~c!0<vi(m/s) <600; horizontal scale: 20.1<z(m) <2.1. 764 764 Am. J. Phys., Vol. 71, No. 8, August 2003 E. J. D. Vredenbregt and K. A. H. van Leeuwen This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 165.123.34.86 On: Wed, 08 Oct 2014 11:05:44well as to the actual code. This work was sponsored by the Netherlands Foundation for Fundamental Research on Mat-ter~FOM !. a!Electronic mail: e.j.d.vredenbregt@tue.nl 1H. Metcalf and P. van der Straten, Laser Cooling and Trapping ~Springer, New York, 1999 !. 2C. E. Wieman, ‘‘The Richtmyer Memorial Lecture: BoseEinstein Conden- sation in an Ultracold Gas,’’Am. J. Phys. 64, 847–855 ~1996!. 3Ph. W. Courteille, V. S. Bagnato, and V. I. Yukalov, ‘‘Bose-Einstein Con- densation of Trapped Atomic Gases,’’ Laser Phys. 11, 659–800 ~2001!. 4C. Wieman, G. Flowers, and S. Gilbert, ‘‘Inexpensive laser cooling and trapping experiment for undergraduate laboratories,’’ Am. J. Phys. 63, 317–330 ~1995!. 5E. J. D. Vredenbregt, K. A. H. van Leeuwen, and H. C. W. Beijerinck, ‘‘Booster for ultra-fast loading of atom traps,’’ Opt. Commun. 147, 375– 381~1998!. 6M. D. Hoogerland, J. P. J. Driessen, E. J. D.Vredenbregt, H. J. L. Megens, M. P. Schuwer, and H. C. W. Beijerinck, ‘‘Bright thermal beams by lasercooling: A 1400-fold gain in beam ﬂux,’’Appl. Phys. B: Lasers Opt. 62, 323–327 ~1996!. 7J. Tempelaars, R. Stas, P. Sebel, H. Beijerinck, and E. Vredenbregt, ‘‘An intense, slow and cold beam of metastable Ne(3 s)3P2atoms,’’Eur. Phys. J. D18, 113–121 ~2002!. 8S. Kuppens, J. Tempelaars, V. Mogendorff, B. Claessens, H. Beijerinck, and E. Vredenbregt, ‘‘Approaching Bose-Einstein condensation of meta- stable neon: Over 109trapped atoms,’’ Phys. Rev. A 65, 023410 ~2002!.9P. Gould, ‘‘Laser Cooling of Atoms to the Doppler Limit,’’Am. J. Phys. 65, 1120 ~1997!. 10P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, ‘‘Optical Molasses,’’ J. Opt. Soc. Am. B 6, 2084–2107 ~1989!. 11J. Dalibard and C. Cohen-Tannoudji, ‘‘Laser cooling below the Doppler limit by polarization gradients: Simple theoretical models,’’ J. Opt. Soc.Am. B6, 2023–2045 ~1989!. 12A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen- Tannoudji, ‘‘Laser Cooling below the One-Photon Recoil Energy byVelocity-Selective Coherent Population Trapping,’’ Phys. Rev. Lett. 61, 826–829 ~1988!. 13J. Lawall, S. Kulin, B. Saubamea, N. Bigelow, M. Leduc, and C. Cohen- Tannoudji, ‘‘Three-Dimensional Laser Cooling of Helium Beyond theSingle-Photon Recoil Limit,’’ Phys. Rev. Lett. 75, 4194–4197 ~1995!. 14The physics behind the program is explained in somewhat more detail in a pdf-ﬁle that is available for download from the website. This is meant as aguide for the user, and has additional information about the inherent limi-tations of the description it offers. 15Reference 7 has a number of references discussing magneto-optic com-pressors. 16Web-link http://www.phys.tue.nl/aow/Pages/Downloads.htm 17See EPAPS Document No. E-AJPIAS-71-017307 for the simulation soft-ware, accompanying manual, and example problems. A direct link to thisdocument may be found in the online article’s HTML reference section.The document may also be reached via the EPAPS homepage ~http:// www.aip.org/pubservs/epaps.html !or from ftp.aip.org in the directory /epaps. See the EPAPS homepage for more information. MAYER’S ONE GOOD IDEA In an age in which German science was rapidly becoming professionalized, Mayer remained a thorough dilettante. He conducted almost no experiments, and although he had an exact, numericalturn of mind, he neither fully understood mathematical analysis nor ever employed it in his papers.His scientiﬁc style, his status as an outsider to the scientiﬁc community, and his lack of institu-tional afﬁliation were all factors that limited Mayer’s access to inﬂuential journals and publishersand hampered the acceptance of his ideas. Mayer was a conceptual thinker whose genius lay in theboldness of his hypotheses and in his ability to synthesize the work of others. Mayer actuallypossessed only one creative idea–his insight into the nature of force–but he tenaciously pursuedthat insight and lived to see it established in physics as the principle of the conservation of energy. R. Steven Turner, Dictionary of Scientiﬁc Biography ~Charles Scribner’s Sons, New York, 1974 !, p. 240. Submitted by Herman Erlichson. 765 765 Am. J. Phys., Vol. 71, No. 8, August 2003 E. J. D. Vredenbregt and K. A. H. van Leeuwen This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 165.123.34.86 On: Wed, 08 Oct 2014 11:05:44
1.3289588.pdf	Influence of multiple magnetic phases on the extrinsic damping of soft magnetic films Bailin Liu , Yi Yang , Dongming Tang , Jiangwei Chen , Huaixian Lu , Mu Lu , and Yi Shi Citation: J. Appl. Phys. 107, 033911 (2010); doi: 10.1063/1.3289588 View online: http://dx.doi.org/10.1063/1.3289588 View Table of Contents: http://aip.scitation.org/toc/jap/107/3 Published by the American Institute of Physics Inﬂuence of multiple magnetic phases on the extrinsic damping of FeCo–SiO 2soft magnetic ﬁlms Bailin Liu, Yi Yang,a/H20850Dongming T ang, Jiangwei Chen, Huaixian Lu, Mu Lu, and Yi Shi Department of Physics, Nanjing University, Nanjing, Jiangsu 210093, China /H20849Received 4 November 2009; accepted 10 December 2009; published online 9 February 2010 /H20850 In order to investigate the high-frequency damping properties of the ferromagnetic ﬁlm for the electromagnetic shielding applications, a series of /H20849FeCo /H208502x//H20849Fe/H20850x//H20849SiO 2/H208501−3xnanogranular ﬁlms with various volume fractions /H20849x/H20850were fabricated by alternate triple-target magnetron sputtering. The /H20849FeCo /H208500.50 /Fe0.25 //H20849SiO 2/H208500.25 ﬁlm shows excellent soft magnetic properties /H208494/H9266Ms /H110111.48 T, Hce/H110114.3 Oe, /H9267/H110114.3/H9262/H9024m/H20850. In both ferromagnetic resonance /H20849FMR /H20850and frequency-dependent permeability spectra measurements, two resonance peaks of the permeability for this ﬁlm are obtained, which can be attributed to the complicated magnetic structure of FeCo andFe phases in the ﬁlm. This multiphase system makes an additional contribution to the extrinsicdamping. As a result, the higher natural resonance frequency ffor the ﬁlm is up to 2.75 GHz with full width at half maximum of about 2.5 GHz for the imaginary part /H9262/H20648; meanwhile, the real part /H9262/ is as high as 650 while f/H110211.3 GHz. These ﬁlms could be novel candidates for the electromagnetic shielding applications. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3289588 /H20852 I. INTRODUCTION With the development of telecommunication technology and highly integrated electronic devices, electromagneticshielding has been intensively studied in the past years tosatisfy the requirements of reducing undesirable electromag-netic radiation and protecting delicate components from pos-sible electromagnetic interference. Meanwhile, these re-searches have been motivated by persisting concerns aboutthe possible impact of electromagnetic energy on people’shealth while using mobile phones and computers. 1,2Different from conventional bulk magnetic materials, magnetic thinﬁlms show more interesting physical phenomena and poten-tial values for electromagnetic applications. Two importantparameters are responsible for the shielding properties of themagnetic ﬁlms in gigahertz regime. The ﬁrst is the real partof permeability /H20849 /H9262//H20850in the high frequency range, which de- termines the impedance properties of ﬁlms. High value /H9262/ could make the incident microwave penetrate into the ﬁlms easily and turn into another energy form. Soft magneticgranular thin ﬁlms, which contain FeCo crystal phase andinsulators /H20849such as SiO 2,A l 2O3, Hf–O, etc. /H20850, have a promis- ing potential application of the electromagnetic shielding be-cause of their combined properties of high saturation magne-tization and suitable in-plane anisotropy ﬁeld, which canprovide a high permeability in gigahertz range. 3–5 The second factor for the shielding application is the damping performance in the high frequency range. Accord-ing to its origins, the damping can be decomposed into in-trinsic and extrinsic parts. Both the two-magnon scatteringand the local anisotropy dispersion contribute signiﬁcantly tothe extrinsic part of the damping, which could be representedby line broadening of the frequency-dependant permeabilityspectrum for the imaginary part. 6According to the results inRefs. 7and8, for the larger grain inhomogeneities in the ﬁlm, local anisotropies make more important contributions tothe line broadening of permeability spectrum than the two-magnon scattering. In order to obtain the larger extrinsicdamping, the enhancement of local anisotropy in the ferro-magnetic ﬁlms is very important and promising. More atten-tions have been paid to the local anisotropy in the singlemagnetic phase ﬁlm; 9however, the combined impact on the extrinsic damping brought by the magnetic multiphase sys-tem of the ferromagnetic ﬁlm has been less considered andreported recently. The controllable magnetic multiphase sys-tem can make a new additional contribution to the extrinsicdamping, and the excellent soft magnetic properties of theﬁlms can be successfully maintained at the same time. In the present work, our efforts have been devoted to the microstructure and dynamic performance of /H20849FeCo /H20850 2x/ /H20849Fe/H20850x//H20849SiO 2/H208501−3xnanogranular thin ﬁlms with various vol- ume fractions /H20849x/H20850, which were fabricated by designed alter- nate triple-target magnetron sputtering and contained sepa- rate FeCo and Fe magnetic phases. Two resonance peaks,appeared in both ferromagnetic resonance /H20849FMR /H20850and frequency-dependent permeability spectra measurements,could be ﬁtted very well by the Landau–Lifschitz–Gilbertequation. Due to the complicated system of FeCo and Femagnetic phases, these ﬁlms exhibit more attractive micro-wave absorption performance for high-frequency shieldingapplications. II. EXPERIMENTS The /H20849FeCo /H208502x//H20849Fe/H20850x//H20849SiO 2/H208501−3xdiscontinuous multilayer nanogranular thin ﬁlms with various volume fractions /H20849x/H20850 and the thickness of about 120 nm were prepared by a dc/rf magnetron sputtering method on a rotating glass substratewith a buffering Ta layer /H20849about 100 nm /H20850. Three separate targets, Fe 50Co50/H20849dc/H20850,F e /H20849rf/H20850, and SiO 2/H20849rf/H20850, were set to sputter alternately with the same power of 40 W. The base pressurea/H20850Author to correspondence should be addressed. Tel.: 86-25-83593011. FAX: 86-25-83593011. Electronic mail: malab@nju.edu.cn.JOURNAL OF APPLIED PHYSICS 107, 033911 /H208492010 /H20850 0021-8979/2010/107 /H208493/H20850/033911/4/$30.00 © 2010 American Institute of Physics 107 , 033911-1 was lower than 10−5Pa. Sputtering was carried out in highly pure Ar gas with the pressure of 0.5 Pa. The chemical com-position of the ﬁlm was controlled by the time of the sub-strate staying on each target, and the composition of FeCoand Fe was kept at the same proportion. The volume ratio ofcomponents in the ﬁlm was estimated by the deposition rateof each target. During sputtering, an external static magneticﬁeld of about 150 mT was applied to induce a uniaxial mag-netic anisotropy. The static magnet properties were measuredusing a vibrating sample magnetometer /H20849VSM /H20850. The compo- sition and microstructure of the ﬁlms was, respectively, de-termined by the x-ray ﬂuorescence spectrometer and the highresolution transmission electronic microscopy /H20849HRTEM /H20850. Microwave magnetic properties were characterized by theFMR measurement using a shorted waveguide at 9.78 GHz.The permeability frequency spectra were measured by abroadband one-port line permeameter in combination withthe Agilent network analyzer from 200 MHz to 8 GHz. 10A four-point probe was served for the determination of theelectrical resistivity. All measurements were carried out atroom temperature. III. RESULTS AND DISCUSSIONS Figure 1displays the bright-ﬁeld HRTEM images for the /H20849FeCo /H208500.50 /Fe0.25 //H20849SiO 2/H208500.25ﬁlm. It can be seen that the FeCo and Fe grains were embedded uniformly in the amorphous SiO 2matrix. According to the electron diffraction patterns, the nanocrystalline of the FeCo and Fe is of bcc polycrystal-line structure. In Fig. 2, FeCo and Fe nanocrystalline grains exhibit clearly a random orientation with average size Dlower than 5 nm, and the thickness of SiO 2layer is not more than 10 nm. The ultraﬁne metal-insulating nanogranularstructure of the ﬁlm should be responsible for the magneticsoftness.Magnetization curves for the /H20849FeCo /H20850 0.50 /Fe0.25 //H20849SiO 2/H208500.25 thin ﬁlm are shown in Fig. 3. The hysteresis loop along the easy axis, which is parallel to the applied magnetic ﬁeldduring deposition, is nearly rectangular. The coercivity H cof the easy and hard axis is as small as 4.3 and 5.0 Oe, respec-tively. The saturation magnetization 4 /H9266Msis up to 1.48 T. The static in-plane anisotropy ﬁeld Hk-statis about 23 Oe, which is calculated from11 Hk-stat=2/H20885 0Hup /H20851mea/H20849H/H20850−mha/H20849H/H20850/H20852dH, /H208491/H20850 where meaandmhaare the reduced magnetization of the easy axis and hard axis measured by VSM, respectively. The up-per integration boundary H upis much higher than Hk-stat. All static magnetic measurement results mentioned above provethat the ﬁlm possesses excellent soft magnetic properties. As mentioned above, the scales of both metal grains and SiO 2insulation layers are in nano-order and below exchange interaction length. The magnetization will not follow the ran-domly oriented easy axis of the individual grains, but is in-creasingly forced to align parallel by exchange interaction.Consequently, the local magneto-anisotropies of grains andthe demagnetization effect are averaged out over an increas-ing number of grains, such that an in-plane uniaxial aniso-tropy can be induced by an external ﬁeld. 12 Figure 4shows the relationship between electrical resis- tivity /H9267and the volume ratio V of metal grains including FeCo and Fe. The /H9267decreases very rapidly with increasing V and then decreases slowly when V is lager than 0.59. It issuggested that the percolation threshold is around 0.59 forthese ﬁlms. The /H9267of the /H20849FeCo /H208500.50 /Fe0.25 //H20849SiO 2/H208500.25sample reaches 4.3 /H9262/H9024m, which could improve the impedance characteristic of the magnetic ﬁlm. The experimental and calculated FMR spectra of the /H20849FeCo /H208500.50 /Fe0.25 //H20849SiO 2/H208500.25ﬁlm are plotted in Fig. 5. The swept ﬁeld Hand microwave magnetic ﬁeld were parallel to the ﬁlm plane and perpendicular to each other during theFMR measurement. It is very interesting that two resonancepeaks appear when Hwas parallel and perpendicular to the FIG. 1. Top-view HRTEM image and electron diffraction pattern for /H20849FeCo /H208500.50 /Fe0.25 //H20849SiO2/H208500.25ﬁlm. FIG. 2. The crystal orientation for FeCo or Fe nanocrystal grains in /H20849FeCo /H208500.50 /Fe0.25 //H20849SiO2/H208500.25ﬁlm.-140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140-1.0-0.50.00.51.0M/MS H(Oe)Hard axis Easy axis(FeCo)0.50/Fe0.25/(SiO2)0.25 Hk-stat=23Oe 4/s61552Ms=1.48T Hce=4.3Oe Hch=5.0Oe FIG. 3. /H20849Color online /H20850Magnetization curves while magnetic ﬁeld H paral- lels to the easy axis and hard axis of the /H20849FeCo /H208500.50 /Fe0.25 //H20849SiO2/H208500.25thin ﬁlm.033911-2 Liu et al. J. Appl. Phys. 107 , 033911 /H208492010 /H20850 easy axis of the ﬁlm, respectively. Considering that the ﬁlm was prepared by the alternate sputtering of two metal targets, we conclude that two distinct types of magnetic nanocrystal-line phases exist in this ﬁlm, i.e., FeCo and Fe. The phenom-enon of two resonance peaks reﬂects that the magnetic mo-ments in FeCo and Fe grains present, respectively, anindependent precession to a certain extent, although the ex- change interaction exists between two phases and leads totheir magnetic softness. In order to further argue that the ﬁlm is composed of FeCo and Fe phases, the results of the FMR measurementhave been ﬁtted by using d /H9273/H20648/dHrelation obtained from the formula /H208492/H20850in Ref. 13. As depicted in Fig. 5/H20849a/H20850and5/H20849b/H20850, the calculated curves are in good correspondence with themeasurement curves. From the ﬁtting results, we estimatethe respective resonance ﬁelds H resand saturation magneti- zations 4 /H9266Msof FeCo and Fe phases. The results are shown as following: Hres/H20849FeCo /H20850=647 Oe, Hres/H20849Fe/H20850=569 Oe, 4/H9266Ms/H20849FeCo /H20850=18210 Gs, and 4 /H9266Ms/H20849Fe/H20850=20870 Gs. An- other interesting point is that the in-plane anisotropy Hkof 21 Oe, estimated by the last equation in the appendix of Ref.14, is very close to the static anisotropy ﬁeld H k-stat. This suggests that the swept ﬁeld His powerful enough to force the magnetic moments in both FeCo and Fe grains to alignparallel and be saturately magnetized nearly in the same wayas the static magnetization. The complex frequency-dependent permeability /H9262 =/H9262/-i/H9262/H20648of the ﬁlm is shown in Fig. 6. There are also two resonance peaks at about 1.75 and 2.75 GHz, respectively.As reported in Ref. 15, the ﬁlm was fabricated by similar materials and triple-target sputtering method, but there isonly one resonance peak emerging in the permeability spec-tra measurement. The different result could be mainly attrib-uted to a different interval while the substrate rotates fromone target to another. The longer interval of rotating betweensputtering targets in our experiments, which is long enoughfor the crystallization process of Fe and FeCo, provides com-pletely different microstructure evolution of magnetic nano-crystalline phases. Obviously, this is an effective approach totailor the combination and growth of sputtered FeCo and Feclusters by controlling the rotation rate of the substrates. The /H9262/H11011fcurve is perfectly ﬁtted by the formula /H208491/H20850in Ref. 16and shown in Fig. 6. The 4 /H9266Msof FeCo and Fe from the FMR ﬁtting results are used in calculation. The dynamicanisotropy ﬁelds H k-dynand damping factors /H9251are used as ﬁtting parameters, because we presume that the different dy-0.3 0.4 0.5 0.6 0.7 0.8 0. 9010203040ρ(µΩ*m) V(Metal) FIG. 4. The relationship between electrical resistivity /H9267and the total volume ratio 3 /H11003of metal grains /H20849FeCo and Fe /H20850. FIG. 5. /H20849Color online /H20850The experimental and ﬁtted FMR curves while swept ﬁeld H parallels to /H20849a/H20850the easy axis and /H20849b/H20850hard axis of the /H20849FeCo /H208500.50 /Fe0.25 //H20849SiO2/H208500.25thin ﬁlm.11 0-400-2000200400600800 α(Fe)=0.017 α(FeCo)=0.027µ',µ" f(GHz)µ' Experimental µ" Experimental µ' Calculated µ" Calculated 2 48 6 FIG. 6. /H20849Color online /H20850The experimental and ﬁtted complex permeability dependence of frequency for the /H20849FeCo /H208500.50 /Fe0.25 //H20849SiO2/H208500.25thin ﬁlm.033911-3 Liu et al. J. Appl. Phys. 107 , 033911 /H208492010 /H20850 namic anisotropies for two phases have a dominant effect for the existence of two resonance peaks in the /H9262/H11011fcurve. The dynamic anisotropy ﬁelds Hk-dynof FeCo and Fe phases are estimated about 21 and 46 Oe, respectively. Compared withthe anisotropy ﬁelds obtained from the static and FMR mea-surements, there is signiﬁcant difference in H k-dynbetween the FeCo and Fe grains in the /H9262/H11011fcurve measurement. This difference can be qualitatively interpreted as the result of thecompetition between uniaxial anisotropy constant K uand lo- cal average anisotropy constant /H20855K/H20856during dynamic magne- tization. It is well known that the growth of magnetic grains weakens the exchange couple interaction among them andenhance intensively the /H20855K/H20856of the individual grains. 17Ac- cording to the theoretical results shown in Ref. 18, a lower ratio of Ku//H20855K/H20856reﬂects the larger angular dispersion between Kuand the easiest axis of the individual grains. As a result, the magnetic moments in FeCo grains will orient to theuniaxial anisotropy ﬁeld H uwith a considerable deviation angle /H9258, which is schematically illustrated in Fig. 7. In this case, the resonance frequency of the FeCo grains are deter-mined by 14 f=/H9253/H20881/H20849Hu+HLoccos 2/H9258/H20850/H20849Hu+4/H9266Ms+HLoccos2/H9258/H20850, /H208492/H20850 where HLoc=2/H20855K/H20856//H92620Msis the average local anisotropy of FeCo grains. In comparison, the local magnetic moments in Fe grains are rotated to the macroscopic uniaxial anisotropyaxis with a vanishing angular dispersion, which will enhancethe contribution of the uniaxial anisotropy and herewith thedynamic anisotropy ﬁeld H k-dyn. However, it is still difﬁcult to quantitatively explain these amazing but not-accidentallarge dynamic anisotropy ﬁelds using the theory in Ref. 18, Hoffmann’s ripple theory 19and Acher’s DM theory;20more details of the mechanism of the exchange interaction in themagnetic multiphase system are still under investigation. It should be especially mentioned that, compared with the published results in the corresponding researchﬁelds, 4,5,15the full width at half maximum /H20849FWHM /H20850of/H9262/H20648is up to about 2.5 GHz, meanwhile, the real part /H9262/is as high as 650 while f/H110211.3 GHz. This high performance of electro- magnetic shielding applications at the high-frequency hasbeen rarely obtained before. This realization of high perme-ability with large FWHM should be attributed to the complexmagnetic structure, which consists of two magnetic nano- crystalline phases. The coexistence of FeCo and Fe nano-crystalline grains brings more abundant interfaces and in-creases grain inhomogeneities in the ﬁlm, makes anadditional and considerable contribution to extrinsic damp-ing beyond two-magnon scattering, and results in more de-sirable line broadening of the permeability. Introducing mag-netic multiphase system is a feasible and valuable approachto the improvement on the effectiveness of materials for theelectromagnetic shielding applications in a wide band and atthe high frequency. IV. CONCLUSION The /H20849FeCo /H208502x//H20849Fe/H20850x//H20849SiO 2/H208501−3xnanogranular thin ﬁlms deposited by alternate triple-target magnetron sputtering were investigated in this paper. The FeCo and Fe nanocrys-talline grains were embedded uniformly in the amorphousSiO 2matrix. The /H20849FeCo /H208500.50 /Fe0.25 //H20849SiO 2/H208500.25thin ﬁlm ex- hibited excellent soft magnetic properties, such as high 4/H9266Msof 1.48 T, low coercivity Hcof 4.3/5.0 Oe for the easy/hard axis, respectively, and high electrical resistivity of4.3 /H9262/H9024m. In particular, in both FMR and /H9262/H11011fcurve mea- surements, two resonance peaks were found. The origin ofthe two peaks was attributed to the different dynamic re-sponse of the FeCo and Fe phases. As a result, the highernatural resonance frequency fof/H20849FeCo /H20850 0.50 /Fe0.25 //H20849SiO 2/H208500.25 ﬁlm is up to 2.75 GHz with the FWHM about 2.5 GHz of /H9262/H20648; meanwhile, the real part /H9262/is as high as 650 for f /H110211.3 GHz. Such a novel property reﬂects a great potential of the magnetic multiphase ﬁlm for high-frequency electro-magnetic shielding applications. 1S. W. Kim, Y. W. Yoon, S. J. Lee, G. Y. Kim, Y. B. Kim, Y. Y. Chun, and K. S. Lee, J. Magn. Magn. Mater. 316, 472 /H208492007 /H20850. 2S. M. Yang, Y. Y. Chang, Y. C. Hsieh, and Y. J. Lee, J. Appl. Polym. Sci. 110, 1403 /H208492008 /H20850. 3N. D. Ha, A.-T. Le, M.-H Phan, H. Lee, and C. Kim, Mater. Sci. Eng., B 139,3 7 /H208492007 /H20850. 4S. Ge, D. Yao, M. Yamaguchi, X. Yang, H. Zuo, T. Ishii, D. Zhou, and F. Li,J. Phys. D 40, 3660 /H208492007 /H20850. 5F. Xu, X. Zhang, N. N. Phuoc, Y. Ma, and C. K. Ong, J. Appl. Phys. 105, 043902 /H208492009 /H20850. 6K. Seemann, H. Leiste, and A. Kovàcs, J. Magn. Magn. Mater. 320, 1952 /H208492008 /H20850. 7B. K. Kuanr, R. E. Camley, and Z. Celinski, J. Magn. Magn. Mater. 286, 276 /H208492005 /H20850. 8R. D. McMichael, D. J. Twisselmann, and A. Kunz, Phys. Rev. Lett. 90, 227601 /H208492003 /H20850. 9J. B. Youssef and C. Brosseau, Phys. Rev. B 74, 214413 /H208492006 /H20850. 10V. Bekker, K. Seemann, and H. Leiste, J. Magn. Magn. Mater. 270,3 2 7 /H208492004 /H20850. 11A. Neudert, J. Mccord, R. Schäfer, and L. Schultz, J. Appl. Phys. 95, 6595 /H208492004 /H20850. 12G. Herzer, J. Magn. Magn. Mater. 157–158 , 133 /H208491996 /H20850. 13N. X. Sun, S. X. Wang, T. J. Silva, and A. B. Kos, IEEE Trans. Magn. 38, 146 /H208492002 /H20850. 14K. Ounadjela, G. Suran, and F. Machizaud, Phys. Rev. B 40,5 7 8 /H208491989 /H20850. 15S.-I. Aoqui and M. Munakata, Mater. Sci. Eng., A 413–414 ,5 5 0 /H208492005 /H20850. 16J. B. Youssef, P. M. Jacquart, N. Vukadinovic, and H. Le Gall, IEEE Trans. Magn. 38, 3141 /H208492002 /H20850. 17K. Seemann and H. Leiste, J. Magn. Magn. Mater. 321, 742 /H208492009 /H20850. 18G. Herzer, J. Magn. Magn. Mater. 294,9 9 /H208492005 /H20850. 19H. Hoffmann, Thin Solid Films 58, 223 /H208491979 /H20850. 20O. Acher, C. Boscher, B. Brulé, G. Perrin, N. Vukadinovic, G. Suran, and H. Joisten, J. Appl. Phys. 81, 4057 /H208491997 /H20850. FIG. 7. The schematic illustration of distribution and dispersion of the mag- netic moments in Fe and FeCo grains under the competition betweenuniaxial anisotropy and local average anisotropy.033911-4 Liu et al. J. Appl. Phys. 107 , 033911 /H208492010 /H20850
1.4975693.pdf	Non-volatile spin wave majority gate at the nanoscale O. Zografos , S. Dutta , M. Manfrini , A. Vaysset , B. Sorée , A. Naeemi , P. Raghavan , R. Lauwereins , and I. P. Radu Citation: AIP Advances 7, 056020 (2017); doi: 10.1063/1.4975693 View online: http://dx.doi.org/10.1063/1.4975693 View Table of Contents: http://aip.scitation.org/toc/adv/7/5 Published by the American Institute of Physics Articles you may be interested in Nanopatterning spin-textures: A route to reconfigurable magnonics AIP Advances 7, 055601055601 (2016); 10.1063/1.4973387 Operating conditions and stability of spin torque majority gates: Analytical understanding and numerical evidence AIP Advances 121, 043902043902 (2017); 10.1063/1.4974472 Effect of nanostructure layout on spin pumping phenomena in antiferromagnet/nonmagnetic metal/ ferromagnet multilayered stacks AIP Advances 7, 056312056312 (2017); 10.1063/1.4975694 Electrical detection of single magnetic skyrmion at room temperature AIP Advances 7, 056022056022 (2017); 10.1063/1.4975998AIP ADV ANCES 7, 056020 (2017) Non-volatile spin wave majority gate at the nanoscale O. Zografos,1,2,aS. Dutta,3M. Manfrini,1A. Vaysset,1B. Sor ´ee,1,2,4 A. Naeemi,3P . Raghavan,1R. Lauwereins,1,2and I. P . Radu1 1IMEC, Kapeldreef 75, B-3001 Leuven, Belgium 2ESAT, KU Leuven, B-3001 Leuven, Belgium 3Department of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA 4Physics Department, Universiteit Antwerpen, B-2020 Antwerpen, Belgium (Presented 1 November 2016; received 23 September 2016; accepted 1 November 2016; published online 6 February 2017) A spin wave majority fork-like structure with feature size of 40 nm, is presented and investigated, through micromagnetic simulations. The structure consists of three merging out-of-plane magnetization spin wave buses and four magneto-electric cells serving as three inputs and an output. The information of the logic signals is encoded in the phase of the transmitted spin waves and subsequently stored as direction of magnetization of the magneto-electric cells upon detection. The minimum dimen- sions of the structure that produce an operational majority gate are identiﬁed. For all input combinations, the detection scheme employed manages to capture the majority phase result of the spin wave interference and ignore all reﬂection effects induced by the geometry of the structure. © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4975693] The exploration and study of novel non-charge-based logic devices has been a main research focus for over a decade.1The purpose is to identify concepts that can extend the semiconductor industry roadmap beyond the complementary metal oxide semiconductor (CMOS) technology.2Since CMOS scaling, dictated by Moore’s Law,3will reach its limits,1there is a need for logic components that can operate at high frequencies, be extremely compact and also consume ultra-low power.4A variety of magnetic devices have been benchmarked as promising candidates for low power applications.4Spin wave devices hold the promise of ultra-low power per computing throughput.4Additionally, utilizing spin waves, majority-based logic can be constructed and has been proven to be advantageous for beyond-CMOS technologies.5,6These devices have been extensively studied through experiments and micromagnetic simulations at large dimensions (down to tens of microns),7,8however the study of spin wave dynamics and interference at the nanoscale are still lacking. In this work, we investigate through micromagnetic simulations, a fork-like spin wave majority structure with feature size of 40 nm. We aim at designing a nanometer scale structure where excitation of higher-order width modes8can be avoided. The proposed design incorporates the advantages of non-volatile data storage in the ME cell, non-reciprocity via a three-phase clocking scheme9,10and robustness to thermal ﬂuctuations missing in the earlier prior designs.7,8,11The structure consists of three merging perpendicular magnetic anisotropy (PMA) spin wave buses and four magneto-electric (ME) cells serving as three inputs and an output. The geometry of the spin wave majority gate is shown in Fig. 1, where the spacing between each arm is S=88 nm. We employed micromagnetic simulations to investigate this structure, using the micromagnetic solver OOMMF.12The mesh cell size is 2 nm 2 nm12 nm and all the PMA spin wave bus regions are extended before and after the ME cell regions with increased damping to allow for magnetization relaxation and avoid edge reﬂections. Thus, the simulated structure represents an spin wave majority aElectronic mail: Odysseas.Zografos@imec.be 2158-3226/2017/7(5)/056020/6 7, 056020-1 ©Author(s) 2017 056020-2 Zografos et al. AIP Advances 7, 056020 (2017) FIG. 1. Geometry of the spin wave majority gate. Spin waves are excited by the three input ME cells (Inputs 1,2,3) and the majority result of the spin wave interference is detected by the ‘Output’ ME cell. The spacing between each arm is S=88 nm. gate arrangement on inﬁnitely long buses. The extended regions of the structure are not shown in Fig. 1 for ease of representation. The basic computational block of a spin wave logic device is the ME cell that acts as a spin wave transmitter, detector and also serves as a non-volatile memory element.9The ME cells are embedded in the bus and have in-plane magnetization (along ˆx). They are heterostructures consisting of a ferroelectric or piezoelectric material intelayered between two metallic electrodes and a top magnetostrictive ferromagnetic layer. We consider a 80 nm 40 nm12 nm Co 60Fe40/(001) PMN-PT (30 nm thick) as the ME cell het- erostructure (with magnetization saturation M S=800 kA/m, exchange constant A=20 pJ/m, Gilbert damping =0.027, magnetostrictive coefﬁcient =200 ppm, Young’s modulus Y=200 GPa, and piezo- electric coefﬁcient d31=-1000 pm/V). (001) PMN-PT is chosen as the piezoelectric layer due to its high piezoelectric coefﬁcient while Co 60Fe40displays a large magnetostrictive coefﬁcient of 2 104,13and is also compatible with PMN-PT. The spin wave bus material is to be considered a [Co(0.4)/Ni(0.8)] 10 multilayer (with M S=790 kA/m, A=16 pJ/m, =0.01, and anisotropy ﬁeld HK=16.78 kA/m). It is selected as the spin wave bus material due to its inherent interface anisotropy, thus providing a bias-free out-of-plane magnetic conﬁguration. The working principle is based on voltage-controlled strain-induced magnetization switching that excites spin waves and a phase dependent determinis- tic detection scheme, where information is encoded in the phase of the transmitted spin wave and subsequently stored as direction of magnetization of the ME cell (+ ˆxor -ˆx).9,10 An applied voltage across the piezoelectric layer causes an isotropic biaxial strain that gets coupled to the top ferromagnet causing an out-of-plane anisotropy. Above a critical strain, the mag- netization switches from an in-plane to out-of-plane conﬁguration exciting spin waves with the information encoded in the phase of the waves. Meanwhile, the detector ME cell is held out-of-plane via application of voltage until the spin waves arrive. Upon arrival, the voltage is turned off causing a phase-dependent deterministic switching of the magnetization. The temporal m xproﬁle of the spin wave generated by an ME cell is shown in FIG. 2a. We observe that the spin wave created has a wave packet-like form, with multiple frequency components (as shown in the inset of FIG. 2a) and duration shorter than 2 ns. The structure simulated to generate the spin wave in FIG. 2a is depicted in FIG. 2b. An ME cell is activated and generates a spin wave that propagates along a spin wave bus. The magnetization dynamics are monitored after 120 nm. FIG. 2b also shows the spatial m xproﬁle at three different timepoints ( t1,t2,t3). At time t1=0.065 ns, the ME cell has not switched out-of-plane and the spin wave is not formed yet. At time t2=0.77 ns, the spin wave is formed and has propagated at least 120 nm but is almost completely dispersed after t3=1.3 ns. Due to the complex nature of the spin wave, it’s impossible to extract an accurate wavelength but from the m xproﬁle at t2, we can extract its wavelength at the largest amplitude is =210 nm.056020-3 Zografos et al. AIP Advances 7, 056020 (2017) FIG. 2. Spin wave generated by ME cell. a) Temporal m xproﬁle. Inset: Frequency components of propagated spin wave. b) Spatial m xproﬁle of the spin wave bus at three different timepoints as denoted in a). For the initial study of the spin wave majority gate’s performance, we conducted single-arm excitation simulations and monitored the spin wave transmission in the complete structure. FIG. 3, presents the spin wave amplitude (deﬁned asq m2x+m2y) averaged over time (i.e. 3 ns) in logarithmic scale. The amplitude transmission from ‘Input 1’ to ‘Output’ is '93%, deﬁned as the ratio of the average intensity of the output to the average intensity of the input. This efﬁcient transmission is due to the nanoscale dimensions of the structure in combination with the low damping values of the materials assumed. The downside of the efﬁcient transmission is that there is signiﬁcant reﬂections and back-propagations (i.e. '89%, denoted by dashed arrows in FIG. 3). This is due to the geometrical symmetry of the structure (unlike Klinger et al8). The back-propagations increase the complexity of the spin wave dynamics and interference but will not affect the states of ME cells that can be interconnected before the majority gate. The ME cell concept applied in this work ensures logical non-reciprocity14due to a three-phase clocking scheme.9 In order to have a functional spin wave majority gate, we need to ensure: (a) the input ME cells switch from in-plane to out-of-plane correctly and in a similar fashion; (b) the spin waves that arrive at the output region are as close to identical as possible (unbiased inputs); (c) that the output ME cell’s detection operation is launched at the appropriate timepoint. The ﬁrst requirement is satisﬁed056020-4 Zografos et al. AIP Advances 7, 056020 (2017) FIG. 3. Spin wave amplitude transmission for single arm excitation of ‘Input 1’, plotted in a logarithmic scale. Dashed arrows demonstrate the ﬂow of back-propagated spin waves into the other input arms. since, when designing structure, we used the analytical expressions in Engel-Herbert et al15and Kani et al16to calculate the minimum arm spacing that also minimizes their dipolar coupling. This coupling would impede the ME cells to completely switch out-of-plane, thus not work properly. The minimum spacing of the arms is 56 nm and is veriﬁed by simulations. To investigate the second requirement, we study the input signals by the means of the out-of-plane angle ( ) as the angle between magnetization (M) and ˆz. The fork-like structure we employ has a mirror symmetry. However, the signals created by ‘Input 1’ and ‘Input 3’ do not follow that symmetry. The spin wave propagation and dispersion depends on the shape anisotropy variation that the Sparameter induces. This dependence is non- linear as demonstrated in inset (i) of FIG. 4, where the maximum out-of-plane angle of the output magnetization (for each single arm excitation) is plotted over different values of S. FIG. 4(i) shows that, by changing the geometry of the majority gate structure, the spin wave behavior changes. This means that, for each spacing value selected, the structure would have to be ﬁne-tuned (in terms of material parameters and input ME cell positioning) to operate correctly. The latter hinders the robustness of the current geometry and needs to be evaluated further, including different geometry options. However, an accurate robustness evaluation is considered outside the scope of this work. We note that the spacing value Swhere all three input signals have the most similar contributions to the output angle is at S=88 nm. Hence these values were selected for a functional majority gate as they lead towards satisfying the second aforementioned requirement of unbiased inputs. To further optimize the performance of the majority gate, through more micromagnetic sim- ulations, we have deﬁned the length of the spin wave bus that connects ‘Input 2’ to ‘Output’ at FIG. 4. Average out-of-plane angle of the output magnetization when excited by individual single arm excitations in a structure with S=88 nm. Based on for each spin wave signal, we select detection timepoint at tdet=0.8 ns. Inset (i): Maximum of the output magnetization when excited by individual single arm excitations, shows that the selection ofS=88 nm as the arm spacing is the best one for the explored values. Inset (ii): deﬁnition of .056020-5 Zografos et al. AIP Advances 7, 056020 (2017) FIG. 5. Spatial proﬁle of mx magnetization of the majority gate at different timepoints of operation. a) At t=0 ns, the inputs are set to ‘110’. b) At t=0.8 ns before the detection of the output ME cell is enabled, most of the magnetization oscillations are centered around the merging/output region. c) At t=3.2 ns the output magnetization is stabilized to its non-volatile state ‘1’ correctly detecting the majority result. 92 nm and a slightly increased damping of =0.016. Such local engineering of magnetic damping has been extensively studied17and it could be implemented in the spin wave bus by controlled ion bean irradiation. This method ensures the PMA could be preserved whereas the magnetic damping diminishes due to increase surface roughness. With this conﬁguration the requirement of the unbiased inputs is satisﬁed, as FIG. 4 shows that the spin wave signals from each input have almost identical contribution to the output magnetization. The third requirement is satisﬁed by the detection timepoint of tdet=0.8 ns, extracted from FIG. 4 where all three spin wave signals induce equal out-of-plane angle . To verify the operation of the majority gate we need to excite all three inputs simultaneously and monitor the detected result. We deﬁne the logic ‘0’ of the majority gate as the spin wave generated by an ME cell initially set along +ˆx(mx=1) and the logic ‘1’ as the as the spin wave generated by an ME cell set along - ˆx(mx=-1). This deﬁnition is arbitrary. FIG. 5 illustrates an example operation of the spin wave majority gate, where the input are set to ‘110’ (FIG. 5a). After the three inputs are activated, the generated spin waves propagate towards the output and interfere. At time t=0.8 ns (FIG. 5b), the detection is enabled which results in the output ME cell to stabilize at the correct majority result ‘1’ (m x=-1 - FIG. 5c). Finally, to verify the complete logic behavior of the spin wave majority gate we simulate all possible input states. The results of these simulations are summarized in FIG. 6, where we observe that all inputs that have majority of ‘0’ set the output ME cell magnetization along + ˆxand all inputs that have majority of ‘1’ set the output ME cell magnetization along - ˆx. This proves the operation of the proposed design. Another interesting fact depicted in FIG. 6 is that the output magnetization switching behavior is symmetric for symmetric inputs (e.g. for inputs ‘010’ and ‘101’), which enhances the validity of the design as one that enables symmetrical and unbiased inputs. FIG. 6. Average m xof the output for all possible input combinations resulting in the correct majority computation.056020-6 Zografos et al. AIP Advances 7, 056020 (2017) The choice of as spin wave generators and detectors is not limited to ME cells, other effects such as V oltage-Controlled Magnetic Anisotropy (VCMA)18could be used. However, the fact that the proposed majority gate utilizes the ME cell concept,9not only makes it non-volatile (characteristic of critical importance for low-energy applications) but also it provides the necessary means for cascading. Having detected and stored the majority result, the output ME cell could be easily triggered and generate the corresponding spin wave which will be detected by a cascaded ME cell interconnected with the spin wave bus. Additionally, having an ME cell operating voltage of 0.1 V , results in an ultra-low intrinsic energy dissipation per ME cell of 4.5 aJ.19 In conclusion, a fully functional, nanoscale, symmetric, non-volatile spin wave majority gate design utilizing ME cells as inputs and outputs, has been presented. The design was optimized for the correct detection of the majority result, without being disturbed by parasitic spin wave reﬂections and back propagations. The feature size of the design is 40 nm and has a total area of 0.074 m2, making it the smallest reported majority spin wave design to be functionally veriﬁed. Also, the proposed design operates in a3 ns timeframe which is fast compared to other spin-based technologies.4Finally, the combination of the proposed majority gate along with the ME cell inverter9and majority-based logic synthesis,6can enable integrated circuit possibilities that exhibit ultra low-energy and small area characteristics. SUPPLEMENTARY MATERIAL See supplementary material for an example of input-output operation utilizing ME cells and more information on the mesh cell size and the frequency components of the propagated spin wave. 1V . Zhirnov, R. Cavin, J. Hutchby, and G. Bourianoff, Proc. IEEE 9, 1934 (2003). 2J. Hutchby, G. Bourianoff, V . Zhirnov, and J. Brewer, IEEE Circuits and Devices Magazine 18, 28 (2002). 3G. Moore, Electronics 38, 114 (1965). 4D. E. Nikonov and I. A. Young, Proc. IEEE 101, 2498 (2013). 5O. Zografos, P. Raghavan, L. Amar `u, B. Sor ´ee, R. Lauwereins, I. Radu, D. Verkest, and A. Thean, in 2014 IEEE/ACM NANOARCH (2014), pp. 25–30. 6L. Amar `u, P. E. Gaillardon, S. Mitra, and G. D. Micheli, Proc. IEEE 103, 2168 (2015). 7S. Klingler, P. Pirro, T. Br ¨acher, B. Leven, B. Hillebrands, and A. V . Chumak, Appl. Phys. Lett. 105(2014). 8S. Klingler, P. Pirro, T. Br ¨acher, B. Leven, B. Hillebrands, and A. V . Chumak, Appl. Phys. Lett. 106(2015). 9S. Dutta, S.-C. Chang, N. Kani, D. E. Nikonov, I. A. Manipatruni, S. Young, and A. Naeemi, Scient. Rep. 5(2015). 10S. Dutta, D. E. Nikonov, S. Manipatruni, I. A. Young, and A. Naeemi, Appl. Phys. Lett. 107(2015). 11A. Khitun and K. L. Wang, J. Appl. Phys. 110(2011). 12M. Donahue and D. Porter, “Oommf user’s guide, version 1.0,” (1999). 13D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Bendersky, S. E. Loﬂand, M. Wuttig, and I. Takeuchi, Nat. Comm. 2(2011). 14R. Waser, Nanoelectronics and Information Technology (Wiley, 2012). 15R. Engel-Herbert and T. Hesjedal, J. Appl. Phys. 97, (2005). 16N. Kani, S. C. Chang, S. Dutta, and A. Naeemi, IEEE Trans. on Magnetics 52, 1 (2016). 17J. A. King, A. Ganguly, D. M. Burn, S. Pal, E. A. Sallabank, T. P. A. Hase, A. T. Hindmarch, A. Barman, and D. Atkinson, Appl. Phys. Lett. 104, 242410 (2014). 18T. Maruyama, Y . Shiota, T. Nozaki, K. Ohta, N. Toda, M. Mizuguchi, A. A. Tulapurkar, T. Shinjo, M. Shiraishi, S. Mizukami, Y . Ando, and Y . Suzuki, Nat. Nano. 4(2009). 19S. Dutta, R. M. Iraei, C. Pan, D. E. Nikonov, S. Manipatruni, I. A. Young, and A. Naeemi, in IEEE NANO Conference (2016), p. accepted for publication.
1.2955831.pdf	Spin-torque oscillator with tilted fixed layer magnetization Yan Zhou, C. L. Zha, S. Bonetti, J. Persson, and Johan Åkerman Citation: Appl. Phys. Lett. 92, 262508 (2008); doi: 10.1063/1.2955831 View online: http://dx.doi.org/10.1063/1.2955831 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v92/i26 Published by the American Institute of Physics. Related Articles Generation of pure phase and amplitude-modulated signals at microwave frequencies Rev. Sci. Instrum. 83, 064705 (2012) Inhomogeneous mechanical losses in micro-oscillators with high reflectivity coating J. Appl. Phys. 111, 113109 (2012) Searching for THz Gunn oscillations in GaN planar nanodiodes J. Appl. Phys. 111, 113705 (2012) High power pulse compression using magnetic flux compression J. Appl. Phys. 111, 094508 (2012) Development of a pulse programmer for magnetic resonance imaging using a personal computer and a high- speed digital input–output board Rev. Sci. Instrum. 83, 053702 (2012) Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 02 Jul 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsSpin-torque oscillator with tilted ﬁxed layer magnetization Yan Zhou,a/H20850C. L. Zha, S. Bonetti, J. Persson, and Johan Åkermanb/H20850 Department of Microelectronics and Applied Physics, Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden /H20849Received 24 February 2008; accepted 17 June 2008; published online 3 July 2008 /H20850 A spin-torque oscillator with a ﬁxed layer magnetization tilted out of the ﬁlm plane is capable of strong microwave signal generation in zero magnetic ﬁeld. Through numerical simulations, westudy the microwave signal generation as a function of drive current for two realistic tilt angles. Thetilted magnetization of the ﬁxed layer can be achieved by using a material with high out-of-planemagnetocrystalline anisotropy, such as L1 0FePt. © 2008 American Institute of Physics . /H20851DOI: 10.1063/1.2955831 /H20852 Broadband microwave oscillators, such as the yttrium iron garnet /H20849YIG /H20850oscillator, play an important role in com- munications, radar applications, and high-precision instru-mentation. Two drawbacks of the YIG oscillator is its bulknature, which foils any attempt of monolithic integration,and its magnetic tuning, which is both complicated and con-sumes high power. A modern nanoscopic analog of the YIGoscillator is the spin-torque oscillator /H20849STO /H20850: 1–8It is ex- tremely broad band /H20849multioctave /H20850, can achieve high spectral purity, and is magnetically tunable with a similar transferfunction related to ferromagnetic resonance. The STO cur-rently receives a rapidly growing interest thanks to its sig-niﬁcant advantages, such as easy on-chip integration and cur-rent tunability instead of only ﬁeld tunability. However, STOs still typically require a large, static, magnetic ﬁeld for operation; removing the need for this ﬁeld is currently an intensely researched topic. As suggested byRedon et al. , 9,10a perpendicularly polarized ﬁxed layer may drive an in-plane magnetization into an out-of-plane preces-sional state even in the absence of an applied ﬁeld, whichwas recently experimentally demonstrated 11using a perpen- dicularly polarized Co /Pt multilayer as ﬁxed layer. However, due to the axial symmetry of the ﬁxed layer magnetizationand the precession, an additional read-out layer was requiredto break the symmetry and generate any signal, which com-plicates the structure and its fabrication; the signal quality isso far also quite limited compared to conventional STOs. Adifferent solution, suggested by Xiao et al. 12and developed in detail by Barnas et al. ,13is based on a wavy angular de- pendence of the spin torque, obtained by judicially choosingfree and ﬁxed layer materials with different spin diffusionlengths. Boulle et al. recently fabricated such a “wavy torque” STO and demonstrated current tunable microwavegeneration in zero ﬁeld, 14the output signal is again quite limited, partly caused by the associated asymmetric magne-toresistance /H20849MR /H20850. A radically different approach was taken by Pribiag et al. , who introduced a magnetic vortex in a thick free layer and excited zero-ﬁeld gyromagnetic precession ofthe vortex core through the spin torque from a conventionalﬁxed layer. 15While the signal quality of this vortex STO is excellent, its frequency range is quite limited, so far onlydemonstrated below 3 GHz. In this letter, a tilted-polarizer STO /H20849TP-STO /H20850has beenstudied where the ﬁxed layer magnetization /H20849M/H20850is tilted out of the ﬁlm plane. The spin polarization hence has both in- plane components /H20849p x,py/H20850and a component along the out-of- plane direction /H20849pz/H20850. We show that pzcan drive the free layer into precession without the need for an applied ﬁeld, while the in-plane component Mxof the ﬁxed layer magnetization generates a large MR, i.e., a rf output without the need for anadditional read-out layer. While Mmay have any out-of- plane direction in the general situation, we limit our discus-sion to the x−zplane, M=/H20849M x,0,Mz/H20850=/H20841M/H20841/H20849cos/H9252,0,sin /H9252/H20850, and/H9252=36° and 45° /H20849Fig.1/H20850, since these two particular angles can be achieved using different crystallographic orientationsof FePt. The time evolution of the free layer magnetization mˆis found using the standard Landau–Lifshitz–Gilbert– Slonczewski equation, dmˆ dt=−/H9253mˆ/H11003Heff+/H9251mˆ/H11003dmˆ dt+/H9253 /H92620Ms,free/H9270, /H208491/H20850 where mˆis the unit vector of the free layer magnetization, Ms,freeits saturation magnetization, /H9253the gyromagnetic ratio, /H9251the Gilbert damping parameter, and /H92620the magnetic vacuum permeability. Setting the applied ﬁeld to zero andseparating the effect of the demagnetizing tensor into a posi-tive anisotropy ﬁeld along xand a negative out-of-plane de- magnetizing ﬁeld we get H eff=/H20849Hkeˆxmx−Hdeˆzmz/H20850//H20841m/H20841.W e deﬁne positive current as ﬂowing from the ﬁxed layer to the free layer. In this study, the lateral dimension of the NiFe a/H20850Electronic mail: zhouyan@kth.se. b/H20850Electronic mail: akerman1@kth.se. FIG. 1. /H20849Color online /H20850/H20849a/H20850Schematic of a TP-STO. Mis the tilted ﬁxed layer magnetization. The free layer magnetization mis separated from the ﬁxed layer by a nonmagnetic /H20849NM /H20850layer; /H20849b/H20850the coordinate system used in this work. Mlies in the x-zplane with angle /H9252with respect to the x-axis.APPLIED PHYSICS LETTERS 92, 262508 /H208492008 /H20850 0003-6951/2008/92 /H2084926/H20850/262508/3/$23.00 © 2008 American Institute of Physics 92, 262508-1 Downloaded 02 Jul 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsthin ﬁlm free layer is assumed to be an elliptical shape of 130/H1100370 nm2, with a thickness of 3 nm. The thickness of the ﬁxed layer FePt is 20 nm. The values of some parametersused in the calculation are listed as follows:16/H9251=0.01, /H20841/H9253/H20841 =1.76 /H110031011Hz /T,Ms=860 kA /m, and Hk=0.01 T, and Hd=1 T. The quantity /H9270in Eq. /H208491/H20850is the Slonczewski spin transfer torque density, /H9270=/H9257/H20849/H9272/H20850/H6036J 2edmˆ/H11003/H20849mˆ/H11003Mˆ/H20850, /H208492/H20850 where /H9272is the angle between mˆandMˆ,dis the free layer thickness, and /H9257/H20849/H9272/H20850=q+ A+Bcos/H20849/H9272/H20850+q− A−Bcos/H20849/H9272/H20850, /H208493/H20850 where q+,q−,A, and Bare all material dependent parameters.16In our simulations below we use Cu as spacer, Permalloy /H20849Py/H20850as the free layer, and FePt as the ﬁxed layer. Due to the lack of available parameters for the Cu /FePt in- terface, we approximate /H9257/H20849/H9272/H20850in our Py /Cu /FePt stack using literature values for Py /Cu /Co.12,13,16,17This approximation may be justiﬁed if a thin polarizing layer of Co is used at theCu /FePt interface. The following parameters are adopted for calculating the spin transfer torque coefﬁcient /H9257/H20849/H9272/H20850based on the asymmetric Slonczewski model:12,18Py, bulk resistivity /H9267bulk=16/H9262/H9024cm, spin asymmetry factor /H9252=0.77, and spin- ﬂip length lsf=5.5 nm; Cu, /H9267bulk=0.5/H9262/H9024cm and lsf =450 nm; and Co, /H9267bulk=5.1/H9262/H9024cm, /H9252=0.51, and lsf =60 nm. For the Py/Cu interface we assume the interfacial resistance per unit square 0.5 /H1100310−15/H9024m2, interface spin asymmetry factor 0.72, and for the Co /Cu interface we as- sume the interfacial resistance per unit square 0.52/H1100310 −15/H9024m2, interface spin asymmetry factor 0.76. Using the above parameters, we can calculate the coefﬁcients q+, q−,A,B, and the angular dependence of spin transfer torque /H9257/H20849/H9272/H20850based on Eq. /H208497/H20850in Ref. 12. We use the following generalized form for describing the angular dependence of MR:19,20 r=R/H20849/H9272/H20850−RP RAP−RP=1 − cos2/H20849/H9272/2/H20850 1+/H9273cos2/H20849/H9272/2/H20850, /H208494/H20850 where ris the reduced MR, /H9273is an asymmetry parameter describing the deviation from sinusoidal angular dependence,and R Pand RAPdenotes the resistance in the parallel and antiparallel conﬁgurations, respectively. The asymmetric torque and the asymmetric MR are de- rived for in-plane spin polarizations and magnetizations, andshould still hold as long as spin-orbit coupling is weak.While this is true for Py, it might be questionable for FePtdue to its large magnetocrystalline anisotropy. Any deviationdue to strong spin-orbit coupling will not change the generalresult of our study and is likely further weakened by the thinpolarizing layer of Co on top of FePt. Figure 2/H20849a/H20850shows the precession frequency versus drive current density for the two selected angles. We observe pre-cession at both positive and negative currents and the fre-quency increases with the magnitude of the current density,similar to perpendicularly polarized STOs. 10,21,22The preces- sion starts along the equator and continues to follow increas-ing latitudes of the unit sphere throughout the entirefrequency range /H20849fincreases due to the increasing demagne-tizing ﬁeld /H20850until it reaches a static state at the north /H20849south /H20850 pole for large negative /H20849positive /H20850current /H20851Fig. 2/H20849c/H20850/H20852.W e highlight six orbits /H20849A–F /H20850, which correspond to points in Figs. 2/H20849a/H20850and2/H20849b/H20850.mprecesses in the north hemisphere for negative Jand in the south hemisphere for positive Jin an attempt to align/antialign with M. We hence conclude that the precession is largely dominated by the perpendicularcomponent p zof the spin polarized current and virtually in- dependent of pxand py. The asymmetry of the dependence for different current polarity is due to the asymmetric spintorque form, as shown in the inset of Fig. 2/H20849a/H20850. Figure 2/H20849b/H20850shows the effective MR /H20849MR eff/H20850as a func- tion of current density for the two tilt angles and different choices of /H9273.M R effJ2is a measure of the expected rf output where MR effis the difference between the maximum and minimum resistance values along the orbit normalized by thefullR AP−RP. As the precession orbit contracts with increas- ing /H20841J/H20841, one may expect MR effto be maximum at the equator and exhibit a monotonic decrease with increasing /H20841J/H20841. While symmetric MR /H20849/H9273=0/H20850indeed yields a maximum MR effat the onset of precession, the higher the MR asymmetry, the more FIG. 2. /H20849Color online /H20850/H20849a/H20850Precession frequency vs drive current for /H9252=36° /H20849solid line /H20850and/H9252=45° /H20849dashed dot line /H20850. Inset: Normalized spin torque /H9270*=4ed/H9270//H6036Jvs/H9272./H20849b/H20850Effective MR vs Jfor/H9252=36° /H20849solid line /H20850and /H9252=45° /H20849dashed line /H20850. Inset: Reduced MR vs /H9272./H20849c/H20850Precession orbits on the unit sphere for different Jand/H9252=36°.262508-2 Zhou et al. Appl. Phys. Lett. 92, 262508 /H208492008 /H20850 Downloaded 02 Jul 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsthe MR effpeak gets shifted to higher positive current. For asymmetric angular dependence of MR, it is hence favorableto precess at a ﬁnite latitude with a larger average angle withrespect to M. For optimal output it is consequently desirable to use positive currents and tailor /H9273as to position MR effin the middle of the operating frequency range. While there areno /H9273values reported for NiFe /Cu /FePt, /H9273may range from 0 to 4 in other trilayers involving NiFe.20,23 Despite the large in-plane component of the spin polar- ization, the initial static states are virtually identical to thenorth and south poles where the spin torque and the torquefrom the demagnetizing ﬁeld balance each other. If we fur-ther increase /H20841J/H20841we expect this equilibrium point to move toward /H20849anti /H20850alignment with Mˆ. As shown in Fig. 3,mˆstarts out at /H9258/H110151/H11568and 177° and then gradually follows a curved trajectory to align with Mˆat very large negative current and antialign at very large positive current. The resistance willchange accordingly and at very large currents reach R Pand RAP, respectively. There are several experimental ways to achieve easy- axis tilted hard magnets.24–28For example, an easy-axis ori- entation of 36° can be achieved by growing L10/H20849111/H20850FePt on conventional Si /H20849001 /H20850substrate27or on MgO /H20849111/H20850 underlayer.28The 45° orientation can be achieved by epitaxi- ally growing an L10/H20849101 /H20850FePt thin ﬁlm on a suitable seed layer /H20851e.g., CrW /H20849110/H20850with bcc lattice /H20852at a temperature above T=350/H11568C.24L10FePt has high magnetocrystalline an- isotropy /H20849Ku=7/H11003107ergs /cm3/H20850, high saturation magnetiza- tion /H20849Ms=1140 emu /cm3/H20850, and a high Curie temperature/H20849TC=750 K /H20850. In both cases, a thin Co layer may be deposited on top of the ﬁxed layer to promote a high degree of spin polarization. In summary, the TP-STO yields the combined advantage of zero-ﬁeld operation and high output signal. Both the pre-cession and effective MR dependence on the driving currentand the equilibrium states can be well understood by inves-tigating the precession orbits of the free layer. TP-STOs withtilt angles /H9252=36° and 45° should be possible to fabricate using FePt with high anisotropy and tilted easy axis. We thank M. Stiles for useful discussions. Support from The Swedish Foundation for Strategic Research /H20849SSF /H20850, The Swedish Research Council /H20849VR/H20850, and the Göran Gustafsson Foundation is gratefully acknowledged. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,1/H208491996 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 /H208492000 /H20850. 4S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoe- lkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850425,3 8 0 /H208492003 /H20850. 5W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. 92, 027201 /H208492004 /H20850. 6I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, D. C. Ralph, and R. A. Buhrman, Science 307, 228 /H208492005 /H20850. 7A. N. Slavin and V. S. Tiberkevich, Phys. Rev. B 74, 104401 /H208492006 /H20850. 8J. Z. Sun, Phys. Rev. B 62, 570 /H208492000 /H20850. 9O. Redon, B. Dieny, and B. Rodmacq, U.S. Patent No. 6532164 B2 /H208493 November 2003 /H20850. 10K. J. Lee, O. Redon, and B. Dieny, Appl. Phys. Lett. 86, 022505 /H208492005 /H20850. 11D. Houssameddine, U. Ebels, B. Delaet, B. Rodmacq, I. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda, M. C.Cyrille, O. Redon, and B. Dieny, Nat. Mater. 6, 447 /H208492007 /H20850. 12J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 70, 172405 /H208492004 /H20850. 13J. Barnas, A. Fert, M. Gmitra, I. Weymann, and V. K. Dugaev, Phys. Rev. B72, 024426 /H208492005 /H20850. 14O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C. Deranlot, F. Petroff, G. Faini, J. Barnas, and A. Fert, Nat. Phys. 3, 492 /H208492007 /H20850. 15V. S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Braganca, O. Ozatay, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Nat. Phys. 3,4 9 8 /H208492007 /H20850. 16J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 72, 014446 /H208492005 /H20850. 17J. Barnas, A. Fert, M. Gmitra, I. Weymann, and V. K. Dugaev, Mater. Sci. Eng., B 126, 271 /H208492006 /H20850. 18M. Gmitra and J. Barnas, Phys. Rev. Lett. 99, 097205 /H208492007 /H20850. 19J. Slonczewski, J. Magn. Magn. Mater. 247, 324 /H208492002 /H20850. 20S. Urazhdin, R. Loloee, and W. P. Pratt, Phys. Rev. B 71, 100401 /H208492005 /H20850. 21X. Zhu and J. G. Zhu, IEEE Trans. Magn. 42, 2670 /H208492006 /H20850. 22W. Jin, Y. W. Liu, and H. Chen, IEEE Trans. Magn. 42, 2682 /H208492006 /H20850. 23N. Smith, J. Appl. Phys. 99, 08Q703 /H208492006 /H20850. 24B. Lu and D. Weller, U.S. Patent No. US2007009766-A1 /H2084911 January 2007 /H20850. 25J. P. Wang, W. K. Shen, J. M. Bai, R. H. Victora, J. H. Judy, and W. L. Song, Appl. Phys. Lett. 86, 142504 /H208492005 /H20850. 26T. J. Klemmer and K. Pelhos, Appl. Phys. Lett. 88, 162507 /H208492006 /H20850. 27C. L. Zha, B. Ma, Z. Z. Zhang, T. R. Gao, F. X. Gan, and Q. Y. Jin, Appl. Phys. Lett. 89, 022506 /H208492006 /H20850. 28J. Y. Jeong, J. G. Kim, S. Y. Bae, and K. H. Shin, IEEE Trans. Magn. 37, 1268 /H208492001 /H20850. FIG. 3. /H20849Color online /H20850MR as a function of current density. Inset: The equilibrium states of mˆat different current densities when /H9252=36°. /H208491/H20850J=−0.5 /H11003108A/cm2, /H208492/H20850J=−1/H110031011A/cm2, /H208493/H20850J=0.75 /H11003108A/cm2,/H208494/H20850J=7/H11003108A/cm2,/H208495/H20850J=1/H11003109A/cm2, and /H208496/H20850J=1 /H110031011A/cm2.262508-3 Zhou et al. Appl. Phys. Lett. 92, 262508 /H208492008 /H20850 Downloaded 02 Jul 2012 to 152.3.102.242. 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1.5025691.pdf	Reversal time of jump-noise magnetization dynamics in nanomagnets via Monte Carlo simulations Arun Parthasarathy , and Shaloo Rakheja Citation: Journal of Applied Physics 123, 223901 (2018); doi: 10.1063/1.5025691 View online: https://doi.org/10.1063/1.5025691 View Table of Contents: http://aip.scitation.org/toc/jap/123/22 Published by the American Institute of Physics Articles you may be interested in Optical properties of lonsdaleite silicon nanowires: A promising material for optoelectronic applications Journal of Applied Physics 123, 224301 (2018); 10.1063/1.5025856 Analysis of dependent scattering mechanism in hard-sphere Yukawa random media Journal of Applied Physics 123, 223101 (2018); 10.1063/1.5030504 On matching the anode ring with the magnetic field in an ATON-type Hall effect thruster Journal of Applied Physics 123, 223301 (2018); 10.1063/1.5026486 AuNP-PE interface/phase and its effects on the tensile behaviour of AuNP-PE composites Journal of Applied Physics 123, 214305 (2018); 10.1063/1.5032083 Expansion of a single Shockley stacking fault in a 4H-SiC (11 0) epitaxial layer caused by electron beam irradiation Journal of Applied Physics 123, 225101 (2018); 10.1063/1.5026448 Determination of band alignment at two-dimensional MoS 2/Si van der Waals heterojunction Journal of Applied Physics 123, 225301 (2018); 10.1063/1.5030557Reversal time of jump-noise magnetization dynamics in nanomagnets via Monte Carlo simulations Arun Parthasarathya)and Shaloo Rakheja Department of Electrical and Computer Engineering, New York University, Brooklyn, New York 11201, USA (Received 12 February 2018; accepted 8 May 2018; published online 8 June 2018) The jump-noise is a nonhomogeneous Poisson process which models thermal effects in magnetiza- tion dynamics, with special applications in low temperature escape rate phenomena. In this work,we develop improved numerical methods for Monte Carlo simulation of the jump-noise dynamics and validate the method by comparing the stationary distribution obtained empirically against the Boltzmann distribution. In accordance with the N /C19eel-Brown theory, the jump-noise dynamics dis- play an exponential relaxation toward equilibrium with a characteristic reversal time, which we extract for nanomagnets with uniaxial and cubic anisotropy. We relate the jump-noise dynamics to the equivalent Landau-Lifshitz dynamics up to second order correction for a general energy land-scape and obtain the analogous N /C19eel-Brown theory’s solution of the reversal time. We ﬁnd that the reversal time of jump-noise dynamics is characterized by N /C19eel-Brown theory’s solution at the energy saddle point for small noise. For large noise, the magnetization reversal due to jump-noisedynamics phenomenologically represents macroscopic tunneling of magnetization. Published by AIP Publishing. https://doi.org/10.1063/1.5025691 I. INTRODUCTION Thermal ﬂuctuation of magnetization in nanomagnets is important in the context of superparamagnetism, which is ana-lyzed profoundly in Brown’s seminal work, 1,2and more recently reviewed by Coffey3to estimate the reversal time of magnetization. Thermal ﬂuctuation is more prominent insmaller magnetic volumes, which leads to spontaneous jumps from one stable state to another. From a practical perspective, this hinders the steady miniaturization of patterned spintronicdevice elements such as magnetic tunnel junction (MTJ) based magnetic random access memories (MRAMs). 4,5 The conventional way of modeling magnetization dynamics with thermal effects is by including two distinct and disjoint terms in the effective ﬁeld of the macroscopic equation of motion:1(a) a dissipative ﬁeld and (b) a random thermal ﬁeld. For a uniformly magnetized particle with mag- netization Mof magnitude Ms, the classical equation of motion is given as6,7 dM dt¼/C0c0M/C2Heff/C0a MsdM dtþHT/C18/C19 ; (1) where c0is the gyromagnetic constant, ais the damping con- stant, and HTdenotes an isotropic white Gaussian thermal ﬁeld. The effective ﬁeld Heffincludes the applied ﬁeld and contributions from exchange, anisotropy, and magnetoelasticeffects. 8The damping term, introduced by Gilbert,7accounts for rapid relaxation toward the equilibrium state, while the thermal ﬁeld is responsible for the Brownian motion. However, there is an important assumption in modeling the dynamics classically: the time scale of the thermal ﬁeld1 and the relaxation of angular momentum associated with magnetic moment9should be much shorter than the responsetimes of the system. Simply put, the internal equilibrium of a magnetization orientation should occur much faster than the thermodynamic equilibrium of the ensemble.2The time scale associated with the thermal ﬁeld is of the order of kT/ h¼10/C013s at room temperature ( kTis the thermal energy andhis Planck’s constant), that of the angular momentum relaxation is of the order of 10–15s, and that of the preces- sional motion of magnetization is around 10/C010s.1,9 Therefore, the assumption and the model are reasonable at nominal temperatures. However, the conventional model of magnetization relaxation becomes inadequate for explaining low temperature (10 m K–10 K) phenomena, such as macro- scopic tunneling of magnetization,10,11in which the transi- tion rate between two magnetic states is independent of temperature. By modeling the dynamics due to thermal effects using a single random process composed of discontinuous transi- tions (jumps) with Poisson arrivals, called the jump- noise,12,13a uniﬁed model applicable over a broad range of temperatures is obtained. The jump-noise provides a justiﬁ- cation for quantum tunneling of magnetization without invoking quantum mechanics.11For small noise, the classical Gilbert damping term emerges from average effects of the jump-noise.13,14 The reversal time of magnetization extracted from jump-noise dynamics requires primarily the knowledge of the energy landscape of the magnetic body. As an applica- tion, it can be used to study the retention time and stability of emerging magnetic memories. An example is magneto- electric antiferromagnetic (AFM) memory using (0001) Cr2O3(chromia) thin ﬁlm. In chromia thin ﬁlms, the energy landscape strongly depends on the product of the applied longitudinal electric ﬁeld and magnetic ﬁeld.15–17However, experimental measurements of retention time of AFM domain state are elusive due to the absence of a neta)Author to whom correspondence should be addressed: arun.parth@nyu.edu 0021-8979/2018/123(22)/223901/6/$30.00 Published by AIP Publishing. 123, 223901-1JOURNAL OF APPLIED PHYSICS 123, 223901 (2018) macroscopic magnetic moment in AFMs. The results obtained here can provide useful insight into the design andoptimization of such memory systems. The direct approach to realize jump-noise dynamics is to solve the Kolmogorov-Fokker-Planck equation for thetransition probability density of the stochastic process usinga specialized ﬁnite element and ﬁnite difference method. Forsmall noise, this equation can be reduced to a stochastic pro-cess on energy graphs and can be set up and solved muchfaster. 18This paper focuses on the algorithmic approach to simulate jump-noise and extract the statistics empirically viaMonte Carlo simulations. While Monte Carlo simulationsare computationally expensive, they are fairly easy to imple-ment and can be executed in parallel on a multi-core proces-sor for signiﬁcant speedup. 19,20 The goal of this paper is twofold. First, we present a complete methodology for numerical modeling of jump- noise statistics (Sec. III). Here, we contrast our work against prior works21,22in which simulations are performed using a sub-optimal global mesh. Second, we empirically extract themagnetization reversal time in nanomagnets (Sec. IV) with uniaxial and cubic anisotropy and compare the data withN/C19eel-Brown theory’s solution to analogous classical dynam- ics (Secs. VandVI). While Ref. 11discusses the thermal switching rate, the reversal time of magnetization of jump-noise dynamics is analyzed in our work. II. THE JUMP-NOISE The magnetization dynamics driven by jump-noise is described by the equation12 dm dt¼/C0m/C2heffþX iDmidðt/C0tiÞ/H20687kmk¼1;(2) where m¼M=Ms;heff¼Heff=Ms;t¼ðc0MsÞtare dimen- sionless quantities. If the total magnetic free energy densitygis known as a function of the state m, then h eff¼/C0 r mgðmÞ. The jump-noise is represented by the weighted Dirac comb which expresses random jumps Dmi on the unit sphere kmk¼1 occurring at random time instants ti. The jump-noise is formulated as a Poisson process char- acterized by a transition probability rate function Sbetween state pairs ðm1;m2Þin the phase space kmk¼1. The transi- tion probability rate is given as13 Sðm1;m2Þ¼Bexp/C01 2r2km1/C0m2k2/C20 þeb0 2gðm1Þ/C0gðm2Þ/C8/C9/C21 ;eb0¼l0M2 sV kT;(3) where Bandrare the jump-noise parameters, l0is the vac- uum permeability, Vis the volume of the nanomagnet, and eb0is the energy barrier parameter. The parameters Bandr can be determined experimentally by measuring the escape rate of the magnetization as a function of temperature. From Eq.(3), the scattering rate kfrom a state mfollows:kðmÞ¼þ km0k¼1Sðm;m0Þd2m0: (4) The probability density function fof a jump Dmito occur at time tigiven the state mifollows: fðDmijmiÞ¼Sðmi;miþDmiÞ kðmiÞ: (5) The statistic of the jump instants tiis given by the inter- arrival times in a nonhomogeneous Poisson process Prðtiþ1/C0ti>sÞ¼exp/C0ðtiþs tikðmðtÞÞdt ! : (6) III. NUMERICAL METHOD To simulate the jump-noise, we need to generate the cor- rect statistics of the jumps Dmiand the instants Dtifrom Eqs. (4)–(6)and evolve Eq. (2)in time. The ﬁrst step is to realize a discrete state space of kmk¼1. It is convenient to use spherical coordinates ðh;/Þto represent the components of the state msuch that mx¼sinhcos/;my¼sinhsin/;mz¼cosh. By uniformly sampling h2½0;p/C138and/2½0;2p/C138, we can form a ﬁnite ele- ment mesh on the unit sphere. In Ref. 22, the spherical mesh forms a global state space, which remains ﬁxed regardless of thestate from where jump occurs. Discretizing the space this way isnot ideal, especially when r 2/C281, because majority of the jumps would occur in a small solid angle centered at the currentstate. Moreover, the error due to discretization depends on thestate as the mesh points are unevenly distributed [see Fig. 1(a)]. A better approach, adopted in this work, is to form a local spherical mesh with half angle Has shown in Fig. 1(b). Such mesh inherently keeps the density of states high near the localstate and less far away, which is beneﬁcial because the proba-bility density (5)behaves alike. Tessellation of the sphere by a regular convex polyhedron 23does not exhibit this property. Hcan be estimated by ﬁrst ﬁnding an upper bound on the transition probability rate (3)forkDmk¼2 sinðH=2Þas lnSðm;mþDmÞ Sðm;mÞ/C20/C21 /C20eb0 2kDmkkrgkmax/C02 r2kDmk2;(7) FIG. 1. Discrete state space formed on the unit sphere using (a) global mesh and (b) local mesh. Red dots in the middle of each mesh element represent the states, while the size of the mesh element determines the weight associated.223901-2 A. Parthasarathy and S. Rakheja J. Appl. Phys. 123, 223901 (2018)then setting the right-hand-side to logarithm of the desired suppression. Hmeasures the strength of jump-noise given the parameters randeb0. The random samples from the probability density func- tion in Eq. (5)are generated by a fundamental method called inverse transform sampling. First, label the state space by asingle index jsuch that S¼f ðh j;/jÞ:j¼1;2;…;Mgis the collection of states, where Mis the total number of states. Now, ﬁnd the cumulative distribution function Fas Fj¼Xj k¼1fksinðhkÞdhd/ ; (8) where ddenotes the sample spacing. Generate a random number vfrom the uniform distribution between 0 and 1, and compute the value of jsatisfying Fj’v/C17min jjFj/C0vj: (9) The state mj¼:ðhj;/jÞis the random jump destination. The jump statistic given by Eq. (6)cannot be numeri- cally realized in its current implicit form. This is resolved byhomogenizing the scattering on the state space via a self-scattering (nil jump) process k 0.21The total scattering rate after homogenization is given by kðmÞþk0ðmÞ¼C¼max mkðmÞ: (10) From Eqs. (6)and(10), the time between scattering scan be explicitly expressed as s¼/C01 Cln Prðtiþ1/C0ti>sÞ ½/C138 : (11) By replacing the probability with a uniform random number between 0 and 1, the random duration between the jumps is generated. But, the probability of a non-zero jump is onlykðmÞ=C, which can be discerned by generating another uni- form random number ubetween 0 and 1, and checking if u/C20kðmÞ=C. Lastly, the state mis evolved in time according to Eq. (2)by repeating the following steps: (a) integrate the deter- ministic term in the scattering-free duration t2½t i;tiþ1Þ using a ﬁnite difference method, (b) at t¼tiþ1perform the jump to mjand generate a new duration ssuch that the next jump occurs at tiþ2¼tiþ1þs. IV. EQUILIBRIUM DISTRIBUTION AND REVERSAL TIME EXTRACTION To corroborate the numerical method to emulate jump- noise dynamics, we perform Monte Carlo simulations tostudy the equilibrium distribution of magnetization for nano-magnets possessing magnetocrystalline anisotropy only,under no applied ﬁeld. We consider 1000 samples alignedalong the same lowest energy state m z¼/C01, without loss of generality, at time t¼0, and let the ensemble evolve with time until the absolute ensemble mean jmzj<¼0:001. The simulation is implemented in MATLAB with the help of Parallel Computing Toolbox on a server with 20-core CPU@ 2.3 GHz and 512 GB memory. The time step is 0.05, the angular spacing of mesh is 0 :72/C14, and the suppression factor forHis 10/C08. The computation time for these speciﬁcations scales /C240:1/C2ss, where sis the reversal time (16). Analytically from statistical mechanics, the probability density function of the states min thermal equilibrium is given by the Boltzmann distribution weqðmÞ¼weqðh;/Þ¼1 Zexpð/C0eb0gðh;/ÞÞ; (12) where Zis the normalization constant given by Z¼ð2p 0ðp 0expð/C0eb0gðh;/ÞÞsinhdhd/: (13) From Eq. (12), the equilibrium distribution of mzfollows: weqðmzÞ¼ð2p 0weqðarccos ðmzÞ;/Þd/: (14) For uniaxial anisotropy, gðh;/Þ¼1 2sin2h. For cubic anisot- ropy, gðh;/Þ¼1 2ðsin4hsin22/þsin22hÞ. The simulation results for the equilibrium distribution of mzfor the uniaxial and cubic anisotropy are shown in Fig. 2. The Monte Carlo simulations replicate the Boltzmann distri-bution correctly for both the energy landscapes. This is expected because the transition probability rate in Eq. (3) satisﬁes the detailed balance Sðm 1;m2Þweqðm1Þ¼Sðm2;m1Þweqðm2Þ; (15) for all ðm1;m2Þ2kmk¼1. The small yet notable error in the simulations is affected by one or more of the sample size and mesh spacing. In N /C19eel-Brown theory,1the reversal time of magnetiza- tionscharacterizes the longitudinal relaxation of ensemble magnetization as FIG. 2. Equilibrium distribution of mzfor (a) uniaxial anisotropy and (b) cubic anisotropy. The parameters are eb0¼10,B¼1, and r¼0:2.223901-3 A. Parthasarathy and S. Rakheja J. Appl. Phys. 123, 223901 (2018)jmzjðtÞ/C25e/C0t=s;t/C29s: (16) Even for jump-noise dynamics, Eq. (16) holds true because the Markov chain represented by it satisﬁes thedetailed balance condition, and therefore has an exponentialrate of convergence to the stationary distribution (12). 24So, scan be estimated from the asymptotic value of sðtÞ ¼/C0t=ln½jmzj/C138from simulations as featured in Fig. 3, by fol- lowing the same procedure mentioned for extracting theequilibrium distribution. V. EQUIVALENT CLASSICAL DYNAMICS It is essential to see how the reversal time of magnetiza- tion extracted from jump-noise dynamics compares with thereversal time obtained from N /C19eel-Brown theory. This requires establishing an equivalence between the jump-noiseand classical magnetization dynamics. Equation (1)can be rewritten in the dimensionless Landau-Lifshitz form, 2excluding the thermal ﬁeld, as dm dt¼/C01 1þa2m/C2heffþam/C2ðm/C2heffÞ ½/C138 : (17) In the absence of applied ﬁeld, heff¼/C0 r gðh;/Þis orthogo- nal to the radial direction m,s o m/C2ðm/C2heffÞ¼ðm/C1heffÞm/C0kmk2heff¼r g:(18) From Eqs. (17) and(18), the component of d m=dtalong rg, in the absence of applied ﬁeld, gives the damping terma=ð1þa 2Þ. Indeed for the jump-noise dynamics (2), the average component of jump-noise along rggives rise to an equivalent damping term locally. The expected value of jump-noise at a given state mis expressed as EdDm dt/C12/C12/C12/C12m"# ¼ð Dm0Sðm;mþDm0Þd2Dm0: (19) For small noise H/C281, the phase space of jumps Dm0can be approximated by the local tangent plane. The energy differ-ence gðmþDm 0Þ/C0gðmÞin Eq. (3)can be truncated up to second order of Taylor series, which then can be used to eval-uate the integral in Eq. (19) deﬁnitely by forming a bivariate Gaussian function. The derivation is similar to that for ﬁrstorder approximation of the energy difference in Ref. 14.T h einclusion of second-order term is important because the term kDm 0k2=r2in Eq. (3)is quadratic. The ﬁnal expression of the expected jump-noise is obtained as EdDm dt/C12/C12/C12/C12m"# ¼/C01 2r2eb0kðmÞW/C01rg; (20) kðmÞ¼B2pr2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ detWp exp1 8ðreb0Þ2rgTW/C01rg/C20/C21 ; (21) W¼I 2þ1 2r2eb0Hg; (22) where I2is the 2 /C22 identity matrix and His the Hessian operator. In local coordinates of the tangent space ofmðh;/Þ, the gradient and Hessian of gare written as rg¼g hcschg//C2/C3T; (23) Hg¼ghh gh//C0cothg/ gh//C0cothg/g//þcoshsinhgh/C20/C21 ; (24) where each letter of subscript denotes partial derivative with respect to that variable. The equivalent damping term for jump-noise can now be determined as aðh;/Þ 1þaðh;/Þ2’aðh;/Þ¼/C0rgT krgk2EdDm dt/C12/C12/C12/C12m"# ¼1 2r2eb0kðh;/ÞrgTW/C01rg krgk2: (25) Unless Wequals identity, the expected value of jump- noise would also yield an orthogonal component along m/C2rg¼½/C0cschg/gh/C138T, which is the precessional motion. The physical signiﬁcance of this component is a smallcorrection to the gyromagnetic ratio [see Eq. (1)]b yaf a c t o ro f c cðh;/Þ¼1/C01 2r2eb0kðh;/Þðm/C2rgÞTW/C01rg km/C2rgk2:(26) VI. REVERSAL TIME: N /C19EEL-BROWN THEORY VERSUS JUMP-NOISE DYNAMICS SIMULATIONS In N /C19eel-Brown theory, the reversal time in the high energy barrier limit eb0/C291 is approximated by inverse of the smallest non-vanishing longitudinal eigenmode of theFokker-Planck equation of the Landau-Lifshitz dynamics. 2,3 The expressions of reversal time for the uniaxial and cubic anisotropy, suniandscub, respectively, in dimensionless units are reproduced from Ref. 3as suni¼ﬃﬃﬃppðaþa/C01Þ ccﬃﬃﬃﬃﬃﬃﬃﬃﬃ2eb0p 1þ2 eb0þ7 e2 b0/C20/C21 expeb0 2/C18/C19 ; (27) scub¼pðaþa/C01Þ ﬃﬃﬃ 2p ccﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 9þ8 a2r þ1 ! A4ﬃﬃﬃ 2p 9aeb0/C18/C19 expeb0 2/C18/C19 ; AðKÞ¼exp1 pð1 0lnf1/C0exp/C0Kðg2þ1=4Þ/C2/C3 g g2þ1=4dg"# : (28)FIG. 3. Existence of an asymptotic time constant for the longitudinal relaxa- tion of magnetization for (a) uniaxial anisotropy, and (b) cubic anisotropy. The parameters are eb0¼10,B¼1, and r¼0:05 to 0.25 in steps of 0.05 from topmost to bottom-most plot.223901-4 A. Parthasarathy and S. Rakheja J. Appl. Phys. 123, 223901 (2018)The expression of scubhere is multiplied by 2 because the reversal time for cubic anisotropy in the literature2,3,25 corresponds to the relaxation associated with surmounting one energy barrier from mz¼1t o mx;my¼61 (see Fig. 4). Since the energy landscape is symmetric about mz¼0 and the barriers are high, the longitudinal relaxation from mz¼1 tomz¼/C01 is associated with overcoming two successive energy barriers with intermediate orientations at mx;my¼61. The equivalent damping term aand the gyromag- netic correction factor ccfor jump-noise dynamics are not constants but depend on the state mðh;/Þ. To obtain the corresponding N /C19eel-Brown theory’s solution of the reversal time, we calculate the numerical bounds of suni (27) andscub(28) by varying a(25) andcc(26) on the phase space. Upon inspection, when the noise is small,the lower bound occurs close to energy maximum(h¼p=2) for s uniand close to energy saddle points (h¼p=4;3p=4;/¼0;p=2;p;3p=2) for scub.F o rl a r g e energy barrier, the internal equilibrium within an energywell occurs much faster than the equilibrium between energy wells, leaving the “ﬂow” across energy barrier to be concentrated near the point of least height, 2thereby giving the lower bound. The upper bound occurs close toenergy minima for both s uniandscubbecause the damp- ing is minimum around it. The reversal time extracted from Monte Carlo simula- tions of jump-noise dynamics and the numerical bounds ofthe reversal time obtained from N /C19eel-Brown theory’s solu- tion to equivalent Landau-Lifshitz dynamics are shown inFig.5. For both uniaxial and cubic anisotropy, the simulation points coincide with the lower bound for smaller values of r and deviate from it for larger values while remaining withinthe bounds. Thus, the reversal time of jump-noise dynamics for small noise is characterized by saddle point values of the equivalent classical dynamics parameters aand c c. The reversal time of jump of dynamics is also always smallerthan that of equivalent classical dynamics characterized byenergy minima values of aandc c. The qualitative explana- tion for this is that jump-noise phenomenologically accountsfor tunneling unlike classical dynamics, and so yields alower reversal time.For large noise, the second order approximation of the energy difference breaks down, and consequently the expres- sions of aand c ccould yield unphysical results. Additionally, the expected value of jump-noise would pro- duce component along the radial direction which cannot be modeled by the Landau-Lifshitz equation. For large scatter- ing rate, the right hand side of Eq. (25) could exceed unity, thereby giving a complex damping parameter. The magnitude of noise for desired suppression of prob- ability is determined from the noise strength Husing Eq. (7). Alternatively, from Eqs. (5) and (20), the value of r2eb0kW/C01rgkgives a good measure of noise. However, when comparing different energy landscapes, the magnitude of noise must be measured relative to the distance between the energy minima and maxima. When the noise is large enough to cross the barrier in few jumps rather than gradu- ally, the equilibrium occurs as a simultaneous process within and in between energy wells; therefore, N /C19eel-Brown theory is not valid. Since the angular distance between the energy minima and maxima is larger in uniaxial (90/C14) than in cubic (55/C14) anisotropy, for a r2eb0product say, 0 :252/C210¼36/C14, evidently the reversal time of jump-noise dynamics for cubic anisotropy shows larger deviation from N /C19eel-Brown theory’s solution in Fig. 5. VII. CONCLUSION In this paper, we develop numerical methods to extract the reversal time of magnetization of jump-noise dynamics in nanomagnets with uniaxial and cubic anisotropy via Monte Carlo simulations. The reversal time of nanomagnets due to thermal ﬂuctuation predicted from jump-noise dynam- ics is more accurate because it accounts for tunneling effects by its very formulation. Modeling of the jump-noise requires primarily the knowledge of the energy landscape of theFIG. 4. Cubic anisotropy energy landscape. Both radial distance and bright- ness of color-map indicate the energy. There are six equivalent energy min- ima along mx;my;mz¼61.FIG. 5. Reversal time of magnetization for (a) uniaxial anisotropy, and (b) cubic anisotropy. Solid lines are numerical bounds of N /C19eel-Brown theory’s solutions (27) and(28) to equivalent Landau-Lifshitz dynamics speciﬁed by Eqs. (25) and(26). The parameters are eb0¼10 and B¼1.223901-5 A. Parthasarathy and S. Rakheja J. Appl. Phys. 123, 223901 (2018)system and not the microscopic origin of the underlying phe- nomena. Results show that the reversal time gathered from N/C19eel-Brown theory’s solution to analogous classical dynam- ics (a) at the energy saddle point characterizes the reversal time of jump-noise dynamics for small noise, (b) at theenergy minima is always larger than the reversal time of jump-noise dynamics. For large noise, the magnetization reversal due to jump-noise dynamics has no classical ana- logue and phenomenologically represents macroscopic tunneling of magnetization. ACKNOWLEDGMENTS This work was supported in part by the Semiconductor Research Corporation (SRC) and the National ScienceFoundation (NSF) through ECCS 1740136. S. Rakheja also acknowledges the funding support from the MRSEC Program of the National Science Foundation under Award No. DMR-1420073. 1W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963). 2W. Brown, IEEE Trans. Magn. 15, 1196 (1979). 3W. T. Coffey and Y. P. Kalmykov, J. 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1.4763544.pdf	REVIEWS OF ACOUSTICAL PATENTS Sean A. Fulop Dept. of Linguistics, PB92 California State University Fresno 5245 N. Backer Avenue, Fresno, California 93740-8001 Lloyd Rice 11222 Flatiron Drive, Lafayette, Colorado 80026 The purpose of these acoustical patent reviews is to provide enough information for a Journal reader to decide whether to seek more information from the patent itself. Any opinions expressed here are those ofthe reviewers as individuals and are not legal opinions. Printed copies of United States Patents may be ordered at $3.00 each from the Commissioner of Patents and Trademarks, Washington, DC 20231. Patents are available via the Internet at http://www.uspto.gov . Reviewers for this issue: GEORGE L. AUGSPURGER, Perception, Incorporated, Box 39536, Los Angeles, California 90039 SEAN A. FULOP , California State University, Fresno, 5245 N. Backer Avenue M/S PB92, Fresno, California 93740-8001 JEROME A. HELFFRICH, Southwest Research Institute, San Antonio, Texas 78228 DAVID PREVES, Starkey Laboratories, 6600 Washington Ave. S., Eden Prairie, Minnesota 55344 CARL J. ROSENBERG, Acentech Incorporated, 33 Moulton Street, Cambridge, Massachusetts 02138 NEIL A. SHAW, Menlo Scientiﬁc Acoustics, Inc., Post Ofﬁce Box 1610, Topanga, California 90290 ERIC E. UNGAR, Acentech, Incorporated, 33 Moulton Street, Cambridge, Massachusetts 02138 ROBERT C. WAAG, Department of Electrical and Computer Engineering, University of Rochester, Rochester, New York 14627 8,162,076 43.35.Zc SYSTEM AND METHOD FOR REDUCING THE BOREHOLE GAP FOR DOWNHOLEFORMATION TESTING SENSORS Ruben Martinez et al ., assignors to Schlumberger Technology Corporation 24 April 2012 (Class 175/40); ﬁled 2 June 2006 This patent teaches about using hydraulically actuated pads for plac- ing various sensors in contact with a borehole during oil well drillingoperations. Speciﬁcally, the authors target so-called ‘‘Measurement While Drilling’’ operations, in which the sensors are actually operated while the drill is rotating and mud is being pumped past the sensors. Thus, a large part of the patent is devoted to maintaining pressure against the borehole wall without creating too much pressure, which will lead to extensive transducer wear. In this design, the sensors are mounted on hinged pads that swing out from the body of the drill, as shown in the ﬁgure, which isa cross-sectional view of the sensor part of a drill string. The patent also teaches the use of reamers around the sensors to scrape away the mud cake while a measurement is being made.—JAH 8,179,020 43.35.Zc VIBRATORY ACTUATOR AND DRIVE DEVICE USING THE SAME Yusuke Adachi et al., assignors to Panasonic Corporation 15 May 2012 (Class 310/323.16); ﬁled in Japan 14 June 2007 This patent discloses the design of a piezoelectrically driven linear translation motor. In such a motor, the piezoelectric actuator typically is driven so as to execute an elliptical motion where the actuator contacts the part to be translated. The actuator contacts the part being translated for a small part of a cycle, translates it longitudinally, and then pulls away and resets for another cycle. The authors claim that one shortcomingof the standard approach is that grooves are worn into the translated part where the actuator contacts it (the contacts are labeled 8a and 8b in the ﬁgure, where the part being translated is not shown). Their solution is to stagger the distribution of the contact points, as shown in the ﬁgure. The authors claim that if the separation of the staggered contacts 8a and 8b is large enough this not only reduces groove depth but increases the stability of the actuator, helping to prevent it from rotating or tipping.—JAH 4086 J. Acoust. Soc. Am. 132(6), December 2012 0001-4966/2012/132(6)/4086/16/$30.00 VC2012 Acoustical Society of America8,189,811 43.38.Ew SYSTEM AND METHOD FOR PROCESSING AUDIO SIGNALS Roy R. Tillis, Columbus, OH 29 May 2012 (Class 381/96); ﬁled 16 July 2010 One of the requirements for a U.S. patent is that the invention be non-obvious. This patent exceeds that requirement by a wide margin—it deﬁes rational explanation. Block 12 actually contains a second loudspeaker 14 with voice coil connections 3 and 4. The function of the hidden loud- speaker is not to produce sound but rather to drive a pickup coil, whoseoutput appears at terminals 1 and 2. That’s right—motional feedback is derived not from the loudspeaker but from a kind of doppelganger speaker. The patent argues that, since a moving sound source generates Doppler dis- tortion, the output of full-range loudspeaker 8 must include Doppler distor- tion. (Pay close attention now.) It follows that the electrical output from the second loudspeaker’s pickup coil must be an accurate representation of the acoustic output from the main speaker, including Doppler distortion.‘‘The difference between the original audio signal and the processed signal is Doppler distortion. This difference is subtracted from the original audio signal to produce an output with minimized or cancelled Doppler distor- tion.’’ Electronic Doppler distortion? Wishful wizardry remains alive and well in the ranks of amateur loudspeaker designers.—GLA 8,194,868 43.38.Hz LOUDSPEAKER SYSTEM FOR VIRTUAL SOUND SYNTHESIS Ulrich Horbach and Etienne Corteel, assignors to Harman International Industries, Incorporated 5 June 2012 (Class 381/59); ﬁled 3 January 2008Those who have heard demonstrations of surround sound via wave ﬁeld synthesis know that some effects are uncannily realistic. However, even a small installation (less than 360 degrees) requires many, many sig- nal channels and an equal number of identical loudspeakers. Some simpli- ﬁcation can be realized by using larger panel-type loudspeakers driven by multiple, closely spaced exciters. Unfortunately, frequency response char- acteristics produced by individual exciters in the same panel can varywidely, and room effects add further contamination. This patent describes an equalization method that is said to counteract both effects. Multiple impulse responses are derived from measurements at multiple microphone locations. These are compared with ideal impulse responses and then, using an iterative algorithm, ﬁnite impulse response ﬁlters are created to provide the required corrections. High frequency and low frequency ranges are equalized separately, then combined.—GLA 8,160,285 43.38.Ja WAVEGUIDE UNIT Michael Gjedbo Kragelund, assignor to Mike Thomas ApS 17 April 2012 (Class 381/338); ﬁled in Denmark 13 September 2005 Diffraction edge 8, one of three such edges as shown in the ﬁgure, but which can vary between 2 and 7 edges, diffracts the sound emanating from dome unit 10. The scattering of sound from the diffraction edges, the number of which and location of such edges (from f¼c/(AþB/C0C)), improves the ‘‘harmonization of the emitted sound’’ since ‘‘the improved sound reproduction is due to the fact that the diffraction edges will deﬂect the sound so that the on-axis sound pressure will have a maximum damp- ening without dampening the off-axis level.’’ The on-axis level is attenu-ated due to path length differences from the diffraction edges. How this affects the level at angles off-axis is not described, or ‘‘these geometric dimensions for the waveguide provides an improved sound distribution that is substantially linear effect characteristic of a high frequency dome transducer is provided.’’ Many commercially available loudspeaker enclo- sures use a device similar to the invention, except they have a smooth proﬁle.—NAS 8,189,841 43.38.Ja ACOUSTIC PASSIVE RADIATING Roman N. Litovsky and Faruk Halil Bursal, assignors to Bose Corporation 29 May 2012 (Class 381/349); ﬁled 27 March 2008 In comparison with a conventional ducted vent, a passive radiator introduces all sorts of problems. However, when a very small box must be tuned to a very low frequency, the passive radiator is much more efﬁcient in its use of available box volume. Therefore, it would seem to be a goodchoice for loudspeakers in small, hand-held devices. But such devices not only have limited internal volume, they have limited surface area available 4087 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 4087for sound radiation. Well, when the passive radiator produces maximum output at resonance the loudspeaker cone hardly moves at all, and higher frequencies are reproduced mostly by the loudspeaker with very little con- tribution from the passive radiator. Could the operation of the two cones somehow be combined? Possibly yes, and that is what Bose has patented here. A lightweight, miniature loudspeaker 32A is mounted in the center of passive radiator 16A. Near the resonance frequency the entire assembly moves, as indicated by (exaggerated) dotted lines 16X, 16Y. The patentdoes not seem to cover the limiting case in which the loudspeaker and the passive radiator are one and the same.—GLA 8,191,674 43.38.Ja ACOUSTIC LOADING DEVICE FOR LOUDSPEAKERS Ambrose Thompson, assignor to Martin Audio Limited 5 June 2012 (Class 181/186); ﬁled in United Kingdom 21 April 2005 This patent is a rarity. It describes an actual, working device rather than a half-formed preliminary concept. The invention is a high frequency horn whose directional response can be steered in the ﬁeld by adjusting pivoting vane 26. The idea seems simple, but numerous trade-offs are involved, and the patent goes into optimum geometry in some detail. Comparative frequency response curves show that lateral off-axis response can be substantially increased by setting the vane appropriately.—GLA 8,194,905 43.38.Ja COHERENT WAVE FULL SPECTRUM ACOUSTIC HORN Gordon Alfred Vinther, Sr., Provincetown, MA 5 June 2012 (Class 381/342); ﬁled 13 February 2008 This is an invention 50 years behind its time. Like the 1934 Voigt corner horn, it is a vertical horn with a large reﬂector to direct high fre- quencies into the listening area. It features coherent high-frequency and low-frequency sound sources, but so did the Voigt design. Unlike theVoigt horn, it uses a ‘‘Y’’ throat with individual high-frequency and low- frequency drivers, but that idea is no longer new either.—GLA 8,199,961 43.38.Ja SPEAKER DEVICE, INSTALLATION BODY FOR SPEAKER DEVICE, AND MOBILE BODY HAVING SPEAKER DEVICE MOUNTED THEREON Manabu Omoda and Masahiro Watanabe, assignors to JVC Kenwood Corporation 12 June 2012 (Class 381/389); ﬁled in Japan 15 July 2005 The diagram is a horizontal section through the right speaker of a motorcycle stereo system. The speakers are attached to both sides of a 4088 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 4088storage compartment behind the rider. The angle of the speaker bafﬂe, the shape of cowling 3, the geometry of sound exit opening 30, and additional air vent 19 are all designed to minimize air turbulence and deliver reason- ably smooth high frequency response while the motorcycle is moving. The patent describes numerous variants and includes air ﬂow diagrams at various wind velocities.—GLA 8,204,266 43.38.Ja AUDIO DEVICES Jordi Frigola Munoz et al., assignors to SFX Technologies Limited 19 June 2012 (Class 381/335); ﬁled in United Kingdom 21 October 2005 Like many others, this patent is mostly doubletalk. We begin with an electronic device in some kind of housing. It probably is a cellular phone or other small device, but the patent Claims do not limit the size or usage. One surface is driven so that it functions as a planar loudspeaker, ‘‘a ﬁrst acoustic radiator.’’ This is all well known prior art. However, a portable device is not always hand-held—it may be operated while lying on a desk or table. The illustration shows a cellular phone 10 lying on such a horizontal surface 12. Back panel 16 is the loudspeaker. Two little spacers 20 allow the panel to vibrate freely and radiate sound into theshallow cavity below. To prevent buzzes and rattles, the spacers should be resilient. (Let’s make it sound scientiﬁc and specify an elastomeric mate- rial with a Shore A hardness of less than 20.) We now have a practical design, but how can it be patented? By asserting that the acoustically coupled support surface also radiates sound. The surface may be, ‘‘…a wall surface, desktop, ceiling…’’—doesn’t matter; we say it radiates sound and that is that. Congratulations to the patent attorneys.—GLA 8,204,241 43.38.Lc SOUND OUTPUTTING APPARATUS, SOUND OUTPUTTING METHOD, SOUND OUTPUTPROCESSING PROGRAM AND SOUND OUTPUTTING SYSTEM Kohei Asada and Goro Shiraishi, assignors to Sony Corporation 19 June 2012 (Class 381/71.1); ﬁled in Japan 27 December 2006 This patent is written in Pidgin English, which does no credit to Sony. Even allowing for that, the text is discursive and hard to follow. The area of interest is the operation of small, portable music devices. Such a device may be used under layers of clothing or in a deep pocket, in which case the operation of simple controls becomes difﬁcult. (Or, as the patent would say, ‘‘cumbersome.’’) Prior art includes a method that allows the user to execute basic commands by tapping on one headphone,using a simple code. Obviously, special headphones are required and, since the overall playback system will be fairly expensive, these may well provide noise cancellation. Aha! Since a noise-cancelling headphone al- ready includes a microphone, the same microphone can be used to pick up the coded taps. And that is where this patent takes over. Using more than 50 illustrations and two dozen pages of text, the patent explains how such dual functionality might be implemented.—GLA8,189,847 43.38.Si DUAL-FREQUENCY COAXIAL EARPHONES WITH SHARED MAGNET Fred Huang, assignor to Jetvox Acoustic Corporation 29 May 2012 (Class 381/380); ﬁled in Taiwan 21 August 2008 This patent describes a variant of an earlier coaxial earphone design. In this case a single magnet 110 energizes both voice coil gaps. All the parts ﬁt together quite nicely, but the scheme for high frequency reproduc- tion is based on a serious misconception. Energy from small diaphragm 32 is conducted through short horn 251 to impinge upon the rear surfaceof low frequency diaphragm 241 (not identiﬁed), ‘‘…so as to energize the central diaphragm 241 of the low frequency speaker part 2 and to form a same phase as, and to output frequency synchronously with, the low frequency speaker part 2.’’ In the real world, instead of being ener- gized the low frequency diaphragm will simply block high frequencies. The arrangement might be described as a self-defeating coaxial trans- ducer.—GLA 4089 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 40898,190,217 43.38.Si DUAL MODE ELECTRONIC HEADSET WITH LOCAL AND REMOTE FOCUSED MICROPHONES Richard S. Slevin, Los Altos Hills, CA 29 May 2012 (Class 455/575.2); ﬁled 4 January 2010 A headset can be worn to screen out exterior noise while carrying on a telephone conversation. Such a headset includes a microphone ori- ented to pick up the user’s voice. (It might even be a noise-cancelling microphone, although the patent does not mention that possibility.) The patent suggests that, during lulls in the conversation or between calls, theuser may want to hear outside sounds clearly. The problem is solved by adding a second microphone and an automatic mode control to switch between the two.—GLA8,194,875 43.38.Si COMMUNICATION APPARATUS AND HELMET Stephen Alfred Miranda, assignor to Innotech Pty Limited 5 June 2012 (Class 381/74); ﬁled in Australia 11 September 2002 Several existing patents describe communications equipment incor- porated into, or attached to, a safety helmet that might be worn by a rac- ing car driver or motorcyclist. This patent is concerned mainly with ﬁre- ﬁghters. A bone conduction microphone 36 and a loudspeaker 38 are mounted toward the back of the user’s head. The loudspeaker is primarily a bone conduction transducer, but may also radiate airborn sound.—GLA 8,199,942 43.38.Si TARGETED SOUND DETECTION AND GENERATION FOR AUDIO HEADSET Xiadong Mao, assignor to Sony Computer Entertainment Incorporated 12 June 2012 (Class 381/309); ﬁled 7 April 2008 Some modern video games generate full-bodied surround sound through headphones. Even if the particular headphones allow some outside sound to enter, chances are the user will not hear a knock on the door or a telephone ring tone. The earphones described in this patent include one or more micro-phones to pick up external sounds. Those signals can then be combined with the multi-channel program material. The patent includes two little tricks to (hopefully) avoid conﬂicts with prior art. First, the direction of an external sound source can be estimated and replicated in the ampliﬁed surround sound ﬁeld. Second, external sounds are not reproduced in real time—they are brieﬂy stored (‘‘recorded’’) during processing.—GLA 4090 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 40908,199,950 43.38.Si EARPHONE AND A METHOD FOR PROVIDING AN IMPROVED SOUND EXPERIENCE Andrej Petef and Gunnar Klinghult, assignors to Sony Ericsson Mobile Communications AB 12 June 2012 (Class 381/322); ﬁled 22 October 2007 Numerous devices have been designed to provide tactile stimulation synchronized with a musical beat. We have seen vibrating dance ﬂoors, chairs, headrests, and saunas. Are you ready for earphones that literally tickle your ears? By including an electro-active polymer in the sealing cushion of an insertable earbud, the inventors have created just such a gadget.—GLA 8,190,425 43.38.Vk COMPLEX CROSS-CORRELATION PARAMETERS FOR MULTI-CHANNEL AUDIO Sanjeev Mehrotra and Wei-Ge Chen, assignors to Microsoft Corporation 29 May 2012 (Class 704/203); ﬁled 20 January 2006 The invention described in this patent is actually a catalog of possi- ble techniques for encoding and decoding multi-channel audio. The one common feature in all the patent Claims seems to be a cross-correlation parameter in which the cross-correlation ratio between channels can be represented by an imaginary number component plus a real number com- ponent.—GLA8,194,898 43.38.Vk SOUND REPRODUCING SYSTEM AND SOUND REPRODUCING METHOD Teppei Yokota, assignor to Sony Corporation 5 June 2012 (Class 381/310); ﬁled in Japan 22 September 2006 The goal of this patent is set forth as, ‘‘…sound image localization of sound of a channel in which a sound image is localized in a position in a front direction of a listener, such as a center channel where the sound image localization is difﬁcult to obtain, can be improved even when sound of a plurality of channels is reproduced by using the virtual sound pro-cess.’’ Roughly translated, it says: Front speakers should be used to repro- duce front sound sources. In this hybrid setup, however, surround channels are reproduced by ‘‘nearphones.’’ For example, speakers 14FL and 14FR might be located conventionally in the dashboard of an automobile while surround speakers 11SW1 and 11SW2 would be attached to a headrest. Various transfer functions are calculated and used to create phantom sur- round sound sources.—GLA 4091 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 40918,199,941 43.38.Vk METHOD OF IDENTIFYING SPEAKERS IN A HOME THEATER SYSTEM Michael D. Hudson et al., assignors to Summit Semiconductor LLC 12 June 2012 (Class 381/303); ﬁled 22 June 2009 Setting up a pair of stereo speakers would seem to be foolproof, yet the channels are reversed in many home installations. Now consider the daunting prospect of setting up a 7.1 surround sound system. Seven speak- ers must be correctly identiﬁed and connected. Moreover, the location ofeach speaker must be accurately measured so the processing unit can com- pensate for non-ideal azimuth and distance. This patent describes a well thought-out method for automating the entire process. Only the center speaker must be identiﬁed in advance. Its cabinet houses two ultrasonic transducers in addition to loudspeaker components. If desired, the remain- ing six speakers can be identical and interchangeable; each cabinet contains a single ultrasonic transducer plus loudspeaker components. Byemitting and receiving ‘‘pings’’ from various pairs of ultrasonic transducers, the locations of all six speakers can be mapped in relation to the center speaker. The distance from the listener to the center speaker is then meas- ured by emitting a ping from the hand-held remote control and receiving the ping at the center speaker. The patent describes the entire process in great detail.—GLA 8,204,235 43.38.Vk CENTER CHANNEL POSITIONING APPARATUS Yoshiki Ohta, assignor to Pioneer Corporation 19 June 2012 (Class 381/27); ﬁled 30 November 2007 When listening to a pair of loudspeakers reproducing a phantom cen- ter channel, interaural crosstalk becomes a signiﬁcant factor. A seldom- mentioned artifact is a frequency response dip centered near 2 kHz. It might seem desirable to make electronic correction by equalizing program content that is highly correlated between left and right channels. At least one existing patent takes that approach. Well and good, but suppose thelistener (this patent assumes there is only one listener) is not equidistant from left and right loudspeakers. The patent teaches that the two acoustic signals can be synchronized by adding a little delay to one channel. To be on the safe side, the signal level is also trimmed to allow for inverse- square loss. This is novel? Perhaps your reviewer missed something, but surely anyone involved with commercial surround sound systems knows that a typical setup routine includes delay and level matching of all chan-nels. Apart from that, attempting to equalize the 2 kHz dip may not be a good idea because it assumes that all multi-channel program material was originally mixed for discreet left, center, and right loudspeakers. Actually, the de facto working standard in the U.S. for non-cinema sound is two-channel stereo; other formats are upmixed or downmixed as required. (At least 90% of all popular music recordings, in any format, are mixed with a phantom center channel.) So, chances are, the mix is already optimized for two-loudspeaker reproduction.—GLA 8,180,607 43.40.At ACOUSTIC MODELING METHOD Mostafa Rassaian and David William Twigg, assignors to The Boeing Company 15 May 2012 (Class 703/2); ﬁled 15 October 2009 A method is described for use in computing a response of a structure to a known acoustic ﬁeld acting on the structure. The method in essence consists of a means for developing ﬁnite-element models that are computa- tionally less intensive than the use of conventional ﬁnite element methods. A composite of two elements is developed, its centroid is determined, andthe cross-spectral correlation function between the elements is assigned to be the autocorrelation function of the composite centroid.—EEU 8,152,651 43.40.Kd IRON GOLF CLUB WITH IMPROVED MASS PROPERTIES AND VIBRATION DAMPING Ryan L. Roach, assignor to Cobra Golf Incorporated 10 April 2012 (Class 473/329); ﬁled 25 April 2011 As disclosed in U.S. Patent 7,938,738 [reviewed in J. Acoust. Soc. Am. 131, 643 (2012) with which it shares the same drawings], the current patent describes a very similar means for lowering the center of gravityof, and providing for vibration reduction in, an iron type club by using a variable density mix of viscoelastic material and ﬁllers, such as tungsten powder and micro-spheres, in channels behind the club face, which ‘‘pro- vides improved feel, improved weight distribution, and enhanced club per- formance.’’—NAS 8,171,796 43.40.Le ACOUSTIC EMISSION DETECTOR AND CONTROLLER Hiroshi Ueno et al., assignors to JTEKT Corporation 8 May 2012 (Class 73/587); ﬁled in Japan 24 May 2006 An acoustic emission sensor system intended for warning of the destruction of a bearing is based on measurement of a number of acoustic emission signal parameters and on correlations between these. The patent discusses various means for determining the signal parameters and theircorrelations.—EEU 4092 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 40928,171,789 43.40.Tm DYNAMIC BALANCING APPARATUS AND METHOD USING SIMPLE HARMONIC ANGULAR MOTION Wan Sup Cheung et al., assignors to Korea Research Institute of Standards and Science 8 May 2012 (Class 73/462); ﬁled in Republic of Korea 25 March 2008 This patent describes an approach to determining the balance require- ments of rotors by subjecting these to simple harmonic motion rather thanspinning them. The approach is particularly useful for large and massive rotors.—EEU 8,186,490 43.40.Tm PUSHING FORCE DEVIATING INTERFACE FOR DAMPING MECHANICAL VIBRATIONS Tobias Melz et al ., assignors to Fraunhofer-Gesellschaft zur Forderung der Angewadten Forschung E.V 29 May 2012 (Class 188/266.7); ﬁled in Germany 16 April 2004 This patent pertains to a device for attenuating mechanical vibrations by means of energy conversion systems. The basic design is illustrated in the accompanying ﬁgure, which shows a section of a cylindrical arrange- ment along a diameter. Between the base disk 110 and the load-carrying disk 112 are connected piezoelectric systems 114 and 116. In order to pro- tect these systems from lateral motions without affecting vertical motions there is provided a laterally stiff arrangement consisting of a ﬂexible dia-phragm 122 between a ring 124 and tube or rod 126. A tubular element 118 of polyvinyl chloride is intended to preload the piezoelectric elements to optimize their performance. Addition of an accelerometer 128, control electronics 130, and ampliﬁer 132 can provide active control.—EEU 8,191,690 43.40.Tm SHIM STRUCTURE FOR BRAKE SQUEAL ATTENUATION Ramana Kappagantu and Eric Denys, assignors to Material Sciences Corporation 5 June 2012 (Class 188/73.37); ﬁled 15 April 2008 A shim, to be interposed in a brake system, is conﬁgured with tabs or extensions that are designed to vibrate out of phase with the part of thebrake pad that vibrates so as to cause brake squeal. Appropriate tuning of the vibrating portions of the shim may be obtained by use of added masses, protrusions, or localized cutouts.—EEU 8,201,365 43.40.Tm VIBRATION RESISTANT REINFORCEMENT FOR BUILDINGS Osama J. Aldraihem, Riyadh, Saudi Arabia 19 June 2012 (Class 52/167.1); ﬁled 20 April 2010 Energy-absorbing reinforcements to be included (presumably in the con- crete) in buildings, such as those that are to house sensitive equipment, are intended to increase the structural damping. The reinforcements described in this patent consist of piezoelectric rods, conductive ﬁbers, and a plastic ma- trix.—EEU 8,156,793 43.40.Yq GOLF CLUB HEAD COMPRISING A PIEZOELECTRIC SENSOR Charles Edward Golden and Peter J. Gilbert, assignors to Acushnet Company 17 April 2012 (Class 73/65.03); ﬁled 10 March 2009 As disclosed in U.S. Patent 8,117,903 [reviewed in J. Acoust. Soc. Am. (in press) with which it shares the same drawings] piezoelectric ac- celerometer 14 is again mounted within the volume of club head 12 towards the rear of the center of gravity and on the back of the face of the golf club head. As before, the accelerometer and associated electronics may be used to determine swing velocity, velocity at impact, vibration during impact, and the linear and rotational acceleration and deceleration of the club head.—NAS 8,166,826 43.40.Yq VIBRATION CHARACTERISTIC MEASURING DEVICE Hajime Tada and Mikio Arai, assignors to NHK Spring Company, Limited 1 May 2012 (Class 73/663); ﬁled in Japan 7 March 2008 A disc drive suspension or similar device whose vibration character- istics are to be measured, is mounted cantilever-like on a block that is attached to a shaker. The shaker’s axial motion and the motion of the tipof the cantilever in the same direction are detected by laser Doppler vibr- ometers, and the resulting signals are subjected to fast Fourier transform and further automated processing.—EEU 8,193,262 43.55.Ev FINISHING COMPOUND SUITABLE FOR ACOUSTIC SUPPORTS Florence Serre et al., assignors to Lafarge Gypsum International 5 June 2012 (Class 524/12); ﬁled in France 22 April 2008 This sprayed-on application offers a monolithic coating that preserves the sound absorptive properties of the substrate to which it is applied. It contains a foaming agent and can also be troweled or rolled on.—CJR 4093 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 40938,166,621 43.58.Kr METHOD OF STABILIZING A FREQUENCY OF A PIEZOELECTRIC VIBRATION ELEMENT Tsuyoshi Ohshima et al., assignors to Seiko Epson Corporation 1 May 2012 (Class 29/25.35); ﬁled in Japan 3 February 2005 This patent teaches a method of stabilizing the oscillation frequency of a ‘‘piezoelectric vibration element.’’ It looks as though the authors are really targeting quartz crystal oscillators, although the references to quartz are not consistent throughout the text. They are concerned about frequency drifts over periods of months to years in quartz crystals mounted using silicone-based adhesives. They present some data on the frequency drift of quartz oscillators in hermetically sealed enclosures that have beenmounted this way, and make the claim that the drift is due to poly- dimethylsiloxane (PDMS) migrating from the adhesive where the quartz element is bonded to the support, and ending up on the surface of the quartz resonant element. The authors’ data shows a temperature depend- ence to the rate and saturation of the drift phenomenon, which they attrib- ute to the establishment of a monolayer of the PDMS. They then propose a scheme by which the drift can be saturated all at once—by intentionallyplacing a small amount of PDMS on the quartz to begin with! This would seem to be an effective way of reducing the drift in the near term, but one suspects that there may be undesirable residual effects from the ﬁlm’s migration in the future. No ‘‘before and after’’ data are given on the efﬁ- cacy of this treatment.—JAH 8,166,816 43.58.Wc BULK ACOUSTIC WAVE GYROSCOPE Farrokh Ayazi and Houri Johari, assignors to Georgia Tech Research Corporation 1 May 2012 (Class 73/504.12); ﬁled 4 May 2009 This patent discloses the use of a disk resonator vibrating in a bulk acoustic wave mode as a gyroscope. Most acoustic resonator gyroscopes have used the ﬂexural mode of either a disk, cylinder, or hemisphere, so this is a relatively novel approach to sensing rotation. The authors claim that it is desirable to use a bulk acoustic wave mode because of the higherQ (lower damping) that it affords, compared to ﬂexural modes. They claim Q’s of 50,000–100,000 for their invention (in vacuum). The devices are fabricated from silicon-on-insulator wafers using a process which is not explained, but would probably be costly to reproduce in a regular sili- con fabrication process line, as these wafers are relatively costly. Never- theless, the authors claim very good performance of the test devices pro- duced this way, with noise levels and random walk typical of goodthumb-sized tactical grade gyros using spinning masses. It won’t be long before devices like this take over all but the very highest end of the gyro market.—JAH 8,176,786 43.60.Ac METHODS, APPARATUSES, AND SYSTEMS FOR DAMAGE DETECTION Hoon Sohn and Seungbum Kim, assignors to Carnegie Mellon University 15 May 2012 (Class 73/602); ﬁled 29 June 2007 This patent, while aimed at non-destructive testing (NDT), tells a lot about the efforts to take the technique of time-reversal focusing into the industrial world of practical devices. The authors’ aim is to make an NDT measurement device that can be operated autonomously (say by an indus- trial robot or pipeline pig) so as to locate and mark places where ﬂawsexist in the material being inspected. The device consists of a pair of pie- zoelectric transducers for transmitting ultrasonic tone bursts, a time rever- sal digitizer/equalizer/transmitter, and a signal processor/classiﬁer. The system they have devised is aimed at the nondestructive testing of metal and composite parts via the measurement of the time of arrival of Lamb waves, and screening for mode conversion to other Lamb modes. The pro- cess involves: (a) Creating a narrowband toneburst acoustic signal andapplying it to a ﬁrst transducer; (b) receiving the signal at a second trans- ducer; (c) truncating (windowing) the signal in time and equalizing it; (d) transmitting the time-reversed, equalized signal back to the ﬁrst trans- ducer; and (e) applying damage classiﬁcation algorithms. Their analysis of the problem is exemplary, and includes a lot of data from experiments that indicate the beneﬁts of what they are doing.—JAH 8,187,202 43.64.Ha METHOD AND APPARATUS FOR ACOUSTICAL OUTER EAR CHARACTERIZATION Antonius Hermanus Maria Akkermans et al ., assignors to Koninklijke Philips Electronics N.V 29 May 2012 (Class 600/559); ﬁled in the European Patent Ofﬁce 22 September 2005 The acoustical properties of an outer ear are characterized by analyz- ing an acoustic signal received from the outer ear in response to a trans- mitted acoustic signal toward the outer ear comprising at least one of music and speech. The acoustic properties are used in a remote terminal which is part of a telecommunication system that authenticates persons based on alleged identity, characterized acoustical properties, and a data- base of enrolled acoustic properties.—DAP 8,189,830 43.66.Ts LIMITED USE HEARING AID Zezheng Hou, assignor to Apherma, LLC 29 May 2012 (Class 381/312); ﬁled 28 August 2007 A hearing aid with conﬁgurable use time is described. The perform- ance of a digital hearing aid ampliﬁer purposely degrades by a predeﬁned amount each time the hearing aid is restarted or rebooted. Degradation 4094 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 4094comprises reducing gain or narrowing frequency range by predeﬁned amounts. Unimpeded ampliﬁcation may return when the hearing aid is subsequently provided with additional use time.—DAP 8,189,833 43.66.Ts HEARING AID AND A METHOD OF PROCESSING INPUT SIGNALS IN A HEARING AID Kristian Tjalfe Klinkby et al., assignors to Widex A/S 29 May 2012 (Class 381/313); ﬁled 10 April 2008 The outputs of two microphones are combined to form a spatial sig- nal, whose low frequencies are boosted for equalization. The acoustic feed- back path is estimated to form a feedback compensation signal, which iscombined with the equalized spatial signal to form a hearing loss compen- sation signal. The spatial signal is adapted to control directional processing. Another independent claim is the same except two equalized spatial signals are formed from the two microphone outputs and are combined with the feedback compensation signal to form a beamformer.—DAP 8,189,836 43.66.Ts EAR MOLD WITH VENT OPENING THROUGH OUTER EAR AND CORRESPONDINGVENTILATION METHOD Wai Kit David Ho and Wee Haw Koo, assignors to Siemens Medical Instruments Pte. Limited 29 May 2012 (Class 381/318); ﬁled in Germany 1 October 2007 To ventilate the ear without acoustic feedback problems, an in-the- ear or concha hearing aid ear shell comprises a ﬁrst portion that inserts into the ear canal, a second portion that projects into the wearer’s concha, and a vent enclosed with a titanium ring that connects the two segments. A vent opening in the second segment is directed toward the outer ear of the wearer. The titanium ring may project from the surface of the earshell. Mentioned in the speciﬁcation of the patent, but not in the claims, is that an opening is surgically created in the outer ear to mate with the tita- nium ring.—DAP8,194,870 43.66.Ts TEST COUPLER FOR HEARING INSTRUMENTS EMPLOYING OPEN-FIT EAR CANAL TIPS Oleg Saltykov et al., assignors to Siemens Hearing Instruments, Incorporated 5 June 2012 (Class 381/60); ﬁled 8 December 2008 The goal is to block the leakage from the vented hearing aid ear- piece from reaching the hearing aid microphone during testing with vent open on an ear simulator or coupler. An acoustic shield is placed around the open ﬁt receiver/earpiece which is attached to an ear extension coupled with the ear simulator coupler. The shield may be cylindrical andhave a removable cover. A measurement unit may be connected by a cable passing through the shield to an ear simulator coupler inside.—DAP 8,194,899 43.66.Ts METHOD FOR IMPROVING THE FITTING OF HEARING AIDS AND DEVICE FOR IMPLEMENTING THE METHOD Graham Naylor, assignor to Oticon A/S 5 June 2012 (Class 381/312); ﬁled in Denmark 21 January 2000 Environmental data experienced by the hearing aid wearer is col- lected, including sound levels over time and spectral distributions of sound over time. The hearing aid is adjusted based on this data either automati- cally or by a hearing health care dispensing professional. The data may include at least one of light levels, ambient temperature, body tempera- ture, amount of movement, cardiovascular activity, psychological stress and long-term statistical values.—DAP 4095 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 40958,194,900 43.66.Ts METHOD FOR OPERATING A HEARING AID, AND HEARING AID Eghart Fischer et al., assignors to Siemens Audiologische Technik GmbH 5 June 2012 (Class 381/313); ﬁled in Germany 10 October 2006 The hearing aid signal processor tracks and selects an acoustic speaker source present in an ambient sound via comparison to a database of the speech proﬁles of preferred speakers. The sounds of the selected sound source are made prominent in the hearing aid output in relation tothose from unselected sound sources. During the comparison, the probabil- ity of an acoustic signal containing a speaker may be determined. The stored speech proﬁles may be ranked by the hearing aid wearer. A selected speaker may be tracked regardless of position of the hearing aid wearer.—DAP8,194,901 43.66.Ts CONTROL DEVICE AND METHOD FOR WIRELESS AUDIO SIGNAL TRANSMISSION WITHIN THE CONTEXT OF HEARING DEVICEPROGRAMMING Daniel Alber et al., assignors to Siemens Audiologische Technik GmbH 5 June 2012 (Class 381/314); ﬁled in Germany 28 July 2006 Audio and programming data are combined within the packets of a single channel wireless transmission. The audio and data may be transmit-ted in different packets. The audio data may be transmitted in the payload block and the control data may be transmitted in a packet header. The ra- dio may be digital and a short range inductive link.—DAP 8,199,943 43.66.Ts HEARING APPARATUS WITH AUTOMATIC SWITCH-OFF AND CORRESPONDING METHOD Robert Ba ¨uml and Ulrich Kornagel, assignors to Siemens Audiologische Technik GmbH 12 June 2012 (Class 381/312); ﬁled in the European Patent Ofﬁce 23 November 2006 To automatically switch the hearing aid off when it is not being worn, a ﬁrst acoustic signal is generated by a hearing aid transducer and sensed by another hearing aid transducer. A matched ﬁlter tuned to the switch-off signal feeds the signal processor, which then automatically switches off at least a part of the hearing aid. A second acoustic signalmay be generated when the hearing aid is put on the wearer, which auto- matically turns the hearing aid on. The acoustic signals may be ultrasonic or infrasonic so as to not be heard. The signals may be generated by pre- determined temporal or spectral events.—DAP 8,199,945 43.66.Ts HEARING INSTRUMENT WITH SOURCE SEPARATION AND CORRESPONDING METHOD Ulrich Kornagel, assignor to Siemens Audiologische Technik GmbH 12 June 2012 (Class 381/313); ﬁled in Germany 21 April 2006 A hearing aid system provides receive angles associated with source- speciﬁc signals and presents these sources for selecting one source. The selection device is a directional microphone that aligns the hearing aid with the selected sound source. The selection device may be integrated into a remote control and may be wirelessly connected to the presentationdevice. Source selection may be made by a user pressing a button on the selection device or knocking on the hearing aid. Source signals may be presented sequentially. The selection device may graphically display the source signals and their receive angles. Source selection may be automatic if the user looks at a source longer than a predetermined time.—DAP 4096 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 40968,199,946 43.66.Ts HEARING AID WITH RADIO FREQUENCY IDENTIFICATION RECEIVER FOR SWITCHING A TRANSMISSION CHARACTERISTIC Hartmut Ritter and Tom Weidner, assignors to Siemens Audiologische Technik GmbH 12 June 2012 (Class 381/314); ﬁled in Germany 28 July 2006 The transfer function of a hearing aid is altered via a radio frequency (RF) tag signal, sent by an RF tag within a spatial detection zone, and received by an RFID receiver in the hearing aid. One of at least two trans- fer functions is selected based on the tag signal, wherein the ﬁrst and sec- ond transfer functions are assigned to a predeﬁned class and to an individ- ual item, respectively, of tag data items in the RF tags. The hearing aid switches between the ﬁrst transfer function for all telephones and the sec- ond transfer function for an individual adaptation to a particular telephone type. The hearing aid may evaluate the tag signal and select the associateddataset from memory.—DAP 8,199,948 43.66.Ts ENTRAINMENT AVOIDANCE WITH POLE STABILIZATION Lalin Theverapperuma, assignor to Starkey Laboratories, Incorporated 12 June 2012 (Class 381/318); ﬁled 23 October 2007 An adaptive ﬁlter in an apparatus is used to estimate an acoustic feedback path from receiver to microphone and at least one estimated future pole position of the adaptive ﬁlter transfer function is analyzed forstability to indicate whether entrainment is present. The adaptation rate of the ﬁlter is adjusted using the estimated future pole positions. A Schur– Cohn stability test may be used to derive reﬂection coefﬁcients of the adaptive ﬁlter using the estimated future pole positions and these reﬂec- tion coefﬁcients are monitored to indicate entrainment. The apparatus may include various styles of hearing aids.—DAP 8,199,951 43.66.Ts HEARING AID DEVICE Markus Heerlein and Wai Kit Ho, assignors to Siemens Audiologische Technik GmbH 12 June 2012 (Class 381/323); ﬁled 25 June 2005 When a switch that opens and closes an electric circuit in a hearing aid device is closed, the battery drawer inhibits battery removal from breaking contact with the electric circuit. The switch is an on/off cover that rotates about an axle to move between on and off. The switch com- prises male and female conductive contacts attached to the on/off cover and circuit, respectively. The battery may be removed when the switch is open. The on/off cover may be mechanically coupled to the battery drawer.—DAP 8,199,952 43.66.Ts METHOD FOR ADAPTIVE CONSTRUCTION OF A SMALL CIC HEARING INSTRUMENT Artem Boltyenkov et al ., assignors to Siemens Hearing Instruments, Incorporated 12 June 2012 (Class 381/328); ﬁled 31 July 2008 The size of a completely in-the-canal hearing aid is made deliber- ately smaller than the ear canal to create more slit leak so as to reduce 4097 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 4097occlusion. The hearing aid shell is shrunk from tightly ﬁtting in the ear canal to a shape more closely surrounding the internal components. The outer periphery of a mounting element having at least one vent contacts the ear canal walls near the eardrum to secure the hearing aid. The mount- ing element may be a dome-shaped cap with several vents. The initial shell image, internal components, and shrunken shell may be displayed on a computer screen.—DAP 8,204,263 43.66.Ts METHOD OF ESTIMATING WEIGHTING FUNCTION OF AUDIO SIGNALS IN A HEARING AID Michael Syskind Pedersen et al., assignors to Oticon A/S 19 June 2012 (Class 381/313); ﬁled in the European Patent Ofﬁce 7 February 2008 Using a weighted sum of at least two microphone outputs, one front- aiming and one rear-aiming, a direction-dependent time-frequency (T-F) gain is calculated from at least two directional signals, one front-aiming and one rear-aiming, formed from the microphone outputs, each containing a T-F representation of a target signal and a noise signal. T-F representa- tions of the target and noise signals are used to estimate a time-frequency mask, denoting whether target or noise is present, which then determines the direction-dependent T-F gain. T-F coefﬁcients are determined based onwhether the ratio of the envelopes of the T-F representations of the target- direction to the noise-direction directional signal is greater or less than a predetermined threshold.—DAP 8,194,864 43.66.Vt EARHEALTH MONITORING SYSTEM AND METHOD I Steven W. Goldstein et al ., assignors to Personics Holdings Incorporated 5 June 2012 (Class 381/56); ﬁled 30 October 2007 An insertable earbud includes a microphone to measure the actual sound level in the ear canal. The signal from such a microphone might beanalyzed in any number of ways to predict and then counteract potential hearing damage. The patent envisions an elaborate, ongoing process that allows for cumulative exposure, recovery periods, etc. However, any long-term electronic log will include periods when the earbud is not worn. The patent Claims are slippery on this point, but it appears that ‘‘ambient’’ ex- posure is somehow estimated and included in the analysis.—GLA 8,050,934 43.72.-p LOCAL PITCH CONTROL BASED ON SEAMLESS TIME SCALE MODIFICATION AND SYNCHRONIZED SAMPLING RATE CONVERSION Atsuhiro Sakurai et al ., assignors to Texas Instruments Incorporated 1 November 2011 (Class 704/503); ﬁled 29 November 2007 This patent claims an extension of ‘‘seamless time-scale modiﬁca- tion.’’ The prior technique for pitch shifting is said to have been applied only by shifting the entire signal at once. Here it is extended to allow local continuous and seamless pitch shifting over small frames throughout a signal.—SAF 8,073,686 43.72.Ar APPARATUS, METHOD AND COMPUTER PROGRAM PRODUCT FOR FEATURE EXTRACTION Yusuke Kida and Takashi Masuko, assignors to Kabushiki Kaisha Toshiba 6 December 2011 (Class 704/207); ﬁled in Japan 29 February 2008 The objective here is to determine a pitch difference between succes- sive speech signal frames, as a part of pitch tracking methods. The method involves computing the cross-correlation between log frequency spectra of successive signal frames.—SAF 4098 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 40988,086,449 43.72.Ar VOCAL FRY DETECTING APPARATUS Carlos Toshinori Ishii et al ., assignors to Advanced Telecommunications Research Institute International 27 December 2011 (Class 704/207); ﬁled in Japan 31 August 2005 This patent describes a very basic method of detecting ‘‘vocal fry,’’ also known as creaky voice, in a speech signal. The method involves ﬁnd-ing a power peak (glottal pulse) in a frame, then hunting for the next power peak in a subsequent frame, then determining the instantaneous pe- riod as the distance between power peaks. It is hard to believe that no one has done this before, or that it is not sufﬁciently obvious from existing voice processing methods.—SAF 8,078,474 43.72.Gy SYSTEMS, METHODS, AND APPARATUS FOR HIGHBAND TIME WARPING Koen Bernard Vos and Ananthapadmanabhan Aasanipalai Kandhadai, assignors to QUALCOMM Incorporated 13 December 2011 (Class 704/500); ﬁled 3 April 2006 Speech transmission engineers are constantly hunting for their own ‘‘perpetual motion machine,’’ which in their business means sending a wideband coded speech signal over a narrowband channel. As this is, onits face, oxymoronic, ingenious tricks are often dreamt up to approximate the desired outcome. Here we ﬁnd proposals to harness the properties of time-warping, to extract a highband excitation signal from an encoded nar- rowband excitation signal.—SAF 8,200,478 43.72.Ne VOICE RECOGNITION DEVICE WHICH RECOGNIZES CONTENTS OF SPEECH Takashi Ebihara et al ., assignors to Mitsubishi Electric Corporation 12 June 2012 (Class 704/10); ﬁled in Japan 30 January 2009 A speech recognition system is described for use in computer plat- forms having limited memory space. The entire point of this patent is a listing of techniques by which a recognition training algorithm can be set up so as to restrict the maximum sentence or word length without severelydegrading the recognizer’s usefulness.—DLR 8,195,460 43.72.Pf SPEAKER CHARACTERIZATION THROUGH SPEECH ANALYSIS Yoav Degani and Yishai Zamir, assignors to Voicesense Limited 5 June 2012 (Class 704/243); ﬁled 17 June 2008 This patent proposes a system for extracting speaker mood and psy- chological characteristics from F0 (pitch) and other prosodic information extracted from the speech signal. The technology for extracting F0 is assumed. There is no mention of this. There is some discussion with list- ings of the relevant prosodic structures assumed to be of interest, such as pause durations, utterance length, etc. The patent assumes that all messagecontent information is contained exclusively in the segment (phonemic) content, that is, that there will be no ‘‘contamination’’ of the speaker char- acterization by segmental content. The patent proposes speech data collec- tion and hand labeling of the relevant speaker information prior to a train- ing phase. Some speech data libraries do exist containing some such information. Finally, the patent says very little about the actual classiﬁca- tion techniques. There is some mention of a scoring system by whichspeaker behavior patterns would be rated according to their presence in the speech signal.—DLR8,204,747 43.72.Pf EMOTION RECOGNITION APPARATUS Yumiko Kato et al., assignors to Panasonic Corporation 19 June 2012 (Class 704/254); ﬁled in Japan 23 June 2006 This patent describes a system for estimating a speaker’s emotional state based on an analysis of the acoustical content of the speech signal. An F0 (fundamental frequency) analysis is performed to extract pitch andother prosodic content, such as timing information, from the speech signal. The classiﬁcation system then looks for two speciﬁc voicing characteris- tics known as ‘‘husky’’ voice and ‘‘pressed’’ voice, determined by meas- urements of the voicing spectrum in the speech signal. The speaker’s emo- tional state is estimated based on the timing and rate of occurrence of these voice patterns. The patent text makes the explicit argument that these extracted speaker state indicators will be valid regardless of the lan-guage or culture.—DLR 8,148,623 43.75.St APPARATUS FOR ASSISTING IN PLAYING MUSICAL INSTRUMENT Hideyuki Masuda et al., assignors to Yamaha Corporation 3 April 2012 (Class 84/723); ﬁled in Japan 30 August 2005 ‘‘An apparatus is to assist an unskilled player in playing a musical instrument by detecting the quantity of the player’s manipulation against the instrument, modifying the detected manipulation quantity with reference to a recommended manipulation to a degree according to a given assistancecoefﬁcient, and actuating the instrument with the modiﬁed manipulation quantity. For a brass instrument, the apparatus comprises an embouchure sensor (11) and a breath pressure sensor (12) to detect the embouchure and the breath pressure of the player as he/she plays the brass instrument. The detected embouchure and breath pressure are then modiﬁed with reference to a recommended embouchure and breath pressure weighted by a given as- sistance coefﬁcient. The apparatus actuates the brass instrument based onthe modiﬁed embouchure and breath pressure.’’ Another embodiment repla- ces the player with a set of artiﬁcial lips and compressed gas to enable a surrogate play of the musical instrument.—NAS 8,152,590 43.80.Pe ACOUSTIC SENSOR FOR BEEHIVE MONITORING Trenton J. Brundage, Sherwood, OR 10 April 2012 (Class 449/2); ﬁled 3 September 2009 ‘‘A method of and system for using sounds produced by bees ﬂying near a beehive entrance enable a beekeeper to assess the operational pro- ductivity of the beehive. In a preferred embodiment, the method entails 4099 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 4099positioning an acoustic pickup device, such as a microphone, at a location to pick up and provide an audio signal representing sounds produced bybees ﬂying around the beehive entrance. The ﬂying bees produce the sounds either while hovering in the vicinity of the beehive or while launch- ing from locations around the beehive entrance to forage for pollen and nectar. The audio signal is analyzed to distinguish the sound of launching ﬂying bees from the sound of ambient background noise.’’—NAS 8,152,734 43.80.Vj SYSTEM AND METHOD FOR DIAGNOSIS OF BOVINE DISEASES USING AUSCULTATION ANALYSIS Thomas H. Noffsinger et al ., assignors to Pierson Precision Auscultation 10 April 2012 (Class 600/529); ﬁled 7 November 2008 ‘‘A system and method are provided for diagnosis of bovine (30) respira- tory diseases using auscultation techniques. Acoustic characteristics of a recorded spectrogram are compared with existing data enabling a diagnosis to be made for a diseased animal. Lung (32) sounds are obtained by use of an electronic stethoscope, and the sounds are stored as digital data. Signal condi- tioning is used to place the data in a desired format and to remove undesirable noise associated with the recorded sounds. An algorithm is applied to the data, and lung scores are calculated. The lung scores are then categorized into vari- ous levels of perceived pathology based upon baseline data that categorizesthe lung scores. From the lung scores, a caregiver can associate a diagnosis, prognosis, and a recommended treatment. Analysis software generates the lung scores from the recorded sounds, and may also provide a visual display of presumptive diagnoses as well as recommended treatments.’’ The apical lobe 34, which is partially covered by the fourth rib 36, is the preferred location for auscultation. Circle 38 is the preferred location where the stethoscope should be placed, a spot approximately three inches above the right elbow 39.—NAS 8,187,186 43.80.Vj ULTRASONIC DIAGNOSIS OF MYOCARDIAL SYNCHRONIZATION Ivan Salgo et al., assignors to Koninklijke Philips Electronics N.V 29 May 2012 (Class 600/438); ﬁled 2 November 2006 Points on opposite sides of a heart chamber are identiﬁed in an ultra- sound image and then tracked through a portion of the heart cycle. Thechanging positions of lines extending between pairs of the points are accu- mulated and displayed. The display uses color to show the location of a line at a particular point in the cardiac cycle.—RCW 8,187,187 43.80.Vj SHEAR WAVE IMAGING Liexiang Fan et al., assignors to Siemens Medical Solutions USA, Incorporated 29 May 2012 (Class 600/438); ﬁled 16 July 2008 Shear wave velocity is estimated in a region of an ultrasound image that is used to guide the selection of the region. The estimate is validated by other calculations of shear wave velocity, for example, by dividing the initial set of data into subsets from which shear wave velocity is also esti- mated. The estimated shear wave velocity is displayed on a scale of veloc- ities associated with a type of tissue.—RCW 8,187,190 43.80.Vj METHOD AND SYSTEM FOR CONFIGURATION OF A PACEMAKER AND FORPLACEMENT OF PACEMAKER ELECTRODES Praveen Dala-Krishna, assignor to St. Jude Medical, Atrial Fibrillation Division, Incorporated 29 May 2012 (Class 600/443); ﬁled 14 December 2006 A function that evaluates electrode sites and pacemaker conﬁgura- tions uses the volume of blood ejected from the heart as determined byultrasound imaging and uses other parameters such as the activation volt- age of the pacemaker to facilitate conﬁguration of the pacemaker and the placement of the pacemaker electrodes in a patient.—RCW 8,187,192 43.80.Vj METHOD AND APPARATUS FOR SCAN CONVERSION AND INTERPOLATION OF ULTRASONIC LINEAR ARRAY STEERING IMAGING Bin Yao et al ., assignors to Shenzhen Mindray Bio-Medical Electronics Company, Limited 29 May 2012 (Class 600/447); ﬁled in China 29 November 2007 Points depending on the steering angle of an ultrasound beam are calculated for use in interpolating points accurately in the principal direc- tion of the imaging system point-spread function.—RCW 8,187,193 43.80.Vj MINIATURE ACTUATOR MECHANISM FOR INTRAVASCULAR IMAGING Byong-Ho Park and Stephen M. Rudy, assignors to Volcano Corporation 29 May 2012 (Class 600/463); ﬁled 13 January 2010 An actuator mechanism inside the bore of an intravascular ultrasound imaging probe is made with a so-called shape-memory alloy. Ultrasound images are obtained by using the mechanism to move a transducer. Themechanism can be fabricated in a small size that enables the distal end of the probe to have a small diameter.—RCW 4100 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 41008,189,427 43.80.Vj CLUTTER SIGNAL FILTERING FOR DOPPLER SIGNAL Kwang Ju Lee and Jong Sik Kim, assignors to Medison Company, Limited 29 May 2012 (Class 367/87); ﬁled in Republic of Korea 17 December 2008 Two highpass ﬁlters are used. The ﬁrst ﬁlter provides reduced clutter signals that are used with information from the unﬁltered Doppler signal to compute an input signal power to ﬁltered input signal power ratio (IFR). The IFR is then used by a controller to deﬁne a second highpassﬁlter and parameters in a processing sequence that modulates, ﬁlters, and demodulates the Doppler signal.—RCW 8,200,313 43.80.Vj APPLICATION OF IMAGE-BASED DYNAMIC ULTRASOUND SPECTROGRAPHY IN ASSISTINGTHREE DIMENSIONAL INTRA-BODY NAVIGATION OF DIAGNOSTIC AND THERAPEUTIC DEVICES Edmond Rambod and Daniel Weihs, assignors to Bioquantetics, Incorporated 12 June 2012 (Class 600/424); ﬁled 1 October 2008 A micron-sized wire made from a polymer is coupled to a catheter. At the tip of the wire, a 100 to 500 micron diameter metallic cylinder is attached. An area of interest containing the catheter and the cylinder isimaged. Amplitude-modulated ultrasound is used to stimulate the cylinder that then emits a unique acoustic response. The signals from the cylinder are used to determine the location of the catheter.—RCW 8,202,222 43.80.Vj EQUAL PHASE TWO-DIMENSIONAL ARRAY PROBE Jinzhong Yao et al., assignors to Sonoscape, Incorporated 19 June 2012 (Class 600/459); ﬁled 16 October 2007 Received signals in one dimension of a two-dimensional array are summed and signals in the other direction of the array are multiplexed to channels in the front-end of an imaging system.—RCW 4101 J. Acoust. Soc. Am., Vol. 132, No. 6, December 2012 4101Copyright of Journal of the Acoustical Society of America is the property of American Institute of Physics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
1.4955455.pdf	Magnetoplasmonic RF mixing and nonlinear frequency generation C. J. Firby and A. Y. Elezzabi Citation: Appl. Phys. Lett. 109, 011101 (2016); doi: 10.1063/1.4955455 View online: http://dx.doi.org/10.1063/1.4955455 View Table of Contents: http://aip.scitation.org/toc/apl/109/1 Published by the American Institute of Physics Articles you may be interested in A magnetoplasmonic electrical-to-optical clock multiplier Appl. Phys. Lett. 108, 051111051111 (2016); 10.1063/1.4941417Magnetoplasmonic RF mixing and nonlinear frequency generation C. J. Firbya)and A. Y . Elezzabi Ultrafast Optics and Nanophotonics Laboratory, Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta T6G 1H9, Canada (Received 3 May 2016; accepted 24 June 2016; published online 6 July 2016) We present the design of a magnetoplasmonic Mach-Zehnder interferometer (MZI) modulator facilitating radio-frequency (RF) mixing and nonlinear frequency generation. This is achieved by forming the MZI arms from long-range dielectric-loaded plasmonic waveguides containingbismuth-substituted yttrium iron garnet (Bi:YIG). The magnetization of the Bi:YIG can be driven in the nonlinear regime by RF magnetic ﬁelds produced around adjacent transmission lines. Correspondingly, the nonlinear temporal dynamics of the transverse magnetization component aremapped onto the nonreciprocal phase shift in the MZI arms, and onto the output optical intensity signal. We show that this tunable mechanism can generate harmonics, frequency splitting, and frequency down-conversion with a single RF excitation, as well as RF mixing when driven by twoRF signals. This magnetoplasmonic component can reduce the number of electrical sources required to generate distinct optical modulation frequencies and is anticipated to satisfy important applications in integrated optics. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4955455 ] In the pursuit of developing integrated nanoplasmonic circuitry for optical computing applications, high frequencymodulators for encoding and transferring data are crucialcomponents. Electrical control of optical signals is desirablefor integrating both complementary-metal-oxide-semicon-ductor (CMOS) and optical components into a singleplatform. In recent years, a vast array of device architectures, achieved by altering the properties of a nanoplasmonic waveguide’s constituent materials, have been developed tomeet this demand. 1These include the modiﬁcation of refrac- tive indices via the thermo-optic effect,2electro-optic effect,3or free carrier effects.4However, these mechanisms present a fundamental limitation: within the response band-width of the material, only the driving source frequency is mapped onto the optical mode. For on-chip applications, this implies that optical modulation at multiple distinct frequen-cies requires numerous single frequency sources, or a widelytunable on-chip electrical source. These solutions are unin-viting, as the footprint consumed by such electrical compo-nents is large. Thus, one requires a tunable mechanism ofgenerating optical modulation at many distinct frequencies while employing only one or two electrical sources. This limitation can be surpassed by considering a nano- plasmonic modulator composed of magnetic materials, andhence employing nonreciprocal magneto-optic phenomenon. Magneto-optic effects are related to the magnetization vector, M¼hM x;My;Mzi, within the material.5Under the application of a time-varying magnetic ﬁeld, the temporaldynamics of Mevolve in accordance with the highly nonlin- ear Landau-Lifshitz-Gilbert (LLG) equation. 6,7Previous studies examining ferromagnetic response in the microwaveregime have shown that the nonlinear behaviour of M can facilitate harmonic generation when driven by a radio- frequency (RF) signal. 8,9As well, the nonlinearity of theresponse can induce frequency mixing when Mis driven with two RF signals.9,10Furthermore, nonlinear frequency generation can be enhanced through parametric excitation ofthe Suhl instability. 11Thus, the use of these materials in plasmonics is attractive for attaining optical modulation at unique frequencies other than the driving frequency. For integration with plasmonics, bismuth-substituted yttrium iron garnet (Bi:YIG) is an ideal material, as it dis- plays signiﬁcantly lower losses than ferromagnetic metals (Fe, Ni, Co, etc.) at k¼1550 nm,12and has been shown to exhibit a temporal response on a picosecond timescale.13If the Bi:YIG waveguide is magnetized transverse to the propa- gation direction of a guided optical mode, then the Lorentz reciprocity condition is broken, and the mode’s wavevectorbecomes nonreciprocal. The wavevectors for forward ( b fwd) and backward ( bbwd) directed modes are no longer equivalent (i.e., bfwd6¼bbwd).5Thus, a nonreciprocal phase shift (NRPS) can be observed between two counter-propagating opticalmodes, Db¼b fwd/C0bbwd.5This NRPS is a strong function of the transverse magnetization. Application of a sinusoidal magnetic ﬁeld will nonlinearly drive M, and thus the NRPS, as prescribed by the LLG equation. Embedding this nonreci-procal wavevector modulation into an interferometric struc- ture, such as a Mach-Zehnder interferometer (MZI) 14,15or a slit-groove interferometer,16,17allows the dynamic and nonlinear transverse component of Mto be mapped onto the optical intensity at the output port of the MZI. As such, the extreme nonlinearity of the magnetization dynamics will generate additional spectral components in the transmittedintensity signal. In this letter, we present the conceptual design of a mag- netoplasmonic MZI modulator that facilitates nonlinear frequency generation and RF mixing. Long-range dielectric-loaded magnetoplasmonic waveguides (LRDLMPWs) con- sisting of a Bi:YIG core form the basis of such a modulator. The magnetization of the Bi:YIG is driven by RF magnetic a)Electronic mail: ﬁrby@ualberta.ca 0003-6951/2016/109(1)/011101/5/$30.00 Published by AIP Publishing. 109, 011101-1APPLIED PHYSICS LETTERS 109, 011101 (2016) ﬁelds, and the nonlinear magnetization dynamics are transcribed onto the transmitted intensity waveform via the NRPS. This platform is shown to be versatile, tunable, and capable of providing modulation at a number of distinctfrequencies by varying either the applied magnetic ﬁelds or the driving frequency. To employ such nonlinear frequency mixing, we con- sider the modulator geometry depicted in Fig. 1, modeled after Ref. 18. The constituent LRDLMPW arms consist of a Bi:YIG core with an Ag guiding strip, thin ﬁlms of Si 3N4 and Al 2O3, and a SiO 2substrate, as illustrated in the insets of Fig.1. The dimensions (width wiand height di) and refrac- tive indices ( ni) of each material are presented in Table I. The Si 3N4and Al 2O3layers provide the required ﬁeld sym- metry around the Ag for long-range plasmon formation.Practically, the considered 15 nm Ag ﬁlm is the thinnest con- tinuous ﬁlm attainable via sputtering, 19and the Bi:YIG MZI can be realized with the use of pulsed laser deposition orsputtering and precise ion beam milling. Note that in order to reduce optical losses, the Ag strip is removed from the input and output Y-junctions. The 55 lm long photonic Y-couplers adiabatically taper to a LRDLMPW separation of 10lm. Furthermore, the input and output Y-junctions are designed to be slightly asymmetric, such that the total opticalpath length of one MZI branch provides a phase bias of p/2. Correspondingly, the length of the LRDLMPW arms is then set to L p=2¼p/(2Db) to offer a NRPS of p/2. This versatile device architecture has recently been implemented in the design of an optical isolator24and an electrical-to-optical clock multiplier.18 Fully vectorial ﬁnite-difference-time-domain (FDTD) simulations were employed to determine both the NRPS and optical transmission within this MZI geometry. Note that thepredominantly z-polarized TM plasmonic mode, shown in the right inset of Fig. 1, is launched into the MZI. Here, the Bi:YIG core was modeled with the following magnetic prop-erties: a speciﬁc Faraday rotation of h F¼0.25/C14/lm,20a satu- ration magnetization of l0MS¼9 mT,20a Gilbert damping parameter of a¼10/C04,25and a gyromagnetic ratio of c0. Correspondingly, permittivity tensor of the Bi:YIG is12 erMðÞ ¼n2 YIG iknYIG pMz MShF/C0iknYIG pMy MShF /C0iknYIG pMz MShF n2 YIG iknYIG pMx MShF iknYIG pMy MShF/C0iknYIG pMx MShF n2 YIG2 666666643 77777775: (1) With Msaturated in the transverse direction, i.e., M¼h þ M S;0;0i, the NRPS is calculated as Db¼–1.77 rad/ mm for the TM mode. This implies that the arm length is Lp=2¼886.1 lm, which is considerably less than the magne- toplasmonic mode’s long propagation length ofL prop¼3.0 mm. Additionally, with this arm length, driving the magnetization between Mx/MS¼61 will modulate the transmission of the MZI between its minimum and maxi-mum values. As depicted in Fig. 1, the RF driving ﬁelds are induced by means of two parallel transmission lines, 16 lm apart, running adjacent to the LRDLMPW arms. 18These transmis- sion lines are made of Ag and have dimensions of 2 lm /C22lm/C2Lp=2. Propagating a sinusoidal RF current signal I(t) through such transmission lines generates sinusoidally varying magnetic ﬁelds, hðtÞ¼h 0;0;6hzðtÞi, that nonli- nearly drive Min the Bi:YIG LRDLMPWs. The initial magnetization state, M¼h0;þMS;0i, is ﬁxed by an exter- nally applied magnetic ﬁeld oriented in the þy-direction, Hstatic ¼h0;þHy;0i. As such, the nonlinear magnetization dynamics can be determined from the LLG formalism6,7 dM dt¼/C0l0c0 1þa2M/C2HstaticþhtðÞ ðÞ ½/C138 /C0l0c0a MS1þa2 ðÞM/C2M/C2HstaticþhtðÞ ðÞ ½/C138 :(2) Note that although the applied magnetic ﬁelds are not ori- ented transverse to the waveguide, the nonlinear dynamics induce a transverse magnetization component ( Mx), which generates the NRPS within the MZI arms. Mis approxi- mated as uniform over the MZI arm due to the low magnetic ﬁeld variation over the Bi:YIG core cross section, uniformity of the structure in the propagation direction, and short length FIG. 1. Schematic illustration of the magnetoplasmonic MZI geometry for RF mixing and nonlinear frequency generation. The left inset shows a cross section of the LRDLMPW arm structure, while the right inset shows thejE zj2proﬁle, where 26% of the optical power is contained within the Bi:YIG.TABLE I. Material dimensions and refractive indices. Material wi(nm) di(nm) ni Bi:YIG 320 400 2.3 (Ref. 20) Ag 160 15 0.145 þ11.438i (Ref. 21) Si3N4 … 175 1.977 (Ref. 22) Al2O3 … 175 1.746 (Ref. 23) SiO 2 … … 1.444 (Ref. 23)011101-2 C. J. Firby and A. Y . Elezzabi Appl. Phys. Lett. 109, 011101 (2016)of the arms relative to the driving RF wavelengths of interest. In the present calculations, the dynamic magnetization model of Eq. (2)was combined with FDTD simulations of the MZI, and the resultant temporal variations in the trans-mitted intensity were determined. The frequency spectrumof the signal depicts the nonlinear modulation frequencygeneration provided by the LLG dynamics. To demonstrate the versatile range of applications of nonlinear frequency generation in the magnetoplasmonicMZI, we consider several exemplary parameter sets. Eachcase is statically biased at l 0Hy¼10 mT and is driven at a single frequency: fd¼280 MHz (case 1), fd¼420 MHz (case 2), and fd¼1 GHz (case 3). Here, hz(t) takes the form, hzðtÞ¼hzsinð2pfdtÞ. The frequency spectra of the modulated intensity at the MZI output port are plotted as a function of driving ﬁeld amplitude, l0hz, in Figs. 2(a)–2(c) . In all spectra presented in Fig. 2, the DC frequency component, arising from the nonzero transmission occurring when Mx/MS¼0i n the MZI arms, has been removed for clarity. Two additionalspectral features are worth noting. For low amplitude drivingsignals, there exists a spectral component at the driving fre-quency, f d, with sidebands occurring at the ferromagnetic resonance (FMR) frequency, /C23¼c0l0Hy/(2p) (the Larmor frequency),6and at f¼2fd/C0/C23. Note that /C23¼280 MHz in cases 1–3. As the amplitude of the driving signal at fdis increased, the frequency of the sidebands begin to shift in anonlinear manner. This is followed by the appearance of oddharmonics of f d. Since the Bi:YIG is driven by a transverse linearly polarized magnetic ﬁeld, the even order mixingeffects are only present in the longitudinal component, M y, while odd order mixing effects are present in the transverse MxandMzcomponents. As such, this harmonic behaviour is observed in the intensity signal, as the NRPS in the MZIonly maps the temporal variations in M xonto the optical waveform. These features arise from the highly nonlineartrajectory of Munder RF excitation.The frequency spectrum for case 1 is shown in Fig. 2(a). Of particular interest in this situation is the frequency spec-trum with a driving amplitude of l 0hz¼36 mT, which is plotted in Fig. 2(d). Notably, this conﬁguration suppresses the driving frequency, fd¼/C23¼280 MHz, and the largest spectral component is in fact the third harmonic of fdat 840 MHz. The spectral amplitude of the 3 fdfrequency is 40 times greater than that of the driving frequency spectralamplitude at f d. Other sizable spectral components are pro- duced at 112 MHz, 448 MHz, 671 MHz, 1 GHz, 1.231 GHz,and 1.4 GHz; however, ﬁltering schemes can be employed to isolate just the sought after frequency component. Thus, this platform provides an efﬁcient means of generating the thirdharmonic of the input RF signal. In a similar manner, the frequency spectrum for case 2 is displayed in Fig. 2(b). At an amplitude of l 0hz¼18 mT, the driving frequency ( fd¼420 MHz in this case) is again suppressed in the spectrum, as shown in Fig. 2(e). However, with these parameters, the most prominent spectral compo-nents are the two peaks at 158 MHz and 682 MHz, eachspaced a distance of 262 MHz from f d. These peaks are excited with nearly equal amplitude, and are the result of thenonlinear sideband formation within the spectrum. Note thatthe spectral separation is a function of both driving fre-quency and amplitude. All other spectral components have amplitudes less than 10% of the 158 MHz peak. Clearly, the device can be driven at a single frequency, but the outputmodulation is split between two different and distinctfrequency components. In case 3, we consider driving the device at a higher fre- quency of f d¼1 GHz. The resultant spectra as functions of driving amplitude are shown in Fig. 2(c). Speciﬁcally, the spectrum of the transmission signal driven with a peak ﬁeldmagnitude of l 0hz¼21 mT is displayed in Fig. 2(f). In this case, two primary spectral peaks are observed: one at fdand the other at 255 MHz, each with equal amplitude. Note thatthe 255 MHz peak represents only a small deviation from the FIG. 2. Plots of the frequency spectra of the modulated intensity signal at the MZI output versus l0hzforl0Hy¼10 mT and (a) fd¼280 MHz, (b) fd¼420 MHz, and (c) fd¼1 GHz. (d) depicts the spectral proﬁle denoted by the dashed line in (a), for l0hz¼36 mT. Note that the 3 fdcomponent is enhanced. (e) shows the spectrum marked by the dashed line in (b), when l0hz¼18 mT. Note that the dominant spectral components are those at 158 MHz and 682 MHz, due to the nonlinear sideband formation. (f) illustrates the spectrum indicated by the dashed line in (c), for l0hz¼21 mT. Note that the fdand the downcon- verted frequency of 255 MHz are excited with nearly equal amplitude.011101-3 C. J. Firby and A. Y . Elezzabi Appl. Phys. Lett. 109, 011101 (2016)FMR frequency at /C23¼280 MHz. Thus, by driving the device at high frequency, the nonlinear response generates a compo-nent near the signiﬁcantly lower FMR frequency. Such down-conversion behaviour is desirable in complex comput- ing architectures, where the various subsystems operate at different speeds. Driving the MZI with two or more RF signals simultane- ously presents a much more complex situation. Here, we observe similar sideband and harmonic generation for each of the two individual frequencies, but the nonlinear mixing process gives rise to numerous additional spectral compo- nents. Two example spectra are displayed in Figs. 3(a) and 3(b) as functions of the driving ﬁeld amplitude. Again, the DC frequency component has been removed for clarity. Both spectra are taken under a static ﬁeld bias of l 0Hy¼25 mT (i.e., /C23¼700 MHz) and are driven by two frequencies, fd1 andfd2, such that hzðtÞ¼hz½sinð2pfd1tÞþsinð2pfd2tÞ/C138. The two exemplary cases consider fd1¼275 MHz and fd2 ¼420 MHz (case 4), and fd1¼700 MHz and fd2¼1G H z (case 5). Case 4 is depicted in Fig. 3(a). To illustrate the fre- quency content obtained by driving the MZI with two RF signals, the spectrum obtained at a driving ﬁeld amplitude of l0hz¼20 mT is expanded in Fig. 3(c). Notably, as stated earlier, the transverse magnetization component, and hence the intensity transmission from output port of the device, exhibits odd ordered mixing effects. This behaviour is con- ﬁrmed by a perturbation expansion of the LLG equation, pre- dicting the presence of eight ﬁrst- and third-order mixedfrequencies due to the nonlinear mixing. 10Each of these fre- quencies are present in Fig. 3(c), as shown by the peaks at fd1¼275 MHz, fd2¼420 MHz, 3 fd1¼825 MHz, 3 fd2 ¼1.26 GHz, 2 fd1þfd2¼970 MHz, 2 fd1–fd2¼130 MHz, 2 fd2 þfd1¼1.115 GHz, and 2 fd2/C0fd1¼565 MHz. Employing appropriate ﬁltering techniques at the output port can then separate these distinct frequency components of the modu-lated signal, for routing to different optical devices. Furthermore, adjusting the operating parameters can enhance or suppress some of these components, increasing the power contained in a particular frequency of interest. Case 5, shown in Fig. 3(b), illustrates this. Here, we considerthe device driven at f d1¼700 MHz and fd2¼1 GHz, with an amplitude of l0hz¼19 mT. The resulting spectrum is illus- trated in Fig. 3(d). Markedly, this conﬁguration enhances the third-order mixed frequencies at 2 fd1/C0fd2¼400 MHz and 2fd2/C0fd1¼1.3 GHz, as these spectral peaks are the domi- nant features of the spectrum. Thus, power is transferredmore efﬁciently to these two components than to the others. In conclusion, we have presented a platform for transfer- ring nonlinear magnetization dynamics into the opticalregime. By employing a magnetoplasmonic MZI driven byRF current signals, the response of the magnetization in the Bi:YIG waveguide core evolves in a highly nonlinear manner, mapping itself onto the output optical intensity viathe NRPS. 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1.3271827.pdf	Relation between critical current of domain wall motion and wire dimension in perpendicularly magnetized Co/Ni nanowires S. Fukami, Y. Nakatani, T. Suzuki, K. Nagahara, N. Ohshima, and N. Ishiwata Citation: Applied Physics Letters 95, 232504 (2009); doi: 10.1063/1.3271827 View online: http://dx.doi.org/10.1063/1.3271827 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/95/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Distribution of critical current density for magnetic domain wall motion J. Appl. Phys. 115, 17D508 (2014); 10.1063/1.4866394 Electrical endurance of Co/Ni wire for magnetic domain wall motion device Appl. Phys. Lett. 102, 222410 (2013); 10.1063/1.4809734 Effects of notch shape on the magnetic domain wall motion in nanowires with in-plane or perpendicular magnetic anisotropy J. Appl. Phys. 111, 07D123 (2012); 10.1063/1.3677340 Magnetic field insensitivity of magnetic domain wall velocity induced by electrical current in Co/Ni nanowire Appl. Phys. Lett. 98, 192509 (2011); 10.1063/1.3590713 Large thermal stability independent of critical current of domain wall motion in Co/Ni nanowires with step pinning sites J. Appl. Phys. 108, 113914 (2010); 10.1063/1.3518046 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 147.143.2.5 On: Sun, 21 Dec 2014 11:23:09Relation between critical current of domain wall motion and wire dimension in perpendicularly magnetized Co/Ni nanowires S. Fukami,1,a/H20850Y . Nakatani,2T . Suzuki,1K. Nagahara,1N. Ohshima,1and N. Ishiwata1 1NEC Corporation, 1120, Shimokuzawa, Sagamihara, Kanagawa 229-1198, Japan 2University of Electro-Communications, Chofu, Tokyo 182-8585, Japan /H20849Received 11 September 2009; accepted 16 November 2009; published online 7 December 2009 /H20850 We investigated the relation between critical current of domain wall motion and wire dimension by using perpendicularly magnetized Co/Ni nanowires with different widths and thicknesses. Thecritical current, I c, became less than 0.2 mA when w/H11021100 nm, suggesting that magnetic random access memory with domain wall motion can replace conventional embedded memories. Inaddition, in agreement with theory, the critical current density, j c, decreased as wire width decreased and became much less than 5 /H11003107A/cm2when w/H11021100 nm. We also performed a micromagnetic simulation and obtained good agreement between the experiment and simulation, although a fewdiscrepancies were found. © 2009 American Institute of Physics ./H20851doi:10.1063/1.3271827 /H20852 Current-induced domain wall /H20849DW /H20850motion, ﬁrst pre- dicted by Berger, 1has recently attracted much attention not only from fundamental interests2–22but also from industrial viewpoints.23–25A number of experimental2–13and theoreti- cal works14–22have been reported. Here, although most of the works have dealt with in-plane magnetized systems2–7 such as NiFe, perpendicularly magnetized systems8–13,19–22 draw increasing attention now. This is mainly because, in theory, the critical current density of DW motion becomesmuch lower in nanowires with perpendicular magnetic aniso-tropy /H20849PMA /H20850than with in-plane magnetic anisotropy. 19,20 Theoretical studies have also indicated that the critical current density of DW motion in PMA nanowires signiﬁ-cantly depends on their width and thickness; this fact origi-nates from a variation in the magnitude of the hard-axis an-isotropy of DW. 19,20,22Tanigawa et al.12reported that, by using Co/Ni nanowires, the critical current density decreasesas the wire width decreases. However, detailed studies on therelation between the critical current density and nanowiredimension, and comparison with theoretical calculation havenot been reported to date. In addition to the above fundamental standpoints, it is of great interest from an aspect of device applications to inves-tigate the dependence of critical current density on the nano-wire dimension. The authors have proposed a magnetic ran-dom access memory /H20849MRAM /H20850with current-induced DW motion for high-speed scalable memory. 25In this device, the value of write current, Ic, was an important criterion to re- place conventional embedded memories, such as embeddedstatic RAM /H20849eSRAM /H20850and embedded dynamic RAM /H20849eDRAM /H20850, and reducing the write current to less than 0.2 mA was crucial. Therefore, investigating the dependence ofthe critical current, I c, on the nanowire dimension is signiﬁ- cant for practical application. In this study, we fabricatedCo/Ni nanowires with different width and thickness, andevaluated their critical currents, I c, and critical current den- sities, jc, of DW motion. Furthermore, the obtained results were compared with micromagnetic simulations. Samples were fabricated through dc-magnetron sputter- ing, KrF lithography, and ion beam etching. The scanningelectron micrograph /H20849SEM /H20850of the element and schematic diagram of the measurement system are shown in Figs. 1/H20849a/H20850 and1/H20849b/H20850, respectively. The magnetic nanowire consisted of three regions: two ﬁxed regions comprised of a free layer/H20849FL/H20850/pinning layer /H20849PL/H20850at each end and one free region comprised of only a FL in the center. The free region wasapproximately rectangular. Both ends of the nanowire wereconnected to a measurement circuit through two viacontacts. The stack structure of the FL and PLwere /H20851Co/H208490.3 nm /H20850/Ni/H208490.6 nm /H20850/H20852 N/H20849=4 or 5 /H20850/Co/H208490.3 nm /H20850and /H20851Pt/H208491.2 nm /H20850/Co/H208490.4 nm /H20850/H208525, respectively. Typical magnetiza- tion curves of the Co/Ni FL are shown in Figs. 1/H20849c/H20850and1/H20849d/H20850. Its saturation magnetization, Ms, and PMA constant, Ku, were 710 emu /cm3and 4.9 /H11003106erg /cm3, respectively. Also, its coercive ﬁeld, which corresponds to a DW motion a/H20850Electronic mail: s-fukami@bu.jp.nec.com. FIG. 1. /H20849Color online /H20850Structure and magnetic property of the sample. /H20849a/H20850 Plan-view SEM image of the sample. /H20849b/H20850Cross-sectional diagram of the sample with the measurement circuit used in this study. Out-of-plane /H20849c/H20850and in-plane /H20849d/H20850magnetization curves of the Co/Ni continuous ﬁlm.APPLIED PHYSICS LETTERS 95, 232504 /H208492009 /H20850 0003-6951/2009/95 /H2084923/H20850/232504/3/$25.00 © 2009 American Institute of Physics 95, 232504-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 147.143.2.5 On: Sun, 21 Dec 2014 11:23:09ﬁeld originated from an intrinsic pinning of Co/Ni ﬁlm, was 150 Oe. We formed the widths of the free region into 60–220nm by using a double exposure and slimming technique with280–400 nm photomask patterns. DW motion characteristics induced by current pulses were evaluated by measuring DW resistance. 13The measure- ment sequence consisted of two steps: /H208491/H20850initializing and /H208492/H20850 applying current pulses. In the initializing step, a strong posi-tive ﬁeld followed by a weak negative ﬁeld was applied,and two DWs were formed at both ends of the free region. Inthis stage, the resistance of the nanowire, R tot, became R0 +2RDW, where R0is base resistance of the nanowire and RDW is the resistance of the DW. After the injection of DWs, current pulses of 100 ns duration were applied. If the currentdensity of the pulse was higher than the critical current den-sity, one of the two DWs was moved and was annihilatedwith the other DW, accordingly resulting in R tot=R0. The DW motion characteristics were evaluated by repeating theabove two steps with sequential measurement of the nanow-ire’s resistance. In addition, we found that the magnetic ﬁeldrequired to depin these DWs was 150–200 Oe, and this de-pinning ﬁeld was not dependent on the thickness of the Co/PtPLs. The fact that the depinning ﬁeld in the nanowire iscomparable to the DW motion ﬁeld of continuous ﬁlm sug-gests that a dominant mechanism of DW-pinning in the fab-ricated nanowire is an intrinsic pinning of the ﬁlm. The de-tailed measurement scheme and the validity of this methodwere veriﬁed in our previous study. 13 Figure 2shows the measured critical current, Ic,a sa function of nanowire width. It is clear that the critical currentdecreased as the wire width decreased for both samples withdifferent thicknesses. In particular, the critical current wasless than 0.2 mA at widths of less than 100 nm. This indi-cates that MRAM with DW motion can replace the conven-tional embedded memories. In addition, the dependence ofthe critical current on the wire width appears to have anx-intercept in the graph. This suggests that the critical current density, j c, decreased with the decrease of wire width. To analyze the relation between the critical current density and nanowire dimension in more detail, wetranslated the unit of the vertical axis of Fig. 2into critical current density, j c/H20849A/cm2/H20850, as shown in Fig. 3/H20849closed squares /H20850. The critical current density clearly decreased as the wire width decreased in the whole range. The jcis about1/H11003108A/cm2when w/H11011200 nm and much less than 5/H11003107A/cm2when w/H11021100 nm. Next, we performed a micromagnetic simulation and compared the experimental results with the calculation. Weused the Landau–Lifshitz–Gilbert equation with spin-transfertorque terms 15,16in the simulation. Here, we assumed, in- stead of the Co/Ni multilayer, a single magnetic materialwith the following magnetic parameters: saturation magneti-zation, M s, magnetic anisotropy constant, Ku, exchange con- stant, A, damping constant, /H9251, and nonadiabatic constant, /H9252, were 710 emu /cm3, 4.9/H11003106erg /cm3, 1.0/H1100310−6erg /cm, 0.02, and 0, respectively. Here, we have conﬁrmed that /H9251and /H9252does not affect on the simulation results. This fact is con- sistent with a theory based on adiabatic spin-transfer torquemodel. 20,21Also, we assumed the spin polarization, P,t ob e 0.6 because the calculation results of DW motion velocityusing this value agree well with experimental results forCo/Ni wire. 26In this study, we carried out the simulation on two different patterns: perfect wire, which had no defects,and anisotropy-distribution wire, in which the magnitude ofthe magnetic anisotropy was distributed locally. 27Here, the degree of anisotropy-distribution was tuned to reproduce theintrinsic pinning ﬁeld of 150 Oe. The obtained calculation results are also plotted in Fig. 3 /H20849open symbols /H20850. The approximate magnitude of the critical current density and the tendency of a decrease with a de-crease of wire width are the same for both the calculationand experiment. In particular, the simulation usinganisotropy-distribution wire agrees well with the experiment.However, investigating carefully, we saw a difference in re-lation between j candw. For example, in the calculation, the variation in jcis relatively small when w/H11022100 nm; on the other hand, in the experiment, the dependence of jconwis strong even when w/H11022100 nm. The dependence of jcon the thickness, t, is also different. The calculated critical current densities for t=4.8 nm /H20851Fig.3/H20849a/H20850/H20852are about 20% higher than that for t=3.9 nm /H20851Fig.3/H20849b/H20850/H20852, whereas the measured critical current densities are almost the same between t=4.8 and 3.9 nm. We fabricated similar samples with t=3.0 nm /H20849N=3 /H20850, but the measured critical current densities were almost the same as the values shown in Figs. 3/H20849a/H20850and3/H20849b/H20850. For ap- proaching these discrepancies between experiments and cal-culations, we have to develop more accurate models, inwhich the ﬁne structure of samples and experimental condi- FIG. 2. /H20849Color online /H20850Critical current of DW motion with /H20851Co /Ni/H20852Nwire width where N=5 /H20849/H17039/H20850and N=4 /H20849/H17009/H20850. FIG. 3. /H20849Color online /H20850Experimental and simulation results of critical cur- rent density as a function of wire width for N=5 /H20849a/H20850and N=4 /H20849b/H20850.I nt h e micromagnetic simulation, we used two kinds of pattern: perfect wire /H20849/H17005/H20850 and anisotropy-distribution wire /H20849/H12331/H20850.232504-2 Fukami et al. Appl. Phys. Lett. 95, 232504 /H208492009 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 147.143.2.5 On: Sun, 21 Dec 2014 11:23:09tions are included more precisely, which will be addressed in the future. Finally, we discuss the validity of the experimental re- sults and the simulation model. First, the effect of thermalassist originated from Joule heating may be negligibly smallbecause temperature increase in the measurement, which wasestimated by monitoring the resistance of nanowire, was assmall as less than 50 K. Next, the driving force of DW-motion may not be /H9252-term but adiabatic spin-transfer torque because measured critical current did not depend on the ex-ternal magnetic ﬁeld. Therefore, we can consider that theorigin of the observed DW motion is a pure adiabatic spin-transfer torque and the micromagnetic simulation model isvalid. In conclusion, we fabricated Co/Ni nanowires with dif- ferent widths and thicknesses and evaluated their critical cur-rents, I c, and critical current densities, jc. We conﬁrmed that theIcbecame less than 0.2 mA when w/H11021100 nm, suggest- ing that MRAM with DW motion can replace conventionalembedded memories. Also, in agreement with theory, thecritical current density clearly decreased as the wire widthdecreased and became much less than 5 /H1100310 7A/cm2when w/H11021100 nm. Furthermore, these experimental results agreed well with a micromagnetic simulation, although a few dis-crepancies were found. The authors would like to thank Professor T. Ono and Dr. D. Chiba of Kyoto University for thoughtful discussion.A portion of this work was supported by New Energy andIndustrial Technology Development Organization Spintron-ics nonvolatile project. 1L. Berger, J. Appl. Phys. 55,1 9 5 4 /H208491984 /H20850. 2J. Grollier, P. Boulenc, V. Cros, A. Hamzi ć, A. Vaurès, A. Fert, and G. Faini, Appl. Phys. Lett. 83, 509 /H208492003 /H20850. 3N. Vernier, D. A. Allwood, D. Atkinson, M. D. Cooke, and R. P. Cowburn, Europhys. Lett. 65,5 2 6 /H208492004 /H20850. 4A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 /H208492004 /H20850. 5M. Kläui, P.-O. Jubert, R. Allenspach, A. Bischof, J. A. C. Bland, G. Faini, U. Rüdiger, C. A. F. Vaz, L. Vila, and C. Vouille, Phys. Rev. Lett. 95,026601 /H208492005 /H20850. 6M. Hayashi, L. Thomas, Ya. B. Bazaliy, C. Rettner, R. Moriya, X. Jiang, and S. S. P. Parkin, Phys. Rev. Lett. 96, 197207 /H208492006 /H20850. 7L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, and S. S. P. Parkin, Nature /H20849London /H20850443, 197 /H208492006 /H20850. 8M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno, Nature /H20849London /H20850 428, 539 /H208492004 /H20850. 9D. Ravelosona, S. Mangin, J. A. Katine, E. E. Fullerton, and B. D. Terris, Appl. Phys. Lett. 90, 072508 /H208492007 /H20850. 10T. Koyama, G. Yamada, H. Tanigawa, S. Kasai, N. Ohshima, S. Fukami, N. Ishiwata, Y. Nakatani, and T. Ono, Appl. Phys. Express 1, 101303 /H208492008 /H20850. 11T. A. Moore, I. M. Miron, G. Gaudin, G. Serret, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and M. Bonﬁm, Appl. Phys. Lett. 93, 262504 /H208492008 /H20850. 12H. Tanigawa, T. Koyama, G. Yamada, D. Chiba, S. Kasai, S. Fukami, T. Suzuki, N. Ohshima, N. Ishiwata, Y. Nakatani, and T. Ono, Appl. Phys. Express 2, 053002 /H208492009 /H20850. 13T. Suzuki, S. Fukami, K. Nagahara, N. Ohshima, and N. Ishiwata, IEEE Trans. Magn. 45, 3776 /H208492009 /H20850. 14G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 /H208492004 /H20850. 15Z. Li and S. Zhang, Phys. Rev. Lett. 92, 207203 /H208492004 /H20850. 16A. Thiaville, Y. Nakatani, J. Miltat, and N. Vernier, J. Appl. Phys. 95, 7049 /H208492004 /H20850. 17S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 /H208492005 /H20850. 18A. Mougin, M. Cormier, J. P. Adam, P. J. Metaxas, and J. Ferré, Europhys. Lett. 78, 57007 /H208492007 /H20850. 19S. Fukami, T. Suzuki, N. Ohshima, K. Nagahara, and N. Ishiwata, J. Appl. Phys. 103, 07E718 /H208492008 /H20850. 20S.-W. Jung, W. Kim, T.-D. Lee, K.-J. Lee, and H.-W. Lee, Appl. Phys. Lett. 92, 202508 /H208492008 /H20850. 21T. Suzuki, S. Fukami, N. Ohshima, K. Nagahara, and N. Ishiwata, J. Appl. Phys. 103, 113913 /H208492008 /H20850. 22S. Fukami, T. Suzuki, N. Ohshima, K. Nagahara, and N. Ishiwata, IEEE Trans. Magn. 44, 2539 /H208492008 /H20850. 23S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,1 9 0 /H208492008 /H20850. 24D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, Science 309, 1688 /H208492005 /H20850. 25S. Fukami, T. Suzuki, K. Nagahara, N. Ohshima, Y. Ozaki, S. Saito, R. Nebashi, N. Sakimura, H. Honjo, K. Mori, C. Igarashi, S. Miura, N. Ishi-wata, and T. Sugibayashi, Dig. Tech. Pap. - Symp. VLSI Technol. 2009 , 230. 26T. Koyama, D. Chiba, G. Yamada, K. Ueda, H. Tanigawa, S. Fukami, T.Suzuki, N. Ohshima, N. Ishiwata, Y. Nakatani, and T. Ono /H20849to be submit- ted/H20850. 27Y. Nakatani and N. Hayashi, J. Magn. Soc. Jpn. 25, 252 /H208492001 /H20850.232504-3 Fukami et al. Appl. Phys. Lett. 95, 232504 /H208492009 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 147.143.2.5 On: Sun, 21 Dec 2014 11:23:09
1.4896951.pdf	The peculiarities of magnetization reversal process in magnetic nanotube with helical anisotropy N. A. Usov and O. N. Serebryakova Citation: Journal of Applied Physics 116, 133902 (2014); doi: 10.1063/1.4896951 View online: http://dx.doi.org/10.1063/1.4896951 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Tuning the magnetization reversal process of FeCoCu nanowire arrays by thermal annealing J. Appl. Phys. 114, 043908 (2013); 10.1063/1.4816479 Giant magneto-impedance effect in amorphous ferromagnetic wire with a weak helical anisotropy: Theory and experiment J. Appl. Phys. 113, 243902 (2013); 10.1063/1.4812278 Magnetization configurations and reversal of magnetic nanotubes with opposite chiralities of the end domains J. Appl. Phys. 109, 073923 (2011); 10.1063/1.3562190 Magnetization configurations and reversal of thin magnetic nanotubes with uniaxial anisotropy J. Appl. Phys. 108, 083920 (2010); 10.1063/1.3488630 Effects of vortex chirality and shape anisotropy on magnetization reversal of Co nanorings (invited) J. Appl. Phys. 107, 09D307 (2010); 10.1063/1.3358233 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.156.157.31 On: Tue, 25 Nov 2014 04:10:18The peculiarities of magnetization reversal process in magnetic nanotube with helical anisotropy N. A. Usov and O. N. Serebryakova Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation RAS, 142190 Troitsk, Moscow, Russia (Received 8 July 2014; accepted 21 September 2014; published online 2 October 2014) The magnetization reversal process in a soft magnetic nanotube with a weak helical magnetic anisotropy is studied by means of numerical simulation. The origin of a helical anisotropy is a small off-diagonal correction to the magneto-elastic energy density. The change of the externalmagnetic ﬁeld parallel to the nanotube axis is shown to initiate a magnetic hysteresis associated with the jumps of the circular magnetization component of the nanotube at a critical magnetic ﬁeld H s. For a uniform nanotube, the critical magnetic ﬁeld Hsis investigated as a function of geometri- cal and magnetic parameters of the nanotube. Using 2D micromagnetic simulation, we study the behavior of a nanotube having magnetic defects in its middle part. In this case, the jump of the cir- cular magnetization component starts at the defect. As a result, two bamboo domain walls appearnear the defect and propagate to the nanotube ends. Similar effect may explain the appearance of the bamboo domain walls in a slightly non uniform amorphous ferromagnetic microwire with nega- tive magnetostriction during magnetization reversal process. VC2014 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4896951 ] I. INTRODUCTION It is well known1–4that the magnetic properties of amor- phous glass-coated microwires with typical diameter of the metallic nucleus d¼5–30 lm depend on the sign of the satu- ration magnetostriction constant ksof a ferromagnet. Actually, for a Fe-rich microwire with ks>0, the easy ani- sotropy axis is parallel to the wire axis, whereas for a Co- rich microwire with ks<0, the easy anisotropy axis has azi- muthal direction.4,5However, in a Co-rich microwire with negative magnetostriction, the easy anisotropy axis usuallydeviates slightly from strict azimuthal direction. Therefore, most Co-rich wires show a weak helical anisotropy. It has been shown recently 6,7that an adequate description of heli- cal anisotropy can be obtained on the basis of a model, which takes into account the actual distribution of the residual quenching stresses over the wire cross section. The inﬂuenceof a relatively small off-diagonal component of the wire re- sidual stress tensor leads to the deviation of the easy anisot- ropy axis from azimuthal direction. In our opinion, thisnaturally explains the existence of weak helical anisotropy in most Co-rich microwires. Interestingly, the presence of the off-diagonal correction to the tensor of the residual quenching stress also leads to important peculiarities of the wire magnetization reversal process in a longitudinal external magnetic ﬁeld. In fact, in awire with helical anisotropy two possible directions of rota- tion of the circular component of the unit magnetization vec- tor,a u>0 and au<0, become nonequivalent. As a result, when the longitudinal magnetic ﬁeld changes from large pos- itive to negative values, there is a jump of the circular mag- netization component at some critical magnetic ﬁeld Hs.I n turn, this feature of the magnetization reversal process leads to the characteristic jumps of the off-diagonal component ofthe wire magneto-impedance tensor. The latter effect has been observed6experimentally. It is easy to see that the experimental observation of the jumps of the off-diagonal component of the magneto- impedance tensor of Co-rich microwire is possible only ifthere are no bamboo domain walls separating circularly mag- netized domains in the whole range of variation of the external magnetic ﬁeld from large positive values, up to the instabilityﬁeld H s. In other words, the aucomponent of the unit magnet- ization vector does not change sign as a function of the zcoor- dinate along the wire axis at least at distances comparablewith the length of the receiving pick-up coil. Note that in the experiment 6the length of the receiving pick-up coil was Lc/C25 0.5 mm, i.e., signiﬁcantly larger than the diameter of the me-tallic nucleus of the microwire, d¼10.7lm. If there are bam- boo domain walls of appreciable density in the microwire, the electromotive force proportional to the off-diagonal compo-nent of the magneto-impedance tensor will be close to zero. This is because of the averaging of the electromotive force signal over the pick-up coil length. On the other hand, the formation of the bamboo domain walls is very likely during the jump of the circular compo- nent of the wire magnetization at the critical ﬁeld, H¼H s, especially in the presence of the defects or non uniformities distributed along the microwire length. The bamboo domain walls were apparently observed8in amorphous Co-rich microwires, although the reason for this is still not clear. From a theoretical point of view,9the existence of the bam- boo domain walls in Co-rich microwires is energeticallyunfavorable. The behavior of a uniform microwire with helical ani- sotropy has been theoretically studied in detail 6using one- dimensional micromagnetic calculations. In particular, the critical ﬁeld Hsfor the jump of the circular magnetization 0021-8979/2014/116(13)/133902/8/$30.00 VC2014 AIP Publishing LLC 116, 133902-1JOURNAL OF APPLIED PHYSICS 116, 133902 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.156.157.31 On: Tue, 25 Nov 2014 04:10:18component has been investigated,7depending on the material parameters of the microwire. However, one-dimensional cal- culation cannot reveal the details of the magnetization rever-sal process and describe possible formation of the bamboo domain walls as the result of the instability of the circular magnetization. Unfortunately, it is difﬁcult to carry out two-dimensional micromagnetic simulation for a microwire because of its huge diameter, d/C2410lm. Indeed, in micro- magnetic simulation the size of the numerical cell cannotexceed the exchange length, R ex¼ﬃﬃﬃ ﬃ Cp =Ms, where Cis the exchange constant and Msis the saturation magnetization. For typical values of the wire magnetic parameters, C/C25 10/C06erg/cm, Ms¼500–800 emu/cm3, the exchange length is given by Rex/C2520 nm. Therefore, it is about three orders of magnitude smaller than the diameter of the microwire. To study the role of the bamboo domain walls in the wire magnetization reversal process, we carry out two- dimensional numerical simulation for ferromagnetic nano-tube of a submicron diameter having a weak helical anisot- ropy. This approach helps us reduce signiﬁcantly the amount of computation and investigate the inﬂuence of defects onthe magnetization reversal process of ferromagnetic nano- tube in axially applied magnetic ﬁeld. It is known 4,6that in Co-rich microwire, there is an exchange core in a smallregion of the order of R exnear the wire center, which elimi- nates the singularity in the radial distribution of the wire magnetization. However, the magnetization reversal of theexchange core has almost no effect 6on the processes occur- ring in the bulk of the wire due to its very small volume. Therefore, to obtain a qualitative picture of the magnetiza-tion reversal in a Co-rich microwire, it is sufﬁcient to con- sider only the phenomena occurring in a certain range of radii near its surface. Consequently, the details of magnetiza-tion reversal process can be studied considering the proper- ties of a magnetic tube. The properties of ferromagnetic nanowires and nano- tubes of nearly circular cross section were studied both the- oretically and experimentally in Refs. 10–24, but the nanotubes with helical anisotropy have not been consideredin detail yet. Meanwhile, it has been realized recently 6that strictly circumferential anisotropy is a rear possibility, as it corresponds to the easy anisotropy axis pointing exactly inazimuthal direction. In this paper, using two-dimensional numerical simulation we observe the jump of the circular magnetization component in a uniform magnetic nanotubewith helical anisotropy, where the exchange core is com- pletely absent. This fact conﬁrms again that the jump of the circular magnetization component in Co-rich micro-wires is due to the presence of helical anisotropy, but not due to the interaction of the core and the outer shell of the wire. In addition, we investigate the role of magneticdefects in the magnetization reversal process of the nano- tube in applied axial magnetic ﬁeld. It is shown that the presence of magnetic defects distributed along the nanotubeaxis leads to the formation of the bamboo domain walls near the magnetic defects during the magnetization reversal process. This fact may explain the appearance of bamboodomain walls in amorphous Co-rich microwires in a certain range of external magnetic ﬁeld.II. HELICAL MAGNETIC ANISOTROPY For the numerical simulation of the magnetization rever- sal process in magnetic nanotubes with helical anisotropy in this paper, we use the same expression for the magneto-elastic energy density, which was used previously 6to calcu- late the properties of microwires with a weak helical anisotropy wm/C0el¼Ke½~rqqa2 qþ~ruua2 uþ~rzza2 zþ2~ruzauaz/C138:(1) Here, Keis the effective magnetic anisotropy constant, which is assumed to vary within Ke¼104–105erg/cm3,ðaq;au;azÞ are the components of the unit magnetization vector in thecylindrical coordinates with the zaxis along the axis of the nanotube. The reduced diagonal components of the residual stress tensor are of the order of unity, ~r qq;~ruu;~rzz/C241. The reduced off-diagonal stress component is considered as a small correction, j~ruzj/C281. Under the condition ~rqq;~ruu <~rzz,~ruz¼0, the easy anisotropy axis of the nanotube is strictly parallel to the azimuthal direction.4However, for non-zero values of ~ruzthe nanotube has a weak helical ani- sotropy, as the easy anisotropy axis deviates6by a certain angle from strictly azimuthal direction. The magneto-elastic energy density, Eq. (1), can be con- sidered as a model expression that describes magnetic anisot-ropy of a thin magnetic nanotube. This model is rather general. It can describe axial, circumferential, and helical types of magnetic anisotropy of magnetic nanotube on acommon ground. The helical type of magnetic anisotropy can be modeled assuming nonzero value of the off-diagonal term, proportional to the product of a uandazcomponents of the unit magnetization vector. The calculations presented below show that even small off diagonal correction to the magneto-elastic energy density leads to interesting peculiar-ities of the axial magnetization reversal process in a mag- netic nanotube. III. MAGNETIZATION REVERSAL OF A UNIFORM NANOTUBE As we noted in the Introduction, a Co-rich microwire with negative magnetostriction is magnetized in a circulardirection in the bulk. However, near the center of the wire, at small distances of the order of the exchange length, q/C24R ex, the exchange core exists to avoid magnetization singularityatq¼0. Within the exchange core, the a zcomponent of the unit magnetization vector decreases rapidly from unity at q¼0, to zero at q>Rex. In the range of radii q/C29Rex, the exchange interaction is small and its inﬂuence on the proper- ties of a microwire of diameter d¼5–30 lm can be neglected. At the same time, for a magnetic nanotube withan outer radius of the order of hundreds of nanometers the exchange interaction should be taken into account. Let a thin nanotube has outer and inner radii Rand R 1¼R/C0DR, respectively, the tube thickness being DR/C28R. To get the exchange energy of the nanotube in this approxi- mation, one can set au(q)¼sinh(q),az(q)¼cosh(q), where h(q) is the angle between the directions of the z-axis and the unit magnetization vector. For a nanotube of length Lz,133902-2 N. A. Usov and O. N. Serebryakova J. Appl. Phys. 116, 133902 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.156.157.31 On: Tue, 25 Nov 2014 04:10:18neglecting the radial dependence of the tube magnetization, the exchange energy of the nanotube is given by Wexc¼2pLzðR R1qdqC 2dh dq/C18/C19 þsin2h q2"#() /C25pLzCa2 uðR R1dq q¼pLzCa2 ulnR R1: (2) Then, for the exchange energy density of the nanotube one obtains wex¼Wexc V¼pLzCa2 u 2pRDRLzlnR R1/C25C 2R2a2 u; (3) where V¼2pRDRLzis the nanotube volume. Let external uniform magnetic ﬁeld H0is applied along the nanotube axis. Then, the total energy density of a long homogeneous nanotube is given by wtot¼C 2R2a2 uþKe~rqqa2 qþ~ruua2 uþ~rzza2 zþ2~ruzauazhi /C0MsH0az: (4) The necessary condition for a nanotube to be magnetized in the azimuthal direction in the ground state, in the absence of the external magnetic ﬁeld is given by C 2R2<Ke~rzz/C0~ruu ðÞ : (5) For Ke¼5/C2104–105erg/cm3,C¼(0.5–1.0) /C210/C06erg/cm, Eq.(5)holds for the tube radius of the order of hundreds of nanometers, R/C2110/C05cm. It is also assumed that the satura- tion magnetization of the tube is large enough, Ms¼500–800 emu/cm3. Then, the deviation of the unit mag- netization vector in the radial direction is energeticallyunfavorable, a q/C250, due to large demagnetizing factor of a thin nanotube in the radial direction. Consequently, the mag- netostatic energy of a long nanotube can be neglected.Magnetization reversal of a thin nanotube in axial uni- form magnetic ﬁeld occurs by the process of uniform rotation. Minimizing the total energy of the nanotube, Eq. (4), one can obtain the components of the unit magnetization vector au andazas the functions of H0and to determine the instability ﬁeld Hs,w h e nt h e aucomponent changes sign abruptly. Fig. 1 shows the behavior of the components of the unit magnetiza- tion vector for a homogeneous nanotube with magnetic pa- rameters Ke¼5/C2104erg/cm3,C¼0.5/C210/C06erg/cm, and Ms¼500 emu/cm3. The outer radius of the nanotube equals R¼100 nm. The reduced diagonal components of the residual stress tensor are given by ~rqq¼0:9,~ruu¼0:8, and ~rzz¼1:3. The external magnetic ﬁeld decreases from a large positive value, H0¼250 Oe, where the tube is uniformly mag- netized along its axis, to a negative value, H0¼/C0250 Oe. The calculations were performed for different values of the reduced off-diagonal component of the residual stress tensor, ~ruz¼0:02 and 0.1. As Fig. 1shows, the jumps of the circular magnetization component for these cases occur at close values of the external magnetic ﬁeld, Hs/C25/C0 56 Oe, and Hs/C25 /C049 Oe, respectively. Evidently, in a sufﬁciently large positive magnetic ﬁeld the longitudinal component of the unit magnetization vector is positive, az>0. Suppose, for deﬁniteness, that the off- diagonal correction to the residual stress tensor is also posi- tive, ~ruz>0. Then, the off-diagonal term Ke~ruzauaz decreases the total energy of the nanotube, Eq. (4),w h e nt h e circular magnetization component is negative, au<0. Moreover, investigating the behavior of the minima of Eq. (4) one can prove that there exists a critical ﬁeld Hssuch that for H0>Hs, the positive values of aucomponent are unstable. Both signs of the aucomponent are possible only in the range of ﬁelds jH0j<Hs. When the external magnetic ﬁeld decreases up to H0</C0Hs, only positive values of the circular magnet- ization component are stable, au>0, since in this interval of the magnetic ﬁeld the azcomponent is negative, az<0. Consequently, by decreasing the axial magnetic ﬁeld from H0¼250 Oe to a sufﬁciently large negative values, in the criti- cal magnetic ﬁeld H0¼/C0Hstheaucomponent experiences a jump from negative to a positive value. These considerations explain the dependencies az(H0)a n d au(H0) shown in Fig. 1. Putting au¼sinh,az¼cosh, one obtains the total energy of the nanotube, Eq. (4), as a function of the angle h. Fig. 2(a) shows the angular dependence of the reduced total energy of the nanotube wtot(h)/Kefor some characteristic values of the external magnetic ﬁeld. The outer radius of the nanotube and the tube magnetic parameters in Fig. 2are the same as in Fig.1. The value of the off-diagonal component of the resid- ual stress tensor is given by ~ruz¼0:05. For these values of the magnetic parameters, the instability of circular magnetiza- tion component of the nanotube occurs at Hs¼54.2 Oe. Therefore, as curve 1 in Fig. 2(a)shows, in the magnetic ﬁeld H0¼68 Oe >Hs, there is only left minimum hmin1for the function wtot(h). For the values of H0¼46, 0 and /C042 Oe (see curves 2, 3, and 4 in Fig. 2(a)), the total energy wtot(h)h a s two different minima, hmin1andhmin2, because these magnetic ﬁeld values are within the interval jH0j<Hs. Finally, for the magnetic ﬁeld H0¼/C064 Oe </C0Hs(curve 5 in Fig. 2(a)), the total energy of the nanotube has only the right minimum, FIG. 1. The magnetization reversal process in homogeneous nanotube in axial magnetic ﬁeld for different values of the off-diagonal component of the residual stress tensor: (1) ~r/z¼0.02; (2) ~r/z¼0.1. The external mag- netic ﬁeld decreases from H0¼250 Oe to H0¼/C0250 Oe.133902-3 N. A. Usov and O. N. Serebryakova J. Appl. Phys. 116, 133902 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.156.157.31 On: Tue, 25 Nov 2014 04:10:18hmin2. Note that the values of hmin1,2(H0) determine the stable directions of the unit magnetization vector for each value of the external magnetic ﬁeld H0. In Fig. 2(b), the dependencies az(H0)a n d au(H0) are shown with the indication of the corre- sponding minima of the total energy wtot(h). In accordance with the above discussion, the minimum hmin1exists only in the range of ﬁelds /C0Hs<H0,w h i l et h em i n i m u m hmin2exists forH0<Hs. Both minima exist simultaneously within the interval jH0j<Hs. This leads to the dependence of the nano- tube magnetization on the magnetic ﬁeld prehistory, i.e., tothe magnetic hysteresis. The symbols in Fig. 2(b) show the actual evolution of the unit magnetization vector components when external magnetic ﬁeld decreases from H 0¼100 Oe up toH0¼/C0100 Oe. One can see clearly that the jump of the cir- cular magnetization component at H0/C25/C054 Oe is associated with the disappearance of the minimum hmin1atH0</C0Hs. The evolution of the total energy minima shown in Fig. 2(a)is similar to the behavior of the total energy minima of a single-domain magnetic nanoparticle in external uniformmagnetic ﬁeld. 25Interestingly, however, in the given case the magnetic hysteresis is associated with the jumps of thecircular magnetization component. The latter is initiated by the change of the external magnetic ﬁeld parallel to the nano- tube axis. It is also interesting to note that as Fig. 2(b) shows, in weak magnetic ﬁelds, H0/C250, the inequality azau<0i s possible. Consequently, if the component az>0, the negative value for the circular magnetization component is preferable,a u<0, and vice versa. In Fig. 3, the instability ﬁeld Hsis shown depending on the reduced off-diagonal component of the residual stresstensor for different values of the nanotube effective anisot- ropy constant K e. One can see that the instability ﬁeld shows only a weak dependence on the off-diagonal component ~ruz and is mainly determined by the effective anisotropy con- stant Ke. IV. INFLUENCE OF MAGNETIC DEFECTS The non uniform magnetization reversal process in a magnetic nanotube can be studied only by means of two- dimensional numerical simulation.5,13,26It is assumed that the distribution of the magnetization in a nanotube has a cy-lindrical symmetry, so that the unit magnetization vector in cylindrical coordinates is given by ~a¼ða qðq;zÞ;auðq;zÞ;azðq;zÞÞ: (6) It can be shown that in this case the demagnetizing ﬁeld of the nanotube ~H0arising due to the presence of surface and bulk magnetic charges does not depend on the coordinate u, and moreover, the circular component of the demagnetizingﬁeld is zero, H 0u/C170. For the exchange energy density of a nanotube, instead of Eq. (3), one can use well-known general expression.25,27For the density of the magnetic anisotropy energy in the present two-dimensional micromagnetic simu- lation, we use the same Eq. (1). With these remarks, the components of effective magnetic ﬁeld of a nanotube in cy-lindrical coordinates are as follows: M sHeff;q¼C1 q@ @qq@aq @q/C18/C19 /C0aq q2þ@2aq @z2() /C02Ke~rqqaqþMsH0 q; (7) FIG. 2. (a) The reduced total energy of the nanotube as a function of the angle h¼arctan ða/=azÞfor some characteristic values of axially applied magnetic ﬁeld: (1) H0¼68 Oe, (2) H0¼46 Oe, (3) H0¼0, (4) H0¼/C042 Oe, and (5) H0¼/C064 Oe. (b) The components of the unit magnetization vectors corresponding to different minima of the total energy of the nanotube.Symbols show the actual evolution of the unit magnetization vector compo- nents when external magnetic ﬁeld decreases from H 0¼100 Oe to H0¼/C0100 Oe. FIG. 3. The critical magnetic ﬁeld for instability of the circular magnetiza- tion component of the nanotube as a function of the off-diagonal component of the residual stress tensor.133902-4 N. A. Usov and O. N. Serebryakova J. Appl. Phys. 116, 133902 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.156.157.31 On: Tue, 25 Nov 2014 04:10:18MsHeff;u¼C1 q@ @qq@au @q/C18/C19 /C0au q2þ@2au @z2() /C02Ke~ruuauþ~ruzaz ðÞ ; (8) MsHeff;z¼C1 q@ @qq@az @q/C18/C19 þ@2az @z2() /C02Ke~rzzazþ~ruzau ðÞ þMsH0þH0 z ðÞ :(9) The demagnetizing ﬁeld components H0qandH0zcan be cal- culated using the scalar magnetic potential28 H0 q¼/C0@U @q;H0 z¼/C0@U @z; Uq;zðÞ ¼Msð Vdvaqq1;z1ðÞ@ @q1þazq1;z1ðÞ@ @z1/C26/C271 j~r/C0~r1j: (10) Note that the aucomponent does not contribute to the mag- netic potential (10) due to assumed axial symmetry of the magnetization distribution, Eq. (6). To determine stationary magnetization distribution in a nanotube, we solve the dynamic Landau-Lifshitz-Gilbert equation13 @~a @t¼/C0c~a;~Hefhi þj~a;@~a @t/C20/C21 ; (11) where cis the gyromagnetic ratio, and jis the phenomeno- logical damping constant. The unit magnetization vector on the external and internal surfaces of the nanotube satisﬁes the boundary condition, @~a=@n¼0, where ~nis a unit vector normal to the surface of the nanotube. The external magnetic ﬁeld is changed in a small increment, dH0¼1 Oe, ranging from H0¼200 Oe up to H0¼/C0200 Oe. For a given value of the external magnetic ﬁeld, the temporal evolution of the magnetization distribution in a nanotube is calculated according to Eq. (11) until this distribution approaches close enough to a stationary state that satisﬁes the equilibrium equation ½~a;~Hef/C138¼0. To calculate numerically the effective magnetic ﬁeld components (7)–(9), a sufﬁciently long nanotube is subdi- vided into a set of Nq/C2Nztoroidal numerical elements. The dimensions of the numerical cells, dz¼dq¼2.5 nm, are cho- sen sufﬁciently small with respect to the exchange length, Rex/C2520 nm. This ensures the accuracy of the numerical cal- culations performed. Magnetostatic interactions as well asself-energies of the torus-shaped elements are calculated pre- liminarily for a given array in a manner similar to the usual case of small cubic elements. 29The calculations of two- dimensional magnetization distributions in nanotubes were carried out on two-dimensional grids with Nq¼40–60 cells along the radius and Nz¼1600–6000 cells along the nano- tube axis. For the sizes of the toroidal numerical cells men- tioned, the outer nanotube radius varies between R¼100–150 nm, the nanotube length being Lz¼4–15 lm. The thickness of the thin nanotube is given by DR¼10–20 nm.First, we calculated the magnetization reversal process in a homogeneous nanotube in a longitudinal external mag- netic ﬁeld. For the case of homogeneous nanotube, theresults of two-dimensional calculation practically coincide with that of one-dimensional one shown in Fig. 1. The jump of the circular magnetization component in a uniform nano-tube occurs uniformly over its entire length in a ﬁxed mag- netic ﬁeld H s. The only difference in the 1D and 2D calculations is that in the two-dimensional simulation the in-homogeneous transition regions appear near the nanotube ends under the inﬂuence of strong demagnetizing ﬁeld. The characteristic size of these transition regions is of the orderof the nanotube diameter. Therefore, for a sufﬁciently long nanotube the non uniform magnetization distributions formed near the nanotube ends do not make a signiﬁcantcontribution to the total magnetic moment of the nanotube. Then, using 2D modeling we investigated the magnet- ization reversal process in magnetic nanotube with an iso-lated defect. The magnetic defect is modeled assuming that the small off-diagonal component of the residual stress ten- sor in some region z 1<z<z2inside the nanotube has the value ~rð1Þ uz, different from its value ~rð0Þ uzin the main part of the nanotube. Due to the presence of the defect, the jump of the circular magnetization component occurs nonuniformlyalong the nanotube length, as the instability ﬁeld H shas dif- ferent values in the defect, and in the main part of the nanotube. Figs. 4–6show the magnetization reversal process in the magnetic nanotube with the isolated defect located near the nanotube center. The magnetic parameters of the nanotubeare given by K e¼5/C2104erg/cm3,C¼0.5/C210/C06erg/cm, and Ms¼500 emu/cm3. The outer tube radius is equal to R¼100 nm, the thickness of the nanotube is in the range DR¼10–20 nm, the nanotube length Lz¼15 000 nm. The reduced diagonal components of the residual stress tensor are assumed to be ~rqq¼0:9,~ruu¼0:8, and ~rzz¼1:3. The tube is subdivided into Nz¼6000 numerical cells along the nanotube axis. Also, there are 4 or 8 numerical cells along the nanotube radius for the tube thickness of 10 and 20 nm,respectively. The magnetic defect is located in the range of 5000 <z<7500 nm. In the defect area, the reduced value of the off-diagonal component of the residual stress tensor isassumed to be ~r ð1Þ uz¼0:1, whereas it is given by ~rð0Þ uz¼0:02 in the main part of the nanotube. Because of the small thick- ness of the nanotube, the radial dependence of the tube mag-netization is virtually absent. Also, due to the small thickness of the nanotube, the radial component of the nano- tube unit magnetization vector was close to zero, a q/C250. This component is not shown in Figs. 4–6. For clarity, in Figs. 4–6the arrangement of the isolated magnetic defect is marked by vertical lines. As Fig. 4(a)shows, when axial magnetic ﬁeld decreases from H0¼100 Oe to zero, appreciable deviations of the unit magnetization vector components exist within and near themagnetic defect. Nevertheless, the circular magnetization component remains negative, because under the conditions a z>0, and ~ruz>0, the total nanotube energy has a mini- mum at au<0. In Fig. 4(a), one can see also narrow regions near the ends of the nanotube. Here, the demagnetizing ﬁeld133902-5 N. A. Usov and O. N. Serebryakova J. Appl. Phys. 116, 133902 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.156.157.31 On: Tue, 25 Nov 2014 04:10:18has appreciable value and the magnetization distribution is nonuniform. Of course, the demagnetizing ﬁeld is also sig- niﬁcantly involved in the formation of an inhomogeneousmagnetization distribution near the magnetic defect. The magnetization distribution in the nanotube at negative values of the external magnetic ﬁeld, but larger than the instabilityﬁeld H s, is shown in Fig. 4(b). One can see that in negative magnetic ﬁelds the azcomponent changes the sign and becomes negative, az<0. However, aucomponent in the interval of moderate magnetic ﬁelds, H0>/C034 Oe, is stable and remains negative. As Fig. 5shows, a further reduction of the magnetic ﬁeld to H0¼/C035 Oe leads to instability of the nanotube magnetization. The jump of aucomponent arises near the magnetic defect. The transient pictures in Fig. 5(a)show that near the magnetic defect the aucomponent changes the sign from negative to positive. Simultaneously, at the edges of the defect the bamboo domain walls appear. Then, the mag-netization distribution gradually spreads out to the nanotube ends. During the jump of the a ucomponent, the azcompo- nent is also greatly perturbed, but it does not change sign. Note that Fig. 5shows the transition process occurring at the instability ﬁeld Hs¼/C035 Oe. The stationary state of thenanotube just after the magnetization jump is shown in Fig. 6 in a magnetic ﬁeld H0¼/C035.1 Oe. Fig. 6shows also further evolution of the nanotube magnetization distribution. When axial magnetic ﬁeld decreases up to H0¼/C0100 Oe, the longi- tudinal magnetization component gradually approaches thevalue a z/C25/C0 1, while aucomponent approaches to zero. FIG. 5. Evolution of (a) circular and (b) longitudinal magnetization compo- nents in the nanotube with isolated magnetic defect at the instability ﬁeld,H s¼/C035 Oe. FIG. 6. The behavior of the unit magnetization vector components in the nanotube with isolated magnetic defect in the range of axial magnetic ﬁelds H0</C0Hs: (1) H0¼/C035.1 Oe; (2) H0¼/C050 Oe; (3) H0¼/C0100 Oe. FIG. 4. The components of the unit magnetization vector in the magnetic nanotube with isolated defect, (a) in the range of positive magnetic ﬁelds:(1)H 0¼100 Oe; (2) H0¼50 Oe; (3) H0¼0; and (b) in the negative mag- netic ﬁelds before the jump of the aucomponent: (1) H0¼/C020 Oe; (2) H0¼/C030 Oe; and (3) H0¼/C034 Oe.133902-6 N. A. Usov and O. N. Serebryakova J. Appl. Phys. 116, 133902 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.156.157.31 On: Tue, 25 Nov 2014 04:10:18Within the defect and at the nanotube edges, there are still sig- niﬁcant perturbations of the nanotube magnetization. Fig. 7shows the transient jump of the circular magnet- ization component in the nanotube with two isolated mag- netic defects. Magnetic and geometric parameters of the nanotube are the same as in Figs. 4–6, but in this case there are two magnetic defect, located in the areas 3000 <z<5000 nm and 10 000 <z<12 000 nm, respec- tively. For the ﬁrst defect, the reduced value of the off-diagonal component of the residual stress tensor is assumed to be ~r ð1Þ uz¼0:04, for the second defect ~rð2Þ uz¼0:1. In the main area of the nanotube, ~rð0Þ uz¼0:02. For this nanotube, the jump of the circular magnetization component begins on the second, stronger magnetic defect, in the same magnetic ﬁeld H0¼/C035 Oe. During the jump, the bamboo domain walls originate on both defects, but the ﬁnal steady state of the nanotube is virtually the same as that for the nanotube with a single defect. The appearance of the bamboo domain walls at the defect boundaries explains probably a reason why the bam- boo domain walls can be observed in experiment8with amorphous Co-rich microwires. However, in this numerical simulation the arising bamboo domain walls propagate to the nanotube ends, where they merge with the non-uniformmagnetization distributions existing at the nanotube ends. This behavior of the bamboo domain walls is consistent with a theoretical conclusion 9that the bamboo domain walls are energetically unfavorable in a perfect cylindrical microwire. However, in the experiment8the stationary bamboo domain walls are observed in a certain range ofexternal magnetic ﬁelds. It seems reasonable to assume that after nucleation, the bamboo domain walls may stationary exist in a range of external magnetic ﬁelds in a wire havingmagnetic defects that can impede the free domain wall movement. V. CONCLUSION In this paper, it is proved that in a magnetic nanotube with a weak helical anisotropy in applied magnetic ﬁeldtwo possible directions of rotation of the circular magnet- ization component are not equivalent, because they corre- spond to different total energies of the nanotube. As aconsequence, during the magnetization reversal of a homo- geneous nanotube in axially applied external magnetic ﬁeld, the jump of the circular magnetization component ofthe nanotube occurs at the critical ﬁeld H 0¼Hs. The insta- bility ﬁeld Hsis studied in this paper for a homogeneous nanotube depending on the effective magnetic anisotropyconstant and for small values of the off-diagonal compo- nent of the residual stress tensor. By means of two-dimensional numerical simulation, we investigate also the magnetization reversal process in a non uniform nanotube, where a small off-diagonal component of the residual stress tensor varies as a function of the coordi-nate zalong the nanotube axis. It is shown that the presence of magnetic defects leads to a smearing of the instability ﬁeld H s, in accordance with experimental observations.6,7At the instability ﬁeld, the bamboo domain walls arise in the vi- cinity of the localized defect boundaries. In our calculations, the bamboo domain walls propagate to the nanotube ends,where they merge with the ending magnetic structures. It seems reasonable to assume, however, that in a sufﬁciently long nanotube the bamboo domain walls may be stuck on theneighboring defects of a different nature, and exist perma- nently in a certain region of the external magnetic ﬁeld. 1M. Vazquez and A. Zhukov, J. Magn. Magn. Mater. 160, 223 (1996). 2A. Zhukov, M. Vazquez, J. Velazquez, A. Hernando, and V. Larin, J. Magn. Magn. Mater. 170, 323 (1997). 3V. Zhukova, A. Chizhik, A. Zhukov, A. Torcunov, V. Larin, and J. Gonzalez, IEEE Trans. Magn. 38, 3090 (2002). 4A. S. Antonov, V. T. Borisov, O. V. Borisov, A. F. Prokoshin, and N. A. Usov, J. Phys. D: Appl. Phys. 33, 1161 (2000). 5N. A. Usov, J. Magn. Magn. Mater. 249, 3 (2002). 6N. A. Usov and S. A. Gudoshnikov, J. Appl. Phys. 113, 243902 (2013). 7N. A. Usov and S. A. Gudoshnikov, Phys. Status Solidi A 211, 1055 (2014). 8M. Ipatov, A. Chizhik, V. Zhukova, J. Gonzalez, and A. Zhukov, J. Appl. Phys. 109, 113924 (2011). 9N. Usov, A. Antonov, A. Dykhne, and A. Lagar’kov, J. Magn. Magn. Mater. 174, 127 (1997). 10Y. C. Sui, R. Skomski, K. D. Sorge, and D. J. Sellmyer, Appl. Phys. Lett. 84, 1525 (2004). 11W. C. Yoo and J. K. Lee, Adv. Mater. 16, 1097 (2004). 12K. Nielsch, F. J. Casta ~no, C. A. Ross, and R. Krishnan, J. Appl. Phys. 98, 034318 (2005). 13N. A. Usov, A. Zhukov, and J. Gonzalez, J. Magn. Magn. Mater. 316, 255 (2007). 14F. S. Li, D. Zhou, T. Wang, Y. Wang, L. J. Song, and C. T. Xu, J. Appl. Phys. 101, 014309 (2007). 15P. Landeros, S. Allende, J. Escrig, E. Salcedo, D. Altbir, and E. E. Vogel, Appl. Phys. Lett. 90, 102501 (2007). 16M. Daub, M. Knez, U. Goesele, and K. Nielsch, J. Appl. Phys. 101, 09J111 (2007). 17N. A. Usov, A. P. Chen, A. Zhukov, and J. Gonzalez, J. Appl. Phys. 104, 083902 (2008). 18I. Betancourt, G. Hrkac, and T. Schreﬂ, J. Appl. Phys. 104, 023915 (2008). 19P. Landeros, O. J. Suarez, A. Cuchillo, and P. Vargas, Phys. Rev. B 79, 024404 (2009). 20O. Albrecht, R. Zierold, S. Allende, J. Escrig, C. Patzig, B. Rauschenbach, K. Nielsch, and D. Gorlitz, J. Appl. Phys. 109, 093910 (2011). 21N. A. Usov, A. Zhukov, and J. Gonzalez, Phys. Status Solidi A 208, 535 (2011). 22M. P. Proenca, J. Ventura, C. T. Sousa, M. Vazquez, and J. P. Araujo,Phys. Rev. B 87, 134404 (2013). FIG. 7. Transient magnetization process for circular magnetization compo- nent in the nanotube with two isolated magnetic defects at the instability ﬁeld, Hs¼/C035 Oe.133902-7 N. A. Usov and O. N. Serebryakova J. Appl. Phys. 116, 133902 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.156.157.31 On: Tue, 25 Nov 2014 04:10:1823T. B €ohnert, V. Vega, A. K. Michel, V. M. Prida, and K. Nielsch, Appl. Phys. Lett. 103, 092407 (2013). 24C. Bran, E. M. Palmero, R. P. del Real, and M. Vazquez, Phys. Status Solidi A 211, 1076 (2014). 25A. Aharoni, Introduction to the Theory of Ferromagnetism (Clarendon Press, Oxford, 1996).26N. A. Usov, A. Zhukov, and J. Gonzalez, IEEE Trans. Magn. 42, 3063 (2006). 27W. F. Brown, Jr., Micromagnetics (Wiley-Interscience, New York, 1963). 28A. I. Akhiezer, V. G. Bar’yakhtar, and S. V. Peletminskii, Spin Waves (North-Holland, Amsterdam, 1968). 29M. E. Schabes, J. Magn. Magn. Mater. 95, 249 (1991).133902-8 N. A. Usov and O. N. Serebryakova J. Appl. Phys. 116, 133902 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.156.157.31 On: Tue, 25 Nov 2014 04:10:18
1.1626123.pdf	Transient energy growth for the Lamb–Oseen vortex Arnaud Antkowiaka)and Pierre Brancher Institut de Me ´canique des Fluides de Toulouse, Alle ´e du Professeur Camille Soula, 31400 Toulouse, France ~Received 14 May 2003; accepted 22 September 2003; published online 13 November 2003 ! The transient evolution of inﬁnitesimal ﬂow disturbances which optimally induce algebraic growth in the Lamb–Oseen ~Gaussian !vortex is studied using a direct-adjoint technique. This optimal perturbationanalysisrevealsthattheLamb–Oseenvortexallowsforintenseampliﬁcationofkineticenergyfortwo-dimensionalandthree-dimensionalperturbationsofazimuthalwavenumber m51.In both cases, the disturbances experiencing the most growth initially take the form of concentratedspirals at the outer periphery of the vortex which rapidly excite bending waves within the vortexcore.Inthelimitoflargewavelengths,theoptimalperturbationleadstoarbitrarilylargegrowthsviaan original scenario combining the Orr mechanism with vortex induction. © 2004 American Institute of Physics. @DOI: 10.1063/1.1626123 # The stability properties of vortices have received consid- erable attention in recent years partly because of a renewedinterest in the dynamics of trailing vortices behind aircrafts.More speciﬁcally, the strong and persistent counter-rotatingvortex pair generated at the trailing edge of airplane wingsrepresent a potential hazard to forthcoming planes thus lim-iting take-off and landing cadences in airports. It has beenshown in the last decades that these vortices are unstable tolong- 1and short-wave instabilities2due to the underlying strain ﬁeld induced by the companion vortex. Moreover, thepresence of an axial ﬂow is at the origin of other instabilitymechanisms. 3 By contrast, an isolated vortex with no axial ﬂow and monotonically decreasing positive vorticity, hereafter calledan axisymmetric monopole, is linearly stable with respect totwo-dimensional ~2D!and three-dimensional ~3D!perturba- tions ~see, for instance, the temporal stability analysis of Fabre and Jacquin 4!. In particular, it is stable with regard to both the centrifugal and inﬂection-point Rayleigh criteria.Stability analyses of this kind of vortex generally focus on2D perturbations. In the inviscid case, a deformed vortexrelaxes toward an axisymmetric state after an exponential~Landau !damping followed by algebraic decay at long times of the initial asymmetric perturbations. 5,6At large but ﬁnite Reynolds numbers, asymmetric perturbations asymptoticallydecay on a Re 1/3time scale via a shear–diffusion mechanism.7,8 Interesting algebraic evolution of 2D disturbances has also been reported in the case of inviscid hollow hurricane-like vortices: 9,10long time asymptotics has revealed the pos- sibility for linear growth of the perturbation kinetic energyeven if the ﬂow is exponentially stable. But this mechanism is only active under the necessary condition that the basicﬂow angular velocity has a local maximum other than at thevortex axis, which is not the case for the axisymmetricmonopole. Yet a generalized stability analysis of monopolarvortices maintained by radial inﬂow has also revealed tran- sient growth for 2D spiral-shaped perturbations. 11Moreover, the same authors have found that the linear response of theseﬂows to random forcing involved a similar spiral-shapeddominant structure. 12Finally, recent theoretical studies13 have suggested that interactions between a vortex and 3Dexternal turbulence could excite bending waves, via a domi-nant linear process that may eventually destroy the vortexafter about 10 rotation times in the nonlinear regime. In that context our objective in this Letter is to present preliminary results revealing the potential for intense tran-sient ampliﬁcation of kinetic energy for speciﬁc perturba-tions ~optimal perturbation !in the linear regime. It is argued that this transient growth could eventually trigger a nonlineartransition in an otherwise linearly stable vortex. The present work analyzes the temporal evolution of in- ﬁnitesimal 3D perturbations with velocity components in cy-lindrical coordinates u(r, u,z,t)5(ur,uu,uz)Tin a steady in- compressible axisymmetric vortex ﬂow U(r)5(0,rV,0)T. The basic ﬂow under consideration here is the Lamb–Oseenmodel, with angular velocity V(r)5 @12exp(2r2)#/r2and associated axial vorticity Z(r)52exp( 2r2). Here space and time have been respectively nondimensionalized by the vor-tex radius r 0and the ~maximum !angular velocity at the axis V0. The Reynolds number based on these characteristic scales is Re 5V0r02/n, where ndenotes the kinematic viscos- ity. Linearizing the Navier–Stokes equations around this ba-sic ﬂow, it is possible to eliminate the perturbation pressure a!Electronic mail: antko@imft.frPHYSICS OF FLUIDS VOLUME 16, NUMBER 1 JANUARY 2004 LETTERS The purpose of this Letters section is to provide rapid dissemination of important new results in the ﬁelds regularly covered by Physics of Fluids. Results of extended research should not be presented as a series of letters in place of comprehensive articles.Letters cannot exceed four printed pages in length, including space allowed for title, ﬁgures, tables, references and an abstractlimitedtoabout100words. Ordinarily,thereisathree-monthtimelimit,fromdateofreceipttoacceptance,forprocessingLetter manuscripts. Authors must also submit a brief statement justifying rapid publication in the Letters section. L1 1070-6631/2004/16(1)/1/4/$22.00 © 2004 American Institute of Physics and axial velocity to get a complete description of the per- turbation in terms of v˜5(ur,uu)T. Then, injecting a classical normal modes decomposition, v˜(r,u,z,t)5v(r,t) 3exp@i(kz1mu)#, wherek~real!andm~integer !are, respec- tively, the axial and azimuthal wavenumbers, yields the fol-lowing system for v, rewritten in compact form: F ~v!5L]v ]t1Cv21 ReDv50, ~1! with the associated boundary conditions that the perturbation is regular at r50 and tends to 0 at inﬁnity. Derivation of ~1! is straightforward.4Dis a viscous diffusion operator and the operatorLresults from the elimination of pressure and axial velocity from the original linearized Navier–Stokes equa-tions. Disturbance and basic ﬂow are coupled through theadvection operator C. Classical linear stability theory focuses on the long time behavior of the normal modes by assuming exponential timedependence of the form v(r,t)5v(r)e 2ivt. The analysis then reduces to an eigenvalue problem for the complex pul-sations v, which are all stable for the Lamb–Oseen vortex.4 Nevertheless, it is noteworthy that the advection operator C is highly non-normal, except in the trivial case k5m50o r in the special case of solid-body rotation. This property, heredue to differential rotation, implies that short time transientampliﬁcation can be anticipated. 14 This conjecture can be addressed by computing the op- timal perturbation, i.e., the initial condition which maximizesthe energy gain G( t)5Et/E0during a ﬁnite time interval @0,t#, where the perturbation energy at time tis given by Et51 2E 0‘ ~u¯rur1u¯uuu1u¯zuz!rdrU t. Here the overbars indicate transpose conjugate quantities. Different techniques can be used to determine the opti- mal initial conditions.15–18The formalism employed in the present work comes from optimal control theory. It has beensuccessfully used to compute the optimal perturbation inswept boundary layers. 19Since we follow closely the proto- col described in Corbett and Bottaro,19we only give a syn- thetic presentation of this approach in the following. The optimization problem lies in maximizing the energy growthG(t)~theobjective !at a given time tunder the con- straintsof respecting ~1!and the associated boundary condi- tions. The initial condition v0is used as a controlto be ad- justed in order to meet the objective. This constrainedoptimization problem can be solved by considering theequivalent unconstrained problem for the Lagrangian func-tional: L ~v,v0,a,c!5G~t!2^F~v!,a&2~H~v,v0!,c!, introducing the adjoint variables a(r,t)5(a,b)Tandc(r) 5(c,d)Twhich play the ro ˆle of Lagrange multipliers. Here H(v,v0)5v(r,0)2v0(r) corresponds to the constraint that the initial condition v(r,0) matches the control v0(r). The inner products appearing in the functional are~p,q!5E 0‘ p¯"qrdr1complex conjugate, ^p,q&5E 0t ~p,q!dt. The task is then to determine v,v0,aandcwhich render L stationary, i.e., corresponding to a local extremum. Setting tozero variations of Lwith respect to these variables yields boundary conditions and the following ~adjoint !system for the variable a: F 1~a!52L]a ]t1C1a21 ReDa50, ~2! whereC1is the adjoint operator of C. It also yields transfer relations between the direct and adjoint variables at times t 50 andt5tas well as the expression of the optimal pertur- bation. The reader is referred to the paper by Corbett andBottaro 19for the details of the derivation. The computation of the optimal perturbation is carried out via the followingiterative algorithm: from an initial guess ~random noise !v 0 the direct system ~1!is integrated to t5t; transfer relations are then applied to provide initial conditions for thebackward-in-time integration of the adjoint system ~2!until t50 thus providing improved initial conditions for the next iteration. In practice this procedure converges within 4 to 6iterations ~i.e.,G( t) varies less than 1022). The spatial treatment of the direct and adjoint systems is based on a pseudospectral Chebyshev method.20The equa- tions are discretized on the Gauss–Lobatto grid algebraicallymapped on the semi-inﬁnite physical domain. 20All compu- tations are done using MATLABand the DMSuite package de- veloped by Weideman and Reddy.21A special trick of the method has been to take advantage of the variables paritythus allowing to reduce the number of collocation points fora given accuracy. 4Convergence tests have been performed FIG. 1. Optimal energy growth and corresponding optimal time ~in rotation periods !versus axial wavenumber.L2 Phys. Fluids, Vol. 16, No. 1, January 2004 A. Antkowiak and P. Brancherby varying the stretching of the mapping and the number of collocation points from 40 to 120 without any dramaticchanges in the results. We next discuss preliminary results obtained for the par- ticular case m51. The evolution of the optimal growth with respect to the axial wavenumber kis reported in Fig. 1, to- gether with the corresponding time toptat which it occurs. It can be seen that considerable growth can be reached, even atmoderate Reynolds numbers. A remarkable feature is thepresence of a relative maximum in energy near k.1.4 inde- pendently of the Reynolds number, indicating some threedimensional core sized mechanism efﬁcient in redirectingenergy from the mean ﬂow to the perturbation. The energyvalue at this peak scales with the Reynolds number. Figure 2shows the optimal disturbance structure corresponding to thismaximum. This perturbation is at t50 composed of a set of spiraling vorticity sheets with a left-handed orientation thatevolve so as to produce a strong bending wave within thevortex core. Due to three-dimensionality, the dynamics ofsuch a perturbation is quite intricate ~stretching and tilting ! and is not yet fully understood. Nevertheless, this dynamicsmight involve an analog of the 3D mechanism analyzed byFarrell and Ioannou. 22These authors present a generalization of the so-called Orr and lift-up mechanisms in plane shearﬂows which could constitute an interesting basis for the de-tailed analysis of the present results. Though stretching and tilting vanish as large wave- lengths are approached, the potential for substantial transientgrowth still exists. More speciﬁcally, the 2D limit exhibits astriking feature: the growth increases linearly 24with terminal timet~Fig. 3 !. Figure 4 depicts the evolution of a typical 2D optimal perturbation. The associated vorticity ﬁeld initiallytakes the form of spirals that tend to thicken and to lie furtherfrom the vortex core as tis increased ~data not shown !. Thisﬁeld satisﬁed the linearized vorticity equation: ~3! where three parts have been underbraced: an advection part which materially advects the vorticity perturbation, an induc-tion part corresponding to redirection of vorticity from themean ﬂow to the disturbance ~both parts coming from the linearization of the advection term in the complete equation ! and a diffusion term. Let us examine how these terms inter-act as time evolves. The initial structure of the optimal per-turbation is a set of vorticity sheets in the form of leading spirals ~by opposition to trailingspirals, as for the advection of a passive scalar spot !. This initial condition is located at the limb of the vortex, where the induction term is negli-gible. As time ﬂows ~middle of Fig. 4 !, the initial leading spirals are advected and unfolded via an analog of the Orrmechanism. This process results in a local reorganization ofthe external perturbation vorticity that promotes vortex in-duction on the vortex axis as the spirals unroll. This originalglobal sequel of the Orr mechanism initiated at the outerperiphery of the vortex thus eventually leads to a contamina-tion of the vortex core by exciting translational ~bending ! modes: quickly, an inner bipolar vortical structure grows, andat larger times most of the kinetic energy is associated withthis ‘‘translation.’’ Maximum growth is reached at terminaltime, before the resulting unblended spirals are stirred backinto trailing spirals. Though the whole process is clearly in-viscid, viscosity plays a ro ˆle in the selection of the initial characteristic radial scale of the optimal disturbance ~the greater the Reynolds number, the thinner the vorticitysheets !. We now present a simple model intended to mimic the combined effects of advection and induction, and to illustratethe initial destructive interference between vorticity spirals.In this model, the evolution of points vortices advected by a1/rﬂow initially organized along spirals is examined, and the resulting induced velocity at the center is evaluated.Starting with two ﬁlaments rolled up in spiral form, the ac-tion of the mean external shear ﬂow ( .1/r) is to materially advect the vorticity and to concentrate the spiral. Figure 5represents the evolution of resulting radial velocity at thecenter, which is a measure of the induction term. Its action isnegligible at initial time, due to destructive interference ofintertwined spirals. But, as time evolves, the spirals become FIG. 2. Isosurfaces of axial vorticity for the optimal 3D case. The levels correspond to 680% of maximum vorticity, at initial time ~left!and optimal time ~right!. FIG. 3. Evolution of growth with terminal time ~in rotation periods !in the 2D case at Re 51000. FIG. 4. Cross section of axial vorticity in the 2D case. The contour plot levels are 660% of maximum absolute vorticity.L3 Phys. Fluids, Vol. 16, No. 1, January 2004 Transient energy growth for the Lamb –Oseen vortexunwound.As a consequence, their action focuses on the cen- ter and redirects vorticity from the mean ﬂow to the distur-bance. The important point of the present Letter is that m51 disturbances injected in a vortex are subject to transient am-pliﬁcation. The physical mechanism feeding the transientgrowth is not restricted to a local Orr mechanism, but in-cludes also a global effect of vortex induction. It is notewor-thy that these two mechanisms are not speciﬁc to the Lamb–Oseen vortex, or even to vortices, but are generic to freeﬂows with the two hydrodynamic ingredients: shear and ro-tation. Nevertheless, several questions remain unanswered.First, in the linear regime, what are the respective roles ofstretching and tilting in the 3D case? Is the peak in Fig. 1 theresult of a resonance phenomenon? Moreover, the nonlinearregime of the optimal perturbation will be investigated viadirect numerical simulations in order to address the rel-evance of a ‘‘bypass’’ 14transition scenario in such a ﬂow. Back to aircraft vortices, the similarity between the result ofoptimal evolution ~a core contamination by external distur- bance leading to a translation !and the long-wave erratic dis- placements of experimental vortices, a phenomenon knownasvortex meandering , 23also encountered in tornado- and hurricane-like ﬂows,11appears puzzling and worthy of fur- ther investigation. Finally, an exhaustive parametric study iscurrently under way in order to investigate other azimuthalwavenumbers and the inﬂuence of base ﬂow diffusion. 25 1S. C. Crow, ‘‘Stability theory for a pair of trailing vortices,’’AIAA J. 8, 2172 ~1970!. 2T. Leweke and C. H. K. Williamson, ‘‘Cooperative elliptic instability of a vortex pair,’’ J. Fluid Mech. 360,8 5~1998!. 3E. W. Mayer and K. G. Powell, ‘‘Viscous and inviscid instabilities of a trailing line vortex,’’ J. Fluid Mech. 245,9 1~1992!.4D. Fabre and L. Jacquin, ’’Viscous instabilities in trailing vortices at large swirl numbers,’’ J. Fluid Mech. ~to be published !. 5R. J. Briggs, J. D. Daugherty, and R. H. Levy, ‘‘Role of Landau damping in cross-ﬁeld electron beams and inviscid shear ﬂow,’’ Phys. Fluids 13, 421~1970!. 6D. A. Schecter, D. H. E. Dubin, A. C. Cass, C. F. Driscoll, I. M. Lansky, and T. M. O’Neill, ‘‘Inviscid damping of asymmetries on a two-dimensional vortex,’’ Phys. Fluids 12, 2397 ~2000!. 7A. J. Bernoff and J. F. Lingevitch, ‘‘Rapid relaxation of an axisymmetric vortex,’’ Phys. Fluids 6, 3717 ~1994!. 8K. Bajer, A. P. Bassom, and A. D. Gilbert, ‘‘Accelerated diffusion in the centre of a vortex,’’ J. Fluid Mech. 437,3 9 5 ~2001!. 9R. A. Smith and M. N. Rosenbluth, ‘‘Algebraic instability of hollow elec- tron columns and clylindrical vortices,’’ Phys. Rev. Lett. 64,6 4 9 ~1990!. 10D. S. Nolan and M. T. Montgomery, ‘‘The algebraic growth of wavenum- ber one disturbances in hurricane-like vortices,’’ J. Atmos. Sci. 57, 3514 ~2000!. 11D. S. Nolan and B. F. Farrell, ‘‘Generalized stability analyses of asymmet- ric disturbances in one- and two-celled vortices maintained by radial in-ﬂow,’’ J. Atmos. Sci. 56, 1282 ~1999!. 12D. S. Nolan and B. F. Farrell, ‘‘The intensiﬁcation of two-dimensional swirling ﬂows by stochastic asymmetric forcing,’’J.Atmos. Sci. 56, 3937 ~1999!. 13T. Miyazaki and J. C. R. Hunt, ‘‘Linear and nonlinear interactions between a columnar vortex and external turbulence,’’ J. Fluid Mech. 402,3 4 9 ~2000!. 14L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, ‘‘Hy- drodynamic stability without eigenvalues,’’ Science 261,5 7 8 ~1993!. 15K. M. Butler and B. F. Farrell, ‘‘Three-dimensional optimal perturbations in viscous shear ﬂow,’’ Phys. Fluids A 4,1 6 3 7 ~1992!. 16D. G. Lasseigne, R. D. Joslin, T. L. Jackson, and W. O. Criminale, ‘‘The transient period for boundary layer disturbances,’’ J. Fluid Mech. 381,8 9 ~1999!. 17P. Luchini, ‘‘Reynolds-number-independent instability of the boundary layer over a ﬂat surface: Optimal perturbations,’’J. Fluid Mech. 404,2 8 9 ~2000!. 18P. Luchini andA. Bottaro, ‘‘Go ¨rtler vortices:Abackward-in-time approach to the receptivity problem,’’ J. Fluid Mech. 363,1~1998!. 19P. Corbett and A. Bottaro, ‘‘Optimal linear growth in swept boundary layers,’’ J. Fluid Mech. 435,1~2001!. 20B. Fornberg, A Practical Guide to Pseudospectral Methods ~Cambridge University Press, Cambridge, 1995 !. 21J. A. C. Weideman and S. C. Reddy, ‘‘A MATLABdifferentiation matrix suite,’’ACM Trans. Math. Softw. 26,4 6 5 ~2000!. 22B. F. Farrell and P. J. Ioannou, ‘‘Optimal excitation of three-dimensional perturbations in viscous constant shear ﬂow,’’ Phys. Fluids A 5, 1390 ~1993!. 23L. Jacquin, D. Fabre, P. Geffroy, and E. Coustols, ‘‘The properties of a transport aircraft wake in the extended near ﬁeld:An experimental study,’’AIAA Paper No. 2001-1038 ~2001!. 24At least within the numerical limits. 25Preliminary results reveal that this inﬂuence is marginal, at least for the range of Reynolds numbers considered here. FIG. 5. Illustration of the initial destructive interference of vorticity spirals.L4 Phys. Fluids, Vol. 16, No. 1, January 2004 A. Antkowiak and P. Brancher
1.5123469.pdf	Appl. Phys. Lett. 115, 182408 (2019); https://doi.org/10.1063/1.5123469 115, 182408 © 2019 Author(s).Magnetic domain size tuning in asymmetric Pd/Co/W/Pd multilayers with perpendicular magnetic anisotropy Cite as: Appl. Phys. Lett. 115, 182408 (2019); https://doi.org/10.1063/1.5123469 Submitted: 06 August 2019 . Accepted: 20 October 2019 . Published Online: 30 October 2019 D. A. Dugato , J. Brandão , R. L. Seeger , F. Béron , J. C. Cezar , L. S. Dorneles , and T. J. A. Mori Magnetic domain size tuning in asymmetric Pd/Co/W/Pd multilayers with perpendicular magnetic anisotropy Cite as: Appl. Phys. Lett. 115, 182408 (2019); doi: 10.1063/1.5123469 Submitted: 6 August 2019 .Accepted: 20 October 2019 . Published Online: 30 October 2019 D. A. Dugato,1,2J.Brand ~ao,2R. L. Seeger,1F.B/C19eron,3 J. C.Cezar,2 L. S.Dorneles,1 and T. J. A. Mori2,a) AFFILIATIONS 1Departamento de F /C19ısica, Universidade Federal de Santa Maria (UFSM), 97105-900 Santa Maria RS, Brazil 2Laborat /C19orio Nacional de Luz S /C19ıncrotron (LNLS), Centro Nacional de Pesquisa em Energia e Materiais (CNPEM), 13083-970 Campinas SP, Brazil 3Instituto de F /C19ısica Gleb Wataghin (IFGW), Universidade Estadual de Campinas (UNICAMP), 13083-859 Campinas SP, Brazil a)Electronic mail: thiago.mori@lnls.br ABSTRACT Magnetic multilayers presenting perpendicular magnetic anisotropy (PMA) have great potential for technological applications. On the path to develop further magnetic devices, one can adjust the physical properties of multilayered thin ﬁlms by modifying their interfaces, thusdetermining the magnetic domain type, chirality, and size. Here, we demonstrate the tailoring of the domain pattern by tuning theperpendicular anisotropy, the saturation magnetization, and the interfacial Dzyaloshinskii-Moriya interaction (iDMI) in Pd/Co/Pd multilayerswith the insertion of an ultrathin tungsten layer at the top interface. The average domain size decreases around 60% when a 0.2 nm thick W layer is added to the Co/Pd interface. Magnetic force microscopy images and micromagnetic simulations were contrasted to elucidate the mechanisms that determine the domain textures and sizes. Our results indicate that both iDMI and PMA can be tuned by carefully changingthe interfaces of originally symmetric multilayers, leading to magnetic domain patterns promising for high density magnetic memories. Published under license by AIP Publishing. https://doi.org/10.1063/1.5123469 The precise control of the nucleation processes of magnetic domain patterns is essential to achieve adequate functionality and per-formance for modern technologies. Much progress has been achieved recently as the stabilization of chiral structures such as skyrmions has been demonstrated either in nanostructures or in multilayer thin ﬁlmspresenting perpendicular magnetic anisotropy (PMA), even at room temperature and zero magnetic ﬁeld. 1–4Mainly being observed in sys- tems with ferromagnetic/heavy metal (FM/HM) interfaces, which caneasily be integrated in current technologies, these achievements have opened an avenue toward the use of PMA multilayers in future spin- tronics devices. 5Indeed, PMA multilayers are a very fertile ground for studying magnetic interactions, since several physical properties of the FM/HM interface can be tuned in order to tailor the magnetic domain pattern. However, the role of these magnetic interactions indetermining the domain’s properties must be well understood before f u r t h e rm a g n e t i cd e v i c ed e v e l o p m e n t . 6–8 Magnetic anisotropy ( K), saturation magnetization ( Ms), and exchange stiffness ( Aex) determine the magnetic domain wall type (N /C19eel or Bloch), chirality, and size. Their role in the magnetic conﬁgurationestablishment in PMA multilayers has been studied for years.9–14More recently, the observation that magnetic skyrmions may be stabilized byDzyaloshinskii-Moriya interaction (DMI), arising from broken inver-sion symmetry 15and spin orbit coupling (SOC) in the case of FM/HM interfaces,16has given the DMI a major role in the study of domain wall patterns.17,18 In this sense, several combinations of FM and HM have been tried to fabricate asymmetric PMA multilayers (HM A/FM/HM B) searching for speciﬁc conditions to host chiral skyrmions preferably stabilized at room temperature and small magnetic ﬁelds.12,16,19–21On the other hand, small asymmetries introduced to originally symmetricmultilayers have also been demonstrated to be a good strategy to tunethe DMI in PMA multilayers. 22–24 Here, we tune the magnetic properties of originally symmetric Pd/Co/Pd multilayers by inserting an ultrathin W layer in the systemtop interface (Pd/Co/W/Pd). The PMA presents a minimum when anultrathin W layer is inserted. Using magnetic force microscopy(MFM) images acquired at the as-grown state, alongside with micro-magnetic simulations, we show that the respective interfacial DMI Appl. Phys. Lett. 115, 182408 (2019); doi: 10.1063/1.5123469 115, 182408-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apl(iDMI) is around three times higher than that observed for thicker W layers or the reference symmetric Pd/Co/Pd. This strategy allows us to obtain a 60% decrease in the average domain size at room tempera- ture, demonstrating an important route to tune magnetic multilayersfor high density magnetic memory devices. To study the magnetic domain pattern evolution with a varying asymmetry at the Co/Pd top interface, we grew multilayers based on a Pd(1 nm)/Co(0.5 nm)/W(t)/Pd(1 nm) structure, with a nominal thick-ness t¼0, 0.1, 0.2, 0.3, or 1.0 nm. The multilayers were deposited on silicon substrates by magnetron sputtering from metallic targets, atroom temperature and 3 mTorr of argon atmosphere, and repeated 15 times [ Fig. 1(a) ]. Saturation magnetization and anisotropy ﬁeld ( H k)w e r e extracted from magnetic hysteresis curves measured in a LakeShorevibrating sample magnetometer (VSM), yielding the perpendicularanisotropy constant value, K eff¼MsHk/2.10Magnetic domain pattern images of the as-grown multilayers were acquired by magnetic force microscopy (MFM) with a NanoSurf Flex scanning probe microscopeoperating in the dynamic force mode. We used Multi75-G MFM(75 kHz) tips from Budget Sensors, which are coated by a cobalt alloypresenting magnetic moment and coercivity of roughly 10 –16Am2and 0.03 T, respectively. The images were acquired at room temperature and zero magnetic ﬁeld, with the tip-surface distance about 60 nm.The magnetic domain homogeneity was conﬁrmed through theobservation of 5 images over distances of 1 mm between them. In addition, the experimental MFM images were compared with those obtained by micromagnetic simulations. For the modeling, we used the Mumax3 GPU-accelerated pro- gram to solve the time-dependent Landau-Lifshitz-Gilbert (LLG) equation to obtain the relaxation of the magnetization distribution. 25 The micromagnetic simulations were performed on an area of 5/C25lm2discretized in cells of 3 /C23/C27.5 nm3and using an effective medium approach to model the multilayer ﬁlm as a single uniform layer.26TheMsandKeffvalues extracted from the VSM measurements served as input, while we varied the iDMI contribution to understand its inﬂuence on the domain pattern formation without the applied magnetic ﬁeld. Starting with a random initial magnetization, the equi- librium condition was obtained by minimizing the LLG energy terms with a relaxation time of 100 ns. The magnetic ground state represents the domain stability for each set of magnetic parameters. The energy of the effective iDMI was evaluated by comparing the simulated ground states with the corresponding MFM images using a methodol- ogy similar to what has been reported in the recent literature.4,16,19,27,28 Both out-of-plane and in-plane magnetic hysteresis loops indicate that all the multilayers present perpendicular magnetic anisotropy [ Figs. 1(b) and1(c)]. The extracted experimental values Ms/C24545 kAm–1and Keff/C240.2 MJm–3, observed for the reference sample, are in accordance with the values found in the literature for Pd/Co/Pd multilayers.29While the reference sample exhibits out-of- plane remanence very close to Ms, the remanence decreases for a very thin (0.1–0.2 nm thick) W layer and increases again, recovering a loop with nearly full remanence for the sample with a 1 nm W layer. The W layer insertion leads to a saturation magnetization decrease, estimated by considering the entire Co volume [ Fig. 1(d) ]. The decline of the total magnetic moment may arise mainly from two coexisting mechanisms: (1) the formation of a magnetic dead layer due to alloying or interdiffusion at the interface30and (2) the reduction of magnetic proximity effect contribution to magnetization since, con- trary to Pd, the spin and orbital magnetic moments of W may couple antiparallel to 3d metals.31Besides, both Hkand Keffexhibit a mini- mum value for t ¼0.2 nm [ Fig. 1(e) ] even though Msdecreases for thicker W layers. This PMA reduction with ultrathin W layer insertion can arise from an irregular Co/W-Pd interface, since such a thin layer should not percolate and can generate roughness instead. A rough Co/Pd interface is known to lessen the interface anisotropy and, consequently, the PMA.32At the same time, such a discontinuous W-Pd layer may lead to competing interfacial effects as Co/W and Co/ Pd interfaces should behave differently. This scenario can also contrib- ute to lower the PMA since the CoPd alloying, which is known to contribute to the strong anisotropy in Co/Pd multilayers,33is restricted by the coexistence of W along the interface. Without the W layer, the MFM image shows a pattern of stripes and skyrmion-like circular domains that are normally observed in Co/ Pd multilayers with thin Co thicknesses [ Fig. 2(a) ].34,35However, small labyrinth domains arise and the domain density increases signiﬁcantly for t¼0.2 nm, reaching a magnetic domain periodicity ( k) of about 280 nm [ Figs. 2(b) and1(c)]. Hereafter, we deﬁne kas the distance between two adjacent peaks in the magnetization proﬁle and domain size as the full width at half maximum of a peak. While this system presents the lowest Keffvalue along with an Msaverage value, increas- ing to the 0.3 nm W layer yields a slightly larger Keffcombined to a Ms FIG. 1. Multilayer structure and magnetic properties. (a) Structure schematic. (b) and (c) Hysteresis loops recorded with the magnetic ﬁeld applied along the out-of- plane and in-plane directions, respectively. (d) Saturation magnetization; (e) blacksquares: anisotropy ﬁeld and open blue circles: anisotropy constant as a function ofthe W layer thickness.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 115, 182408 (2019); doi: 10.1063/1.5123469 115, 182408-2 Published under license by AIP Publishingdecrease of around 25%, resulting in both the domain size and period- icity enlargement [ Fig. 2(d) ] .T h ed o m a i ns i z ec o n t i n u e st oi n c r e a s e for t¼1 nm, as the saturation magnetization continues decreasing [Figs. 2(e) and2(f)]. In this case, the domain shape is similar to the reference sample, with a large periodicity of 670 nm and a domaindensity of /C2460% lower than with the 0.2 nm W layer. In order to verify the role of the magnetic parameter in the mag- netic domain pattern formation, we carried out micromagnetic simu-lations using the Mumax3 code. In a ﬁrst attempt, we only used theexperimental values extracted from the magnetization curves for K eff andMs, a ﬁxed exchange stiffness A ¼12/C210–12Jm–1, and damping a¼0.3. These preliminary simulated domain patterns exhibited dis- tinct ground states compared to the ones observed in the measuredMFM images, mainly for the samples with the ultrathin (0.1 and0.2 nm) W layer. To reproduce the main features of the experimental images, a non-null interfacial Dzyaloshinskii-Moriya interaction— within 0.3 and 1.3 mJm –2—had to be added to the simulated system. Very good agreement with the experimental images is achieved with the additive iDMI, even for the nominally symmetric Pd/Co/Pdsample ( Fig. 3 ). Although it should have a null iDMI in the ideal case, where the bottom and top interfaces contribute with the same amplitude but opposite sign, as represented in Fig. 4(a) , the different qualities between the Pd/Co and Co/Pd interfaces may lead to smallvalues of iDMI. 4,36,37On the other hand, the combination of a bottom Pd/Co with a top Co/W interface is expected to yield a resulting nega- tive iDMI [ Fig. 4(b) ]. This situation is similar to the iDMI reported for the Ru/Co/W system,24since both Co/Pd and Co/Ru interfaces pre- sent the same signal and similar amplitudes of iDMI.38Indeed, in the case of Ru/Co/W/Ru with varying W thicknesses, an iDMI peak hasalso been reported when the W thickness is about 0.2 nm. 24In Ref. 24, the authors studied quasisymmetric multilayers with non-null iDMI focused on the isolated skyrmion nucleation and its behavior in thepresence of an out-of-plane magnetic ﬁeld. Here, we show thatthe interface engineering strategy of adding a “dusting” interlayer atthe FM/HM interface can also be used to tune the magnetic domain size of worm-like patterns at zero magnetic ﬁeld. According to our micromagnetic simulations, the small iDMI observed for the symmetric sample rises about 3 times with the FIG. 2. Experimental magnetic force microscopy images acquired with zero mag- netic ﬁeld. (a), (b), (d), and (e) Pd/Co/Pd reference sample and multilayers with 0.2,0.3, and 1.0 nm of W at the Co/Pd interface, respectively. (c) and (f) MFM proﬁlemeasured along the straight lines highlighted on the MFM images in (b) and (e), respectively, where the periodicity kis deﬁned as the distance between two adjacent peaks. The scale bar in the images is 1 lm, and the color scale ranges from blue (amplitude /C01, magnetization downward) to red (amplitude þ1, magneti- zation upward). FIG. 3. Zero magnetic ﬁeld micromagnetic simulated domain patterns with Msand Kefftaken from VSM and non-null iDMI. (a), (b), (d), and (e) Pd/Co/Pd reference sample and multilayers with 0.2, 0.3, and 1.0 nm of W at the Co/Pd interface, respectively. (c) and (f) MFM proﬁle measured along the straight lines highlighted on the MFM images in (b) and (e), respectively. The scale bar in the images is1lm, and the color scale ranges from blue (amplitude /C01, magnetization down- ward) to red (amplitude þ1, magnetization upward).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 115, 182408 (2019); doi: 10.1063/1.5123469 115, 182408-3 Published under license by AIP Publishinginsertion of an ultrathin layer of W (0.1–0.2 nm) at the top interface, as it is shown in Fig. 4(c) . It is very interesting that a further minor increase in the W layer thickness (0.3 nm) leads to an iDMI almost as small as for the reference sample value. Note that this iDMI peak occurs for the same W thickness as the observed anisotropy minimum[seeFig. 1(e) ]. The iDMI decline for thicker W layers is suggested to be due to the magnetic dead layer present when the W forms a contin-uous layer, which leads to the ferromagnetic layer degradation. 24,30 The thicker dead layer diminishes the orbital hybridization and conse- quently the SOC and magnetic exchange in both interfaces, which are important ingredients required for a strong iDMI.39Notwithstanding, the formation of very distinct domain patterns in the 0.1–0.2 nm rangeof W occurs due to atypically low K effvalues and an additive iDMI. Indeed, the geometric properties such as domain size and periodicity also present discrepant values in this range [ Figs. 4(d) and 4(g)]. Similar to the results reported in Ref. 30,h e r ei ti sa l s ol i k e l yt h a tt h e small ratios between PMA and iDMI lead to smaller domain sizes as aresult of the reduced energy of domain walls. 40 In conclusion, we investigated the inﬂuence of a W layer, inserted at the top interface of a nominally symmetric Pd/Co/Pd multilayer, on the physical properties of the ferromagnetic Co layer as a function ofits thickness. From hysteresis loops, we extracted the saturation mag- netization Ms, anisotropy ﬁeld Hk, and hence the perpendicular magnetic anisotropy Keff.B o t h MsandKeffdecay for thicker W layers. Most notably, a minimum of the anisotropy is observed with the inser- tion of an ultrathin 0.2 nm thick W layer. MFM images were acquired to obtain the magnetic domain pat- terns at zero ﬁeld and room temperature. Labyrinth domains were imaged, revealing a strong dependence of the size and periodicity onthe W thickness. In particular, a domain size decrease of around 60%was obtained at 0.2 nm W, which coincides with the minimum per-pendicular anisotropy, indicating that the physical properties of the multilayers play a direct role in the features of the magnetic domains. To understand the magnetic domain formation, micromagnetic simulations were carried out and the results were compared with the experimental ﬁndings. By adjusting the physical parameters obtainedfor each W thickness in the modeling, the experimental observationswere reproduced by taking into account the interfacial Dzyaloshinskii-Moriya interaction. The iDMI reaches a peak at 0.2 nm W and is remarkably reduced for thicker W layers. Very importantly, the small ratio between PMA and iDMI within the W thickness range0.1–0.2 nm leads to very small domain sizes, which can be interestingfor applications such as high density hard disk drives. The strategy oftuning magnetic domains by changing the interfaces of originally symmetric multilayers is promising on the path to develop devices based on skyrmions and chiral domain walls. This study was ﬁnanced in part by the Coordenac ¸~ao de Aperfeic ¸oamento de Pessoal de N /C19ıvel Superior-Brasil (CAPES)-Finance Code 001, by the Fundac ¸~ao de Amparo /C18aP e s q u i s ad oE s t a d od eS ~ao Paulo-S ~ao Paulo, Brasil (FAPESP)-Project No. 2012/51198-2, and by the Conselho Nacional de Desenvolvimento Cient /C19ıﬁco e Tecnol /C19ogico- Brasil (CNPq). J.C.C., L.S.D., and F.B. acknowledge grants provided byCNPq: Project Nos. 309354/2015-3, 302950/2017-6, and 436573/2018-0, respectively. F.B. acknowledges grant by FAPESP: No. 2017/10581-1. The samples were grown at the Microfabrication Laboratory-Brazilian Nanotechnology National Laboratory (LNNano). 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1.3143042.pdf	Tuning magnetization dynamic properties of Fe – SiO 2 multilayers by oblique deposition Nguyen N. Phuoc, Feng Xu, and C. K. Ong Citation: Journal of Applied Physics 105, 113926 (2009); doi: 10.1063/1.3143042 View online: http://dx.doi.org/10.1063/1.3143042 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/105/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic properties of amorphous Co 0.74 Si 0.26 Si multilayers with different numbers of periods Low Temp. Phys. 36, 821 (2010); 10.1063/1.3499251 Reversibility and coercivity of Fe-alloy/ Fe : SiO 2 multilayers J. Appl. Phys. 107, 09E710 (2010); 10.1063/1.3360768 Influence of multiple magnetic phases on the extrinsic damping of FeCo – SiO 2 soft magnetic films J. Appl. Phys. 107, 033911 (2010); 10.1063/1.3289588 Induced ferromagnetism in Mn 3 N 2 phase embedded in Mn / Si 3 N 4 multilayers J. Appl. Phys. 106, 043912 (2009); 10.1063/1.3203997 Magnetic properties of YIG ( Y 3 Fe 5 O 12 ) thin films prepared by the post annealing of amorphous films deposited by rf-magnetron sputtering J. Appl. Phys. 97, 10A319 (2005); 10.1063/1.1855460 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.39.62.90 On: Thu, 21 Aug 2014 09:29:11Tuning magnetization dynamic properties of Fe–SiO 2multilayers by oblique deposition Nguyen N. Phuoc,1,a/H20850Feng Xu,1and C. K. Ong2 1Temasek Laboratories, National University of Singapore, 5A Engineering Drive 2, Singapore 117411, Singapore 2Department of Physics, Center for Superconducting and Magnetic Materials, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore /H20849Received 16 February 2009; accepted 4 May 2009; published online 10 June 2009 /H20850 The static and dynamic magnetic properties of Fe–SiO 2multilayers fabricated onto Si /H20849100 /H20850 substrates by a radio frequency sputter-deposition system are investigated as functions of depositionangle and Fe thickness. By changing the oblique deposition angle, one can effectively tune theferromagnetic resonance frequency from 1.7 to 3.5 GHz. In addition, the frequency linewidth issigniﬁcantly changed with the oblique deposition angle when the Fe layer is thick, but it is almostconstant in the case of small Fe thickness. © 2009 American Institute of Physics . /H20851DOI: 10.1063/1.3143042 /H20852 I. INTRODUCTION High frequency soft magnetic thin ﬁlms, which were widely used in many electromagnetic devices such as mag-netic recording heads, wireless inductor cores, microwavenoise ﬁlters and absorbers, have recently been studiedextensively. 1–14In many modern devices, the operating fre- quencies have reached gigahertz bands. Hence, the magneticthin ﬁlms are required to have a broadband at very highfrequencies to be used as microwave noise ﬁlters for futureapplications. For such a purpose, the ferromagnetic reso-nance /H20849FMR /H20850frequency of magnetic thin ﬁlms should be in the gigahertz range. It is well known that for in-plane mag-netized ﬁlms without external applied ﬁeld, the FMR fre-quency is strongly dependent on the uniaxial magnetic aniso-tropy ﬁeld /H20849H K/H20850and the saturation magnetization /H20849MS/H20850 according to Kittel’s equation as follows:15 fFMR=/H9253 2/H9266/H20881HK/H20849HK+4/H9266MS/H20850, /H208491/H20850 where /H9253is the gyromagnetic ratio /H20849/H9253=1.76 /H11003107Hz /Oe/H20850. To obtain high FMR frequency, one should increase the satu- ration magnetization and/or the uniaxial anisotropy ﬁeld.Therefore, among many studies on soft magnetic thin ﬁlmsfor high frequency application, considerable effort has beenfocused on controlling the uniaxial magnetic anisotropy ofthe ﬁlms. 3–14In the literature, there are several methods to increase the uniaxial anisotropy ﬁeld such as patterning thinﬁlms to create shape anisotropy, 3,4using a magnetic ﬁeld application during deposition,5–8ﬁeld annealing,9and using exchange bias coupling between a ferromagnet and an anti-ferromagnet to induce magnetic anisotropy. 10–14 Uniaxial magnetic anisotropy induced in magnetic thin ﬁlms by oblique deposition was discovered in 1959 by Knorrand Hoffman 16and Smith et al. ,17,18who showed that mag- netic anisotropy /H20849even at zero applied ﬁeld /H20850can be inducedin iron and Permalloy ﬁlms by making the metal vapor to hit the substrate at an oblique deposition angle. Using this tech-nique, one can in principle tune the uniaxial magnetic aniso-tropy by changing the oblique deposition angle 16–22and thus change the dynamic properties of the thin ﬁlms. However,there is still little work reporting about the employment ofoblique deposition to tune the high frequency characteristicsof thin ﬁlms. In this work, we therefore investigate the inﬂu-ence of the oblique deposition angle on the high frequencymagnetic characteristics of Fe–SiO 2multilayered thin ﬁlms. II. EXPERIMENTAL DETAILS Samples with the stacks of /H20851Fe–SiO 2/H208525were fabricated onto Si /H20849100 /H20850substrates at ambient temperature using the re- active rf magnetron sputter-deposition system with the basepressure better than 7 /H1100310 −7Torr. The thickness of SiO 2 layer was ﬁxed at 5 nm while the thickness of Fe was changed from 10 to 20 nm. No magnetic ﬁeld was appliedduring the deposition process. The argon pressure was keptat 10 −3Torr during the deposition process by introducing a/H20850Author to whom correspondence should be addressed. Tel./FAX: 65- 65162816/65-67776126. Electronic mail: tslnnp@nus.edu.sg. FIG. 1. Schematic view of the oblique sputtering deposition system.JOURNAL OF APPLIED PHYSICS 105, 113926 /H208492009 /H20850 0021-8979/2009/105 /H2084911/H20850/113926/4/$25.00 © 2009 American Institute of Physics 105 , 113926-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.39.62.90 On: Thu, 21 Aug 2014 09:29:11argon gas at the ﬂow rate of 16 SCCM /H20849SCCM denotes cubic centimeter per minute at STP /H20850. The deposition arrangement is shown in Fig. 1, where the substrates were put at different positions so that the thin ﬁlm can be fabricated at an obliquedeposition angle ranging from 0° to 30° as shown in Fig. 1. The magnetic properties of the ﬁlms were measured by a M-Hloop tracer at room temperature. The permeability spectra over the frequency range from 0.05 to 5 GHz wereobtained by a shorted microstrip transmission-line perturba-tion method using a ﬁxture developed in our laboratory. Fur-ther details of this method can be found in a previouspaper. 23 III. RESULTS AND DISCUSSION Figure 2shows the in-plane hysteresis loops of /H20851Fe/H2084920 nm /H20850/SiO 2/H208495n m /H20850/H208525multilayers deposited onto Si/H20849100 /H20850substrates at various oblique deposition angles. The direction of the in-plane easy axis /H20849EA /H20850was found to be perpendicular to the deposition direction, while the hard axis/H20849HA /H20850is along the deposition direction as indicated in Fig. 1. With increasing the oblique deposition angle the HA M-H loops become sheerer, which is a clear evidence of the effectof induced uniaxial magnetic anisotropy by oblique deposi-tion. The behaviors of the oblique angular dependence of thecoercivity /H20849in both easy and hard axes, H CEAandHCHA/H20850, the saturation ﬁeld /H20849HS/H20850and the uniaxial anisotropy ﬁeld /H20849HK/H20850for different thicknesses of Fe layers are summarized in Fig. 3. Here, the uniaxial anisotropy ﬁeld /H20849HK/H20850was extracted from the slope of rotational-like magnetization curve on the HA.8,12It is clearly observed in Fig. 3that for all the thick- nesses of Fe layers, both the coercivity and the uniaxial an-isotropy ﬁeld are increased with the oblique depositionangle. The change in the uniaxial anisotropy ﬁeld with theoblique deposition angle was previously explained within theframework of a so-called self-shadowing model. 18According to this mechanism, the region of the substrate behind a grow-ing crystallite is prevented from receiving metal vapor be-cause this region is in the “shadow” of the crystallite and asa result, the crystallites agglomerate into two-dimensionalarray of chains whose long axis tends to be perpendicular tothe beam direction. According to Smith et al. , 18the aniso- tropy induced by applying a magnetic ﬁeld during depositionis only an M-induced anisotropy and in the case of oblique- incident ﬁlms, the direction of Mis deﬁned by the crystallite chains so that an M-induced anisotropy should exist even in the absence of an applied ﬁeld. Thus the uniaxial anisotropycan be induced by oblique deposition technique. The real /H20849 /H9262/H11032/H20850and imaginary /H20849/H9262/H11033/H20850permeability spectra of /H20851Fe/H2084920 nm /H20850/SiO 2/H208495n m /H20850/H208525multilayers deposited at various oblique deposition angles are presented in Fig. 4.A si so b - FIG. 2. /H20849Color online /H20850Hysteresis loops of /H20851Fe/H2084920 nm /H20850–SiO2/H208495n m /H20850/H208525mul- tilayers deposited at various oblique deposition angles. FIG. 3. /H20849Color online /H20850Variation of coercivity measured along easy /H20849HCEA/H20850 and hard /H20849HCHA/H20850axes /H20851/H20849a/H20850and /H20849b/H20850/H20852saturation ﬁeld /H20849HS/H20850/H20849c/H20850and uniaxial anisotropy ﬁeld /H20849HK/H20850/H20849 d/H20850on oblique deposition angle for /H20851Fe/H20849xnm/H20850–SiO2/H208495n m /H20850/H208525multilayers with various Fe thicknesses /H20849x/H20850.T h e open symbol in /H20849c/H20850is the value obtained from LLG ﬁtting.113926-2 Phuoc, Xu, and Ong J. Appl. Phys. 105 , 113926 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.39.62.90 On: Thu, 21 Aug 2014 09:29:11served, the peak of the imaginary permeability /H20849/H9262/H11033/H20850is shifted to the higher frequency range indicating that the FMR fre- quency is increased with the oblique deposition angle. Also,the FMR peak gets broader as the oblique deposition angleincreases. For a more quantitative examination of the effectof oblique deposition onto the dynamic properties of the thinﬁlms, an analysis based on the Landau–Lifshitz–Gilbert/H20849LLG /H20850equation was employed. As is well known, the dynamic magnetization behavior of the thin ﬁlm can be described by the LLG equation, 24dM/H6023 dt=−/H9253/H20849M/H6023/H11003H/H6023/H20850+/H9251eff MM/H6023/H11003dM/H6023 dt, /H208492/H20850 where Mrepresents the magnetic moment, His the magnetic ﬁeld,/H9251effis the dimensionless effective damping coefﬁcient /H20849/H9251effis not intrinsic but takes into account all possible effects of damping /H20850, and/H9253is the gyromagnetic ratio. Solving the LLG equation, one can readily obtain the complex permeability as follows:7,10 /H9262/H11032=1+4 /H9266MS/H92532/H208494/H9266MS+HK/H20850/H208491+/H9251eff2/H20850/H20851/H9275R2/H208491+/H9251eff2/H20850−/H92752/H20852+/H208494/H9266MS+2HK/H20850/H20849/H9251eff/H9275/H208502 /H20851/H9275R2/H208491+/H9251eff2/H20850−/H92752/H208522+/H20851/H9251eff/H9275/H9253/H208494/H9266MS+2HK/H20850/H208522, /H208493/H20850 /H9262/H11033=4/H9266MS/H9253/H9275/H9251eff/H208494/H9266MS+HK/H208502/H208491+/H9251eff2/H20850+/H92752 /H20851/H9275R2/H208491+/H9251eff2/H20850−/H92752/H208522+/H20851/H9251eff/H9275/H9253/H208494/H9266MS+2HK/H20850/H208522. /H208494/H20850 Here, MS,HK, and/H9275R/H20849/H9275R=2/H9266fFMR /H20850are the saturation magnetization, uniaxial magnetic anisotropy, and FMR fre- quency, respectively. Considering 4 /H9266MS=21 kG for Fe lay- ers estimated from the hysteresis loops and taking HKand /H9251effas ﬁtting parameters, one can ﬁt the experimental curves in Fig. 4with formulas /H208493/H20850and /H208494/H20850quite well. The uniaxial anisotropy ﬁeld HKderived from the ﬁtting is presented in Fig. 3/H20849c/H20850/H20849open symbols /H20850in comparison with HKobtained from the M-Hloops. It is noticed that there is a discrepancy between HKderived from dynamic curves and HKobtained from the static curves. This discrepancy was similarly ob-served in various soft magnetic thin ﬁlms, 25,26which may be interpreted in term of the Hoffmann’s ripple theory.27Ac- cording to this theory, there is an additional effective isotro-pic ﬁeld that contributes to the anisotropy ﬁeld obtained frompermeability spectra beside the static intrinsic anisotropyﬁeld. This additional effective ﬁeld dependent on a so-calledripple constant may originate from the local randomanisotropies, which are in isotropic distribution 25,26and con- sequently it is not included in HKfrom the static measure- ment as that from the dynamic measurement. As a result,there is a discrepancy between two H Kvalues obtained from two methods. Figure 5shows the dependences of the effective damp- ing coefﬁcient /H9251eff, the frequency linewidth /H9004f, and the FMR frequency fFMRon the deposition angle. All the parameters were derived from the ﬁtting of the curves in Fig. 4for three series of /H20851Fe/H20849xnm/H20850/SiO 2/H208495n m /H20850/H208525multilayers /H20849x=10, 15, and 20 nm /H20850. The frequency linewidth /H9004fis obtained from the following formula:8 /H9004f=/H9253/H9251eff/H208494/H9266MS+2HK/H20850 2/H9266. /H208495/H20850 As seen in Figs. 5/H20849a/H20850and 5/H20849b/H20850, the effective damping coefﬁcient /H9251effand the frequency linewidth /H9004fare increased signiﬁcantly with the oblique deposition angle for the casetFe=20 nm, while they are not changed much when Fe thick- ness is thinner /H20849tFe=10 nm, 15 nm /H20850. This behavior is some- FIG. 4. /H20849Color online /H20850Experimental and calculated permeability spectra for /H20851Fe/H2084920 nm /H20850–SiO2/H208495n m /H20850/H208525multilayers deposited at various oblique depo- sition angles.113926-3 Phuoc, Xu, and Ong J. Appl. Phys. 105 , 113926 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.39.62.90 On: Thu, 21 Aug 2014 09:29:11what similar to the trend of variations of the saturation ﬁeld /H20849HS/H20850and the uniaxial anisotropy ﬁeld /H20849HK/H20850with oblique deposition angle in Fig. 3/H20849c/H20850suggesting that there may pos- sibly be a correlation between the frequency linewidth andthe static magnetic properties. Despite the difﬁculty in eluci-dating such a correlation, one may presumably ascribe thebroadening of the frequency linewidth to the dispersion ofthe magnetic anisotropy, which may be inﬂuenced by thechange of the uniaxial anisotropy ﬁeld as well as the satura-tion ﬁeld. However, other possibilities may also be the rea-son and should not be excluded. As in Fig. 5/H20849c/H20850, the FMR frequency can be tuned from 1.7 to 3.5 GHz by changing the oblique deposition angle.The variation of FMR frequency with the oblique depositionangle calculated from Eq. /H208491/H20850using the H Kobtained from static measurement is also presented in Fig. 5/H20849c/H20850with the open symbols. It is clearly observed that the experimentalbehavior of FMR position is similar to the theory althoughthere are small discrepancies, which were interpreted asabove within the framework of ripple theory. 27The result that the FMR positions can be tuned with the oblique depo-sition angle opens up a possibility to use oblique depositiontechnique to fabricate magnetic thin ﬁlms for microwave ap-plications besides the other traditional methods. However,more effort is needed to extend the FMR frequency to higherfrequency range.IV. SUMMARY AND CONCLUSION In summary, we show in the present work that the mag- netization dynamics of Fe–SiO 2multilayers can be effec- tively tuned by oblique deposition technique. The FMR fre-quency can be changed from 1.7 to 3.5 GHz implying thatoblique deposition technique is a promising method to in-duce magnetic anisotropy for high frequency application.Also, the frequency linewidth is signiﬁcantly changed withthe oblique deposition angle when t Fe=20 nm, while they are not much affected by the oblique deposition angle varia-tion for the case t Fe=10 and 15 nm. ACKNOWLEDGMENTS The authors acknowledge the ﬁnancial support from the Defense Science and Technology Agency /H20849DSTA /H20850of Sin- gapore. 1O. Acher and S. Dubourg, Phys. Rev. B 77, 104440 /H208492008 /H20850. 2O. Acher, V . Dubuger, and S. Bubourg, IEEE Trans. Magn. 44, 2842 /H208492008 /H20850. 3M. Vroubel, Y . Zhuang, B. Rejaei, and J. Burghartz, J. Magn. Magn. Mater. 258–259 , 167 /H208492003 /H20850. 4X. Chen, Y . G. Ma, and C. K. Ong, J. Appl. Phys. 104, 013921 /H208492008 /H20850. 5Y . G. Ma and C. K. Ong, J. Phys. D 40, 3286 /H208492007 /H20850. 6N. D. Ha, M. H. Phan, and C. O. Kim, Nanotechnology 18, 155705 /H208492007 /H20850. 7S. Ge, S. Yao, M. Yamaguchi, X. Yang, H. Zuo, T. Ishii, D. Zhou, and F. Li,J. Phys. D 40, 3660 /H208492007 /H20850. 8F. Xu, X. Chen, Y . G. Ma, N. N. Phuoc, X. Y . Zhang, and C. K. Ong, J. Appl. Phys. 104, 083915 /H208492008 /H20850. 9S. Li, Z. Yuan, J. G. Duh, and M. Yamaguchi, Appl. Phys. Lett. 92, 092501 /H208492008 /H20850. 10B. K. Kuanr, R. E. Camley, and Z. Celinski, J. Appl. Phys. 93,7 7 2 3 /H208492003 /H20850. 11S. Queste, S. Dubourg, O. Acher, K. U. Barholz, and R. Mattheis, J. Appl. Phys. 95, 6873 /H208492004 /H20850. 12Y . Lamy and B. Viala, IEEE Trans. Magn. 42, 3332 /H208492006 /H20850. 13N. N. Phuoc, S. L. Lim, F. Xu, Y . G. Ma, and C. K. Ong, J. Appl. Phys. 104, 093708 /H208492008 /H20850. 14N. N. Phuoc, F. Xu, and C. K. Ong, Appl. Phys. Lett. 94, 092505 /H208492009 /H20850. 15C. Kittel, Phys. Rev. 71, 270 /H208491947 /H20850. 16T. G. Knorr and R. W. Hoffman, Phys. Rev. 113, 1039 /H208491959 /H20850. 17D. O. Smith, J. Appl. Phys. 30, S264 /H208491959 /H20850. 18D. O. Smith, M. S. Cohen, and G. P. Weiss, J. Appl. Phys. 31,1 7 5 5 /H208491960 /H20850. 19J. M. Alameda, F. Carmona, F. H. Salas, L. M. Alvarez-Prado, R. Morales, and G. T. Perez, J. Magn. Magn. Mater. 154, 249 /H208491996 /H20850. 20F. Liu, M. T. Umlor, L. Shen, J. Weston, W. Eads, J. A. Barnard, and G. J. Mankey, J. Appl. Phys. 85, 5486 /H208491999 /H20850. 21F. Tang, D. L. Liu, D. X. Ye, Y . P. Zhao, T. M. Lu, G. C. Wang, and A. Vijayaraghavan, J. Appl. Phys. 93, 4194 /H208492003 /H20850. 22M. T. Umlor, Appl. Phys. Lett. 87, 082505 /H208492005 /H20850. 23Y . Liu, L. F. Chen, C. Y . Tan, H. J. Liu, and C. K. Ong, Rev. Sci. Instrum. 76, 063911 /H208492005 /H20850. 24T. L. Gilbert, IEEE Trans. Magn. 40, 3443 /H208492004 /H20850. 25J. Rantschler and C. Alexander, Jr., J. Appl. Phys. 93, 6665 /H208492003 /H20850. 26G. Suran, H. Ouahmane, I. Iglesias, M. Rivas, J. Corrales, and M. Contr- eras, J. Appl. Phys. 76, 1749 /H208491994 /H20850. 27H. Hoffmann, IEEE Trans. Magn. 4,3 2 /H208491968 /H20850. FIG. 5. /H20849Color online /H20850The dependences of effective damping coefﬁcient /H20849/H9251eff/H20850/H20849a/H20850, frequency linewidth /H20849/H9004f/H20850/H20849b/H20850, and resonance frequency /H20849fFMR /H20850/H20849c/H20850 on oblique deposition angle for /H20851Fe/H20849xnm/H20850–SiO2/H208495n m /H20850/H208525multilayers with various Fe thicknesses /H20849x/H20850. The open symbol in /H20849c/H20850is the value obtained from Eq. /H208491/H20850.113926-4 Phuoc, Xu, and Ong J. Appl. Phys. 105 , 113926 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.39.62.90 On: Thu, 21 Aug 2014 09:29:11
1.3687909.pdf	Sub-nanosecond switching of vortex cores using a resonant perpendicular magnetic field Ruifang Wang and Xinwei Dong Citation: Appl. Phys. Lett. 100, 082402 (2012); doi: 10.1063/1.3687909 View online: http://dx.doi.org/10.1063/1.3687909 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v100/i8 Published by the American Institute of Physics. Related Articles Angular dependence of switching field distribution in (Co/Pd)n multilayer nanostructure arrays J. Appl. Phys. 112, 093918 (2012) Application of local transverse fields for domain wall control in ferromagnetic nanowire arrays Appl. Phys. Lett. 101, 192402 (2012) Dynamics and collective state of ordered magnetic nanoparticles in mesoporous systems J. Appl. Phys. 112, 094309 (2012) Oscillation frequency of magnetic vortex induced by spin-polarized current in a confined nanocontact structure J. Appl. Phys. 112, 093905 (2012) The magneto-optical behaviors modulated by unaggregated system for γ-Fe2O3–ZnFe2O4 binary ferrofluids AIP Advances 2, 042124 (2012) Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 11 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsSub-nanosecond switching of vortex cores using a resonant perpendicular magnetic field Ruifang Wanga)and Xinwei Dong Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, China (Received 22 November 2011; accepted 30 January 2012; published online 21 February 2012) We performed micromagnetic numerical studies on ultrafast switching of magnetic vortex cores (VCs) using a perpendicular magnetic ﬁeld that oscillates at the eigenfrequency of a permalloy nanodisk. Our calculations show that a resonant magnetic ﬁeld with amplitude of 30 mT stimulates strong axially symmetric magnetization oscillation and forces the vortex core to stay at the centerof the nanodisk. The compression of the vortex core by spin wave leads to core reversal at 602 ps. This switching process is mediated by the propagation of a Neel wall across the sample thickness. VC2012 American Institute of Physics . [doi: 10.1063/1.3687909 ] Magnetic vortex1,2is a typical ground state of nanoscale magnetic thin-ﬁlm structures, in which the in-plane magnet-ization curls around a core with diameter of only 10 to 20 nanometers. 3–5Inside the core, magnetization rotates out of the plane, either up or down, characterizing the two possiblecore polarities of a magnetic vortex, which can be coded as “0” and “1” in data storage devices. However, the large energy density in the core presents a large barrier for corepolarity reversal. Recent studies on fast switching of a vortex core (VC) under an in-plane oscillating magnetic ﬁeld (or spin polarized current), 6–11through large amplitude gyration of the VC, have signiﬁcantly promoted the use of the mag- netic vortex in practical applications for magnetic data stor- age devices.12,13Using this approach, the switching speed depends on both the amplitude and frequency of the in-plane oscillating ﬁeld.6,9,10,14,15A magnetic ﬁeld of 50 mT in am- plitude and oscillating at the gyration eigenfrequency of theVC can reverse the core polarity within 100 ps. 14However, under an in-plane magnetic ﬁeld, the VC moves along a spi- ral path from the disk center and accelerates to a criticalspeed of a few hundred meters per second, until the core re- versal occurs through the creation and annihilation of a vortex-antivortex pair. 15,16 While switching the VC using the oscillating in-plane ﬁeld is energy efﬁcient and fast, the large core movement creates a severe obstacle for bit reading in magnetic datastorage devices. Applying a magnetic ﬁeld perpendicular to the nanodisk is a way to switch the VC in-situ . However, a static perpendicular ﬁeld of the order of 500 mT is requiredto ﬂip the core polarity. 17,18In this letter, we demonstrate by numerical calculations that an oscillating perpendicular mag- netic ﬁeld of only 30 mT can not only reverse the core polar-ity within a sub-nanosecond time frame but also force the core to stay at the center of the vortex, when the external ﬁeld is tuned to the magnetization eigenfrequency of thenanodisk. Our numerical 3D micromagnetic simulations 19are car- ried out using the Landau-Lifshitz-Gilbert equation, which issuitable for calculating the magnetization dynamics on a 10 ps temporal scale.14In the modeling, we choose a diameter of 300 nm and a thickness of 20 nm for the Permalloy nano- disk ( Ms¼800 KA =m, exchange constant A¼13 pJ=m, Gilbert damping constant a¼0:01, and anisotropy constant k¼0). The mesh cell size is 2.5 /C22.5/C22.5 nm. To study the vortex frequency mode numerically, a square wave pulse of 100 ps in duration and 30 mT in strength is applied per-pendicular to the nanodisk. The temporal evolution of the perpendicular magnetization component averaged over the whole nanodisk ( hM zi=Ms) is given in Fig. 1. Damped peri- odic oscillations are observed for the transient excitation per- pendicular to the disk, with an early response that also reﬂects the pulse proﬁle. Subsequent Fourier transformation(FT) 20–24onhMzi=Msis given in the inset of Fig. 1, where we identify two eigenfrequencies at 10.4 GHz and 14.3 GHz. The eigenmode images in Fig. 1are obtained by Fourier transforming the time domain signal recorded at each loca- tion into the frequency domain. The FT data are reassembled to display spatially resolved maps of the amplitude. The twoeigenmodes in Fig. 1can be classiﬁed as being in radial mode, which is governed principally by magnetostatic interactions. 21,24–27The eightfold symmetry of the eigen- mode at 10.4 GHz can be attributed to mode coupling27and the deviation from cylindrical symmetry caused by con- structing the disk out of small cubes.26 To excite magnetization oscillation of the nanodisk, we apply a sinusoidal magnetic ﬁeld along the zdirection, namely ~HextðtÞ¼/C0 30sin ð2pftÞ^zmT, where f¼10:4 GHz. The dynamic processes leading to the core reversal can be clearly identiﬁed by the temporal evolution of the zcompo- nent of the magnetization ( mz¼Mz=Ms) of the sample as displayed in Fig. 2. In the static state, the magnetic vortex has positive polarization ( mzis 0.99 at the apex of the VC), and its magnetization shows clear circular symmetry [seeFig. 2(a)]. After applying the resonant perpendicular mag- netic ﬁeld, the magnetic vortex undergoes signiﬁcant oscilla- tion of magnetization, which can be seen in Figs. 2(b)–2(h). The stimulated spin wave reﬂects between the VC and the edge of the nanodisk, and as a result leads to considerable expansion and compression of the VC [see Figs. 2(b)–2(h)].a)Author to whom correspondence should be addressed. Electronic mail: wangrf@xmu.edu.cn. 0003-6951/2012/100(8)/082402/3/$30.00 VC2012 American Institute of Physics 100, 082402-1APPLIED PHYSICS LETTERS 100, 082402 (2012) Downloaded 11 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsThe vortex forms a highly compressed core surrounded by a deep annular dip, where mz</C00:6, at 591.5 ps [see Fig. 2(d)], then the core quickly diminishes until mzat the apex of the VC ( mzavc) reduces to /C01.0 at 602 ps, which marks the accomplishment of the VC switching process. Mean- while, a burst of high frequency spin waves is emitted fromthe core region immediately after the core reversal, because a large amount of exchange energy is released in this pro- cess.28Under the resonant perpendicular magnetic ﬁeld, the VC reverses its polarity again at 637 ps, through a core com- pression process similar to the previous core reversal [see Figs. 2(g) and2(h)]. It is noticeable in Fig. 2that, during the switching process, the spin wave shows good axial symme- try. This axially symmetric spin wave forces the VC to stay at the center of the sample, and the whole system keeps ingood circular symmetry throughout the switching process. To quantitatively study the variation of core size with time, we deﬁne the size of the VC in such a way that theedge and apex of the core has a 30% difference in m z, namely mzcore edge ¼0:7mzavc.29This deﬁnition gives a core diameter of 11.8 nm30in the static state, which agrees well with previous studies on the size of the VC.3,4,31The varia- tion of core size with time is plotted in Fig. 3(a). The core size oscillates nearly in phase with the external ﬁeld beforethe ﬁrst switching of the VC. The higher oscillation rate afterwards can be attributed to the high frequency spin wave emitted from the VC after its switching. Fig. 3(b) also dis- plays the variation of m zavcand the zcomponent of the exchange ﬁeld at the apex of VC ( Hexzavc) with time. Within 588 ps after applying the resonant magnetic ﬁeld, the spinwave stimulated by external ﬁeld drives the size of the VC to ﬂuctuate over a wide range between 4.5 nm and 18.8 nm. However, in this period of time, the VC is able to generate aH exzavcof over 20 T at the apex of the VC, therefore, mzavc is kept in a narrow range of 0.98 to 0.99. After 588 ps, the continuous compression of the VC by the spin wave leads toan abrupt decline in H exzavc, which in turn gives rise to a sharp fall in mzavc. At 602 ps, Hexzavcdrops to /C025 T and mzavcis reduced to /C01.0, which indicates a complete switching of the VC. While the discussion above has shown the dynamics of vortex switching using the temporal evolution of mz, another interesting question is how the VC switching progresses along the disk thickness. In Fig. 4, we present cross-sectional views of the central region of the vortex from 595 ps to 602 FIG. 1. (Color online) Variation of hMzi=Mswith time, after applying a small pulsed perpendicular magnetic ﬁeld to the permalloy nanodisk. The inset shows its Fourier transform and two eigenmode images at eigenfre- quencies of 10.4 GHz and 14.3 GHz. FIG. 2. (Color) Dynamics of the vortex reversal process. In (a)-(h), a visual- ization with topography of mzhas been used. The insets on the upper right dis- play a cutline along the diameter through the VC. The insets on the lowerright show the xcomponent of magnetization (red: m x¼1 and blue: mx¼/C01). (enhanced online) [URL: http://dx.doi.org/10.1063/1.3687909.1 ]. FIG. 3. (Color) (a) Variation of the vortex core diameter (black circles) and external magnetic ﬁeld (blue line) with time. (b) Variation of Hexzavc(black circle) and mzavc(red circle) with time.082402-2 R. Wang and X. Dong Appl. Phys. Lett. 100, 082402 (2012) Downloaded 11 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsps. This plot clearly shows that the VC reversal starts from the top plane [see Fig. 4(b)]. Furthermore, in the VC, spins in the 8 planes form a Neel wall. The core reversal is accom-plished when the Neel wall propagates vertically through the sample [see Figs. 4(b)–4(d)]. Such a switching process by Neel wall propagation is in contrast to Bloch point mediatedswitching under a static perpendicular ﬁeld. 18In the core region, the axial symmetry of magnetization is broken when the Neel wall propagates through the sample thickness. Thislocal break of axial symmetry is necessary for realizing an in-situ VC switching. However, in the region outside the VC, the vortex is still in good circular symmetry during theswitching process. In summary, by employing a perpendicular magnetic ﬁeld that oscillates at the eigenfrequency of the vortex, thecore polarity can be reversed within 602 ps. The resonant perpendicular magnetic ﬁeld stimulates axially symmetric magnetization oscillation which in turn changes the size ofthe VC and keeps the core at the center of the sample. The continuous compression of the VC eventually leads to a quick decline of H exzavc, from over 20 T to under /C025 T, and, therefore, drives the VC to reverse its polarity. The switching process is mediated by a Neel wall that travels ver- tically through the sample. This study is of fundamental in-terest and may be relevant for possible applications of the magnetic vortex in data storage devices. This work is ﬁnancially supported by the National Natu- ral Science Foundation of China under Grant No. 10974163and 11174238 and the Specialized Research Fund for theDoctoral Program of Higher Education in China under Grant No. 20090121120029. 1T. Shinjo, T. Okuno, and R. Hassdorf, Science 289, 930 (2000). 2R. Cowburn, D. Koltsov, A. Adeyeye, M. Welland, and D. Tricker, Phys. Rev. Lett. 83, 1042 (1999). 3A. Wachowiak, Science 298, 577 (2002). 4K. W. Chou, A. Puzic, H. Stoll, D. Dolgos, G. Schu ¨tz, B. Van Waeyen- berge, A. Vansteenkiste, T. Tyliszczak, G. Woltersdorf, and C. H. Back, Appl. Phys. Lett. 90, 202505 (2007). 5M. Buess, Y. Acremann, A. Kashuba, C. Back, and D. Pescia, J. Phys.: Condens. Matter 15, R1093 (2003). 6B. Van Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T. Tyliszczak, R. Hertel, M. Fa ¨hnle, H. Bru ¨ckl, K. Rott, G. Reiss et al.,Nature 444, 461 (2006). 7K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville, and T. Ono, Nature Mater. 6, 270 (2007). 8Y. Liu, H. He, and Z. Zhang, Appl. Phys. Lett. 91, 242501 (2007). 9M. Weigand, B. Van Waeyenberge, A. Vansteenkiste, M. Curcic, V. Sackmann, H. Stoll, T. Tyliszczak, K. Kaznatcheev, D. Bertwistle, G. Woltersdorf et al.,Phys. Rev. Lett. 102, 077201 (2009). 10Q. F. Xiao, J. Rudge, B. C. Choi, Y. K. Hong, and G. Donohoe, Appl. Phys. Lett. 89, 262507 (2006). 11M. Kammerer, M. Weigand, M. Curcic, M. Noske, M. Sproll, A. Vansteenkiste, B. Van Waeyenberge, H. Stoll, G. Woltersdorf, and C. H. Back, Nature Commun. 2, 279 (2011). 12B. Pigeau, G. de Loubens, O. Klein, A. Riegler, F. Lochner, G. Schmidt, L. W. Molenkamp, V. S. Tiberkevich, and A. N. Slavin, Appl. Phys. Lett. 96, 132506 (2010). 13B. Pigeau, G. de Loubens, O. Klein, A. Riegler, F. Lochner, G. Schmidt, and L. W. Molenkamp, Nat. Phys. 7, 26 (2010). 14K.-S. Lee, S.-K. Kim, Y.-S. Yu, Y.-S. Choi, K. Y. Guslienko, H. Jung, and P. Fischer, Phys. Rev. Lett. 101, 267206 (2008). 15K.-S. Lee, S. Choi, and S.-K. Kim, Appl. Phys. Lett. 87, 192502 (2005). 16R. Hertel and C. Schneider, Phys. Rev. Lett. 97, 177202 (2006). 17T. Okuno, K. Shigeto, T. Ono, K. Mibu, and T. Shinjo, J. Magn. Magn. Mater. 240, 1 (2002). 18A. Thiaville, J. Garcı ´a, R. Dittrich, J. Miltat, and T. Schreﬂ, Phys. Rev. B 67, 094410 (2003). 19We used LLG Micromagnetic Simulator ver. 2.63c developed by M. R. Scheinfein to carry out micromagnetic simulations. 20R. D. McMichael and M. D. Stiles, J. Appl. Phys. 97, 10J901 (2005). 21M. Buess, R. Ho ¨llinger, T. Haug, K. Perzlmaier, U. Krey, D. Pescia, M. Scheinfein, D. Weiss, and C. Back, Phys. Rev. Lett. 93, 077207 (2004). 22V. Novosad, F. Fradin, P. Roy, K. Buchanan, K. Guslienko, and S. Bader, Phys. Rev. B 72, 024455 (2005). 23J. Park and P. Crowell, Phys. Rev. Lett. 95, 167201 (2005). 24M. Buess, T. Haug, M. Scheinfein, and C. Back, Phys. Rev. Lett. 94, 127205 (2005). 25K. Y. Guslienko, J. Nanosci. Nanotechnol. 8, 2745 (2008). 26V. Novosad, M. Grimsditch, K. Guslienko, P. Vavassori, Y. Otani, and S. Bader, Phys. Rev. B 66, 052407 (2002). 27M. Grimsditch, L. Giovannini, F. Montoncello, F. Nizzoli, G. K. Leaf, and H. G. Kaper, Phys. Rev. B 70, 054409 (2004). 28S. Choi, K.-S. Lee, K. Guslienko, and S.-K. Kim, Phys. Rev. Lett. 98, 087205 (2007). 29There is an exception at 598.5 ps, and we deﬁne that mzcore edge ¼1.3 mzcorecenter in this case. 30We apply data interpolation for determining the core size. 31A. Hubert and R. Scha ¨fer, Magnetic Domains: The Analysis of Magnetic Microstructures (Springer, New York, 1998). FIG. 4. (Color online) Transient images (cross-section) showing displace- ment of a Neel wall during the VC reversal process. The images were taken at times t¼595 ps (a), 599 :2 ps (b), 600 :6 ps (c), and 602 ps (d). The gray scale reﬂects the zcomponent of magnetization.082402-3 R. Wang and X. Dong Appl. Phys. Lett. 100, 082402 (2012) Downloaded 11 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions
1.4876128.pdf	Spin-orbit interaction tuning of perpendicular magnetic anisotropy in L10 FePdPt films X. Ma, P. He, L. Ma, G. Y. Guo, H. B. Zhao, S. M. Zhou, and G. Lüpke Citation: Applied Physics Letters 104, 192402 (2014); doi: 10.1063/1.4876128 View online: http://dx.doi.org/10.1063/1.4876128 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/104/19?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Tunable magnetization dynamics in disordered FePdPt ternary alloys: Effects of spin orbit coupling J. Appl. Phys. 116, 113908 (2014); 10.1063/1.4895480 Tuning magnetotransport in PdPt/Y3Fe5O12: Effects of magnetic proximity and spin-orbit coupling Appl. Phys. Lett. 105, 012408 (2014); 10.1063/1.4890239 Magneto-optical Kerr effect in L10 FePdPt ternary alloys: Experiments and first-principles calculations J. Appl. Phys. 115, 183903 (2014); 10.1063/1.4872463 Large change in perpendicular magnetic anisotropy induced by an electric field in FePd ultrathin films Appl. Phys. Lett. 98, 232510 (2011); 10.1063/1.3599492 Nanostructure and magnetic properties of polycrystalline FePdPt/MgO thin films J. Appl. Phys. 91, 8813 (2002); 10.1063/1.1453328 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 35.13.245.22 On: Sat, 06 Dec 2014 02:01:53Spin-orbit interaction tuning of perpendicular magnetic anisotropy in L1 0FePdPt films X. Ma,1P . He,2L. Ma,2G. Y . Guo,3,a)H. B. Zhao,4,a)S. M. Zhou,2and G. L €upke1,a) 1Department of Applied Science, College of William and Mary, 251 Jamestown Road, Williamsburg, Virginia 23187, USA 2Shanghai Key Laboratory of Special Artiﬁcial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 3Department of Physics, National Taiwan University, Taipei 10617, Taiwan 4Department of Optical Science and Engineering, Fudan University, 220 Handan Road, Shanghai 200433, China (Received 2 April 2014; accepted 30 April 2014; published online 12 May 2014) The dependence of perpendicular magnetic anisotropy Kuon spin-orbit coupling strength nis investigated in L10ordered FePd 1/C0xPtxﬁlms by time-resolved magneto-optical Kerr effect measurements and ab initio density functional calculations. Continuous tuning of Kuover a wide range of magnitude is realized by changing the Pt/Pd concentration ratio, which strongly modiﬁes nbut keeps other leading parameters affecting Kunearly unchanged. Ab initio calculations predict a nearly quadratic dependence of Kuonn, consistent with experimental data. Kuincreases with increasing chemical order and decreasing thermal spin ﬂuctuations, which becomes more signiﬁcant for samples with higher Pt concentration. The results demonstrate an effective methodto tune K uutilizing its sensitivity on n, which will help fabricate magnetic systems with desirable magnetic anisotropy. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4876128 ] Perpendicular magnetic anisotropy (PMA) deﬁnes the low-energy orientation of the magnetization M(easy axis) normal to the ﬁlm plane, as well as the stability of Mwith respect to external ﬁelds, electric currents, and temperature- induced ﬂuctuations, which is of great interest in magnetoe-lectronics. Materials with large PMA are good candidates for high density memory and spintronics devices, 1–3as they exhibit promising thermal stability and allow going beyondthe superparamagnetic effect, giving access to magnetic media with smaller magnetic domains. 4,5Moreover, PMA- based magnetic tunnel junctions provide high tunnelingmagneto-resistance ratio, 6–8and low critical current for spin- transfer-torque switching,9which are potential candidates for next-generation persistent memory. Considering theseadvantages, tailoring the PMA of magnetic materials as well as elucidating its physical origin becomes important for further technological advancements. PMA can be related to the spin-orbit-coupling (SOC)- induced splitting and shifting of electronic states that depend on the magnetization direction, which results from the simul-taneous occurrence of the spin polarization and SOC. 10–12 An effective method to achieve signiﬁcant PMA utilizes appropriate combinations of 3 dand heavier 4 d,5delements, which merge the large magnetic moment and magnetic sta- bility of 3 dtransition metals (TMs) with the strong SOC of 4d,5dTMs.13–17L10ordered FePt material is one prominent example.18–22TheL10ordered structure consists of alternate stacking of Fe and Pt atomic planes along the face-centered tetragonal (fct) [001] direction. The enlarged Fe-Fe distance,compared with their bulk phase, narrows the bandwidths and enhances the exchange splitting of band structure. 18Inaddition, Pt acquires a sizable spin polarization in contact with Fe and contributes signiﬁcantly to the PMA, due to itslarge SOC. In principle, the tunability of PMA by SOC strength ( n) is signiﬁcant as predicted by various theoretical models. 18–22 However, an experimental method to continuously tailor PMA by exploiting its dependence on nhas not been well demonstrated, and a systematic analysis is still lacking. Thechallenge lies in the fact that PMA is also affected by other physical parameters, such as magnetic moment, lattice con- stant, and bandwidth, 18,20,23,24in addition to n, which may change signiﬁcantly when nis altered by using various metals and alloys.25–29It is difﬁcult to deconvolute the under- lying mechanisms of spin-orbit interaction tuning of PMA inthose systems. L1 0ordered FePd (1/C0x)Ptxalloy is one promis- ing candidate for such purpose of systematic study.30,31The element Pt falls into the same group as Pd in the periodictable, and n (Pt)is stronger than n(Pd). Continuous substitution of Pd by Pt leads to a gradual increase of ninL10ordered FePd (1/C0x)Ptxalloy, while the magnetic moment and lattice structure remains almost the same. In this paper, we demon- strate tailoring PMA through spin-orbit interaction tuning in L10ordered FePd (1/C0x)Ptxternary alloy ﬁlms. We investigate uniform magnetization precession in the frequency range of 14–340 GHz for various samples by time-resolved magneto- optical Kerr effect (TRMOKE) experiments. The measuredPMA increases by almost an order of magnitude from 0.8 to 6.0 (10 7erg/cm3) and the coercivity ﬁeld increases from almost zero to 4.4 kOe, when Pd atoms are completelyreplaced by Pt. Ab initio calculations predict a nearly quad- ratic dependence of PMA ( K u)o n n, consistent with experi- ment. Structure characterization and ab initio calculations show that other physical parameters affecting PMA remain nearly unchanged. The spin-orbit interaction tuning of PMAa)Electronic addresses: gyguo@phys.ntu.edu.tw; hbzhao@fudan.edu.cn; and gxluep@wm.edu. 0003-6951/2014/104(19)/192402/5/$30.00 VC2014 AIP Publishing LLC 104, 192402-1APPLIED PHYSICS LETTERS 104, 192402 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 35.13.245.22 On: Sat, 06 Dec 2014 02:01:53is further enhanced by reducing the chemical disorder and thermal spin ﬂuctuations. A series of L10FePd (1/C0x)Ptxternary alloy ﬁlms with 0/C20x/C201 are deposited on single crystal MgO (001) sub- strates by magnetron sputtering. The FePd (1/C0x)Ptxcompos- ite target is fabricated by placing small Pt and Pd pieces on a Fe target. The base pressure of the deposition system is1.0/C210 /C05Pa and the Ar pressure is 0.35 Pa. During deposi- tion, the substrates are kept at 620/C14C and the rate of deposi- tion is about 0.1 nm/s. After deposition, the samples are annealed in situ at the same temperature for 2 h. The ﬁlm thickness is determined by x-ray reﬂectivity to be1761 nm. The microstructure analysis is performed by using X-ray diffraction (XRD), with Cu Karadiation. In order to measure the PMA, TRMOKE measurements areperformed, in which the equilibrium state is perturbed dur- ing laser pulse excitation, and the resulting magnetization precession dynamics allows us to derive the PMA from pre-cession motion simulated with the Landau-Lifshitz-Gilbert (LLG) equation. 31–36TRMOKE measurements are per- formed in a pump-probe setup using pulsed Ti:sapphire laserwith a pulse duration of 200 fs and a repetition rate of 250 kHz. 31The wavelength of pump (probe) pulses is 400 nm (800 nm). The intensity ratio of the pump to probepulses is set to be about 6:1. A variable magnetic ﬁeld Hup to 6.5 T is applied at an angle of h H¼45/C14with respect to the ﬁlm normal direction using a superconducting magnet. Thegeometry of external magnetic ﬁeld application and magnet- ization precession is depicted in Fig. 1(a). Static magnetiza- tion hysteresis loops are measured by vibrating samplemagnetometer at room temperature.Structure characterization of FePd (1/C0x)Ptxsamples is performed with XRD measurement, as presented in Fig 1(b). Only fct (001) and (002) peaks of FePtPd are observed in the spectrum along with other peaks from MgO substrate, whichindicates the L1 0ordering in the FePtPd alloys. The chemical ordering parameter ( S) is deﬁned as S2¼I001 I002/C16/C17 meas I001 I002/C16/C17 calc¼rFeþrPt PdðÞ/C01; (1) where I001andI002are integrated intensity of fct (001) and (002) peaks in such superlattice systems,37–39andrFe,rPt(Pd) is the probability of the correct site occupation for Fe and Pt(Pd) atoms. ( I001/I002)calcis calculated to be 2.0 for the ﬁlm thickness ranging from 11 to 49 nm.37Thus, Sis about 0.8, indicating a good suppression of disordering defects in these ﬁlms and about 90% of atoms stay at the correct sites. Asdepicted in Fig. 1(b), the peak positions do not shift as x varies from 0 to 1, which indicates that the lattice constant varies by less than 1.0% for different x. It is shown in Ref. 18that tetragonal distortion of the lattice can lift the degen- eracy of the orbital occupancy for delectrons and affect PMA, and the lattice constant c/aratio varies only about 1.4% between FePt and FePd alloys. This is consistent with our characterization results that tetragonal distortion of the lattice for FePtPd alloys is not affected by doping. Figures 1(c)–1(e) display the out-of-plane and in-plane magnetization hysteresis loops for x¼1.0, 0.5, and 0, respectively. As shown in Fig. 1(c),f o r x¼1(L1 0FePt) the out-of-plane hysteresis loop is almost square-shaped with FIG. 1. Schematic of TRMOKE mea- surement geometry (a) and structure characterization results by XRD (b). Static magnetic hysteresis loops meas- ured by vibrating sample magnetometer(VSM) (c)–(e) and TRMOKE data with different doping level of FePtxPd(1 /C0x) under magnetic ﬁeld H¼5 T (f). The solid lines are ﬁtted curves.192402-2 Ma et al. Appl. Phys. Lett. 104, 192402 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 35.13.245.22 On: Sat, 06 Dec 2014 02:01:53coercivity Hc¼0.44 T; but it is hard to reach the saturated magnetization with in-plane ma gnetic ﬁeld, indicating the establishment of high PMA. With decreasing x,Hc decreases due to the reduced PMA. For x¼0i nF i g . 1(e), Hcapproaches zero and the out-of-plane and in-plane hys- teresis loops almost overlap wi th each other, indicating weak PMA. The PMA therefore i ncreases with increasing c o n c e n t r a t i o no fP ti nF e P d (1/C0x)Ptxa l l o yt h i nﬁ l m s .F r o m the experiments, the saturation magnetization Msfor all samples is determined to be 1100 emu/cm3within 10% rela- tive error, close to the bulk value of L10FePt.40 Figure 1(f) shows TRMOKE results of FePd (1/C0x)Ptx ﬁlms with various xatH¼5 T. A uniform magnetization precession is excited as manifested by the oscillatory Kerr signal hk, while the magnetic damping is indicated by the decaying precession amplitude as the time delay increases. As shown in Fig. 1(f), the measured Kerr signal can be well ﬁtted by the following equation: hk¼aþb* exp(/C0t/t0) þA*exp(/C0t/s)sin(2 pftþu), where parameters A,s,f, and u are the amplitude, magnetic relaxation time, frequency, and initial phase of the magnetization precession, respectively.Here, a,b, and t 0are related to the background signal owing to the slow recovery process after fast demagnetization by laser pulse heating. It is well-demonstrated in Fig. 1(f)that the spin precession frequency and magnetic damping effect become larger and stronger for higher doping level xwith the same magnetic ﬁeld H. In order to derive the PMA for FePd 1/C0xPtxsamples at different doping levels, magnetic ﬁeld-dependent TRMOKE measurements are performed. As shown by the TRMOKEresults of x¼0.82 and x¼0.5 samples in Figs. 2(a)and2(b), spin precession frequency increases as Hincreases. The ﬁeld dependence of frequency can be understood by theenhancement of the effective ﬁeld due to larger H. The Hdependences of the precession frequency ffor some typical samples are displayed in Fig. 2(c). We note that fcan be widely tuned from 15 GHz to 340 GHz by varying the mag- netic ﬁeld and doping level. From the LLG equation withGilbert damping parameter a/C281.0, one can obtain the fol- lowing dispersion equation: 2pf¼cH 1H2ðÞ1=2; (2) where H1¼Hcos(hH/C0h)þHKcos2handH2¼Hcos(hH/C0h) þHKcos 2 h,HK¼2Ku/MS/C04pMSwith uniaxial magnetic anisotropy constant Ku, gyromagnetic ratio c,a n d hH¼45/C14. The equilibrium angular position hof the magnetization satis- ﬁes the following equation: sin 2 h¼(2H/HK)sin(hH/C0h). The measured ﬁeld dependence of fcan be well ﬁtted by Eq. (2),a s shown in Fig. 2(c). With the measured MSof 1100 emu/cm3, and the gfactor ﬁtted to be 2.1 60.05 for samples with differ- entx,Kuis derived from the dispersion and displayed in Fig. 2(d) as a function of doping level x.T h em e a s u r e d Ku exhibits a sensitive dependence on the chemical substitution in Fig.2(d). To investigate the physics of spin-orbit interaction tun- ing in FePd 1/C0xPtxdoping system, ab initio density functional calculations is performed to quantitatively determine the key physical parameters (see supplementary material43for calcu- lation details). The calculated n, bandwidth W, and magnetic moment m sat Fe or Pd/Pt site as a function of doping level x is derived and depicted in Figs. 3(a)–3(c). Figure 3(a)shows that nat Pd/Pt site changes from 0.19 to 0.57 eV when x varies from 0 to 1.0. For Pt, Pd, and Fe atoms, nis 0.6, 0.20, and 0.06 eV,30,31respectively, and therefore the effect of Fe atoms is negligible compared with those of Pd and Pt atoms.The change of spin moment and bandwidth are negligible when xis tuned from 0 to 1, as shown in Figs. 3(b) and3(c). FIG. 2. TRMOKE results of FePt0.82Pd0.18 (a) and FePt0.5Pd0.5(b) under different magnetic ﬁelds H. The dependence of precession fre- quency (c) on Hfor samples with x¼0, 0.5, 0.75, 0.9, and 1. The solid lines refer to ﬁtted results. PMA as a function of doping level xof FePt xPd(1/C0x)(d).192402-3 Ma et al. Appl. Phys. Lett. 104, 192402 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 35.13.245.22 On: Sat, 06 Dec 2014 02:01:53The Pt/Pd atom acquires spin polarization around 0.27 (lB/atom) due to the proximity effect. Moreover, the lattice constant remains unchanged for different xfrom both XRD results and theoretical structure analysis of L10FePdPt alloys,18andc/aratio is around 0.94 as shown in Fig. 3(d). Thus, the variation of PMA is mainly controlled by SOC strength through different doping level x. The ndependence of PMA derived from TRMOKE measurements (black squares) is plotted in Fig. 3(e). In our previous work,31we pointed out that the magnetic anisotropy arises from second-order energy correction of SOC in theperturbation treatment. Therefore, the PMA is roughly pro- portional to both nand orbital magnetic moment when nis smaller than the exchange splitting. 10Since the orbital moment is also proportional to nwithin perturbation theory;41the enhanced PMA at higher xis attributed to a larger nof Pt atoms compared with that of Pd atoms and a quadratic scaling law of PMA with nis expected.31 However, in our previous work, the measured Ku was inves- tigated without theoretical calculations and was ﬁtted empiri-cally as a quadratic function of n. 31In this study, the key physical parameters affecting Ku other than n, such as spin moment, are investigated by ﬁrst-principle calculations andstructure characterizations, which indicate that the control of PMA in these ternary alloys is governed by spin-orbit interaction tuning. Moreover, the calculated values of PMA(red spots in Fig. 3(e)) are larger than the measured ones at higher x. Several reasons exist for the discrepancy, such as slight differences in the lattice structure between experimentand theory, and thermal ﬂuctuations and chemical disorder are not accounted for in the theoretical calculations. Hence, there is potential for further enhancing PMA by SOC tuning. One approach to enhance the spin-orbit coupling de- pendence of PMA is increasing the chemical order of FePd 1/C0xPtxalloys, as the large PMA values in the theoretical study are based on an ideally ordered lattice. To estimate the inﬂuence of disorder, we perform TRMOKE experiments on the samples with higher chemical ordering S¼0.85, where the correct site occupation of atoms is increased to 92% from rFe¼rPt(Pd)¼90% ( S¼0.8). The measured PMA values, indicated by the blue rhombus symbols in Fig. 3(e), show a slight increase for x¼0.5 ( n¼0.26) and a noticeable enhancement for x¼1(n¼0.57). This indicates that the chemical ordering of FePd 1/C0xPtxalloys has a strong effecton the tunability of PMA by spin-orbit interaction, as the dis- ordering can affect the electronic band structure.42Also at lower temperature (20 K) the PMA values of FePd 1/C0xPtx alloys can be more enhanced by SOC strength due to the sup- pression of spin thermal ﬂuctuation, as shown in the supple-mentary material. 43Furthermore, other approaches such as modulation of tetragonal distortion and chemical substitution of 3d transition metal in alloys may also be utilized for SOCtuning of PMA. In summary, we demonstrate that PMA in L1 0 FePd (1/C0x)Ptx ternary alloy ﬁlms can be continuously tuned by SOC tuning with appropriate Pt/Pd concentration. In par- ticular, PMA is found to be nearly proportional to n2from ab initio density functional calculations, which is consistent with experimental results. The tunability of PMA is enhanced by increasing chemical order in FePd 1/C0xPtxalloys and by lowering the temperature. The present experimentalresults provide deeper insight into the correlation between PMA and nin magnetic metallic materials and are helpful to explore ideal ferromagnets with desirable PMA for applica-tions of magnetic devices. The TR-MOKE experiments, data analysis, and discus- sions performed at the College of William and Mary were sponsored by DOE through Grant No. DE-FG02-04ER46127. H.Z. acknowledges ﬁnancial support from National Natural Science Foundation of China (Grant Nos. 61222407 and51371052) and NCET (No. 11-0119). G.Y.G. thanks the National Science Council of Taiwan for ﬁnancial supports. 1D. Weller, A. Moser, L. Folks, M. E. Best, W. Lee, M. Toney, M. Schwickert, J.-U. Thiele, and M. Doerner, IEEE Trans. Magn. 36,1 0 (2000). 2D. Alloyeau, C. Ricolleau, C. Mottet, T. Oikawa, C. Langlois, Y. LeBouar, N. Braidy, and A. Loiseau, Nat. 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1.4813488.pdf	Spin-transfer torque magnetization reversal in uniaxial nanomagnets with thermal noise D. Pinna, A. D. Kent, and D. L. Stein Citation: J. Appl. Phys. 114, 033901 (2013); doi: 10.1063/1.4813488 View online: http://dx.doi.org/10.1063/1.4813488 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i3 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 22 Jul 2013 to 143.88.66.66. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsSpin-transfer torque magnetization reversal in uniaxial nanomagnets with thermal noise D. Pinna,1,a)A. D. Kent,1and D. L. Stein1,2 1Department of Physics, New York University, New York, New York 10003, USA 2Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA (Received 10 March 2013; accepted 23 June 2013; published online 15 July 2013) We consider the general Landau-Lifshitz-Gilbert (LLG) dynamical theory underlying the magnetization switching rates of a thin ﬁlm uniaxial magnet subject to spin-torque effects andthermal ﬂuctuations. After discussing the various dynamical regimes governing the switching phenomena, we present analytical results for the mean switching time behavior. Our approach, based on explicitly solving the ﬁrst passage time problem, allows for a straightforward analysis ofthe thermally assisted, low spin-torque, switching asymptotics of thin ﬁlm magnets. To verify our theory, we have developed an efﬁcient Graphics Processing Unit (GPU)-based micromagnetic code to simulate the stochastic LLG dynamics out to millisecond timescales. We explore theeffects of geometrical tilts between the spin-current and uniaxial anisotropy axes on the thermally assisted dynamics. We ﬁnd that even in the absence of axial symmetry, the switching times can be functionally described in a form virtually identical to the collinear case. VC2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4813488 ] I. INTRODUCTION More than a decade has passed since spin-torque effects were demonstrated experimentally by the switching of the magnetization of a thin ferromagnetic ﬁlm when current is passed between it and a pinned ferromagnetic layer.1–4A spin-polarized current passing through a small magnetic conductor will deposit spin-angular momentum into the magnetic system. This in turn causes the magnetic momentto precess and in some cases even switch direction. This has led to sweeping advances in the ﬁeld of spintronics through the development and study of spin-valves and magnetic tun-nel junctions (see, for example, Ref. 6). The theoretical approach to such a problem has conventionally been to treat the thin ferromagnetic ﬁlm as a single macrospin in the spiritof Brown. 5Spin-torque effects are taken into account phenomenologically by modifying the macrospin’s Landau- Lifshitz-Gilbert (LLG) dynamical equation.1A thorough understanding of the phenomena, however, cannot proceed without taking into account the effect of thermal ﬂuctuations. This is of particular experimental relevance since spin-transfer effects on nanomagnets are often conducted at low currents, where noise is expected to dominate. Recent debate in the literature over the proper exponential scaling behaviorbetween mean switching time and current shows how the thermally assisted properties of even the simplest magnetic setups leave much to be understood. 7–12The interplay between spin-torque and thermal effects determine the dynamical properties of recent experimental studies on nanopillar devices.13Except at very high currents where the dynamics are predominantly deterministic, the switching appears to be thermal in nature. Fitting to experimental data requires accurate knowledge of the energetics, which, in therealm of spin-torque, are hard to come by due to the inher- ently non-conservative nature of the spin-torque term. Theoretical progress has been hindered by the computa- tional power needed to run numerical simulations to thedesired degree of accuracy. The LLG equation modiﬁed into its set of coupled stochastic equations can be studied in one of two ways: either by concentrating on the associatedFocker-Planck equation or by constructing a stochastic Langevin integrator to be used enough times to gather sufﬁ- cient statistics on the phenomena. 14The latter approach, however, has been unable to extrapolate to long enough times to capture the dynamical extent of the thermal regime. Recent papers by Taniguchi and Imamura15and Butler,16 following arguments ﬁrst made by Suzuki,17suggest that previous analytics of the thermally assisted dynamics should be revisited. Nonetheless, no numerical simulation has yetbeen able to fully evaluate the accuracy of the Taniguchi and Imamura results without resorting to comparison with the ﬁeld switching model. 18In fact, as will be apparent in what follows, the applied current in the spin-torque model cannot always be interpreted mathematically as an applied magnetic ﬁeld. In our paper, we will show that simulations run har-nessing the vast computational parallelization capabilities intrinsic in Graphics Processing Unit (GPU) technology for numerical modeling can allow a deeper probing of such athermally activated regime. II. GENERAL FORMALISM A simple model of a ferromagnet uses a Stoner- Wohlfarth monodomain magnetic body with magnetizationM. The body is assumed to have a size l malong the eydirec- tion, and size ain both the exandezdirections. The total volume of the object is then V¼a2lm. The energy landscape experienced by Mis generally described by three terms: an applied ﬁeld H, a uniaxial anisotropy energy UKwith easya)Electronic address: daniele.pinna@nyu.edu 0021-8979/2013/114(3)/033901/9/$30.00 VC2013 AIP Publishing LLC 114, 033901-1JOURNAL OF APPLIED PHYSICS 114, 033901 (2013) Downloaded 22 Jul 2013 to 143.88.66.66. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsaxis along ^nKin the ex/C0ezmaking an angle xwith the ez axis, and an easy-plane anisotropy Upin the ey/C0ezplane, with normal direction ^nDperpendicular to ^nK. The magnet- ization Mis assumed to be constant in magnitude and for simplicity normalized into a unit direction vector m¼M=jMj. A spin-polarized current Jenters the magnetic body in the /C0eydirection, with spin polarization factor gand spin direction along ez. The current exits in the same direc- tion, but with its average spin direction aligned to that of M. The self-induced magnetic ﬁeld of the current is ignored here as the dimension ais considered to be smaller than 100nm, where the spin-current effects are expected tobecome dominant over the current induced magnetic ﬁeld. The standard LLG equation used to describe the dynamics is then written as _m¼/C0 c 0m/C2Heff/C0ac0m/C2ðm/C2HeffÞ /C0c0jm/C2ðm/C2^npÞþc0ajm/C2^np; (1) where c0¼c=ð1þa2Þis the Gilbert ratio, cis the usual gyromagnetic ratio 1 :76/C1/C21011ðrad T/C1sÞ, and j¼ð/C22h=2eÞgJis the spin-angular momentum deposited per unit time with g¼ðJ"/C0J#Þ=ðJ"þJ#Þthe spin-polarization factor of inci- dent current J. The last two terms describe a vector torque generated by current polarized in the direction ^np. These are obtained by assuming that the macrospin absorbs angularmomentum from the spin-polarized current only in the direc- tion perpendicular to m. 1 To write Heffexplicitly, we must construct a proper energy landscape for the magnetic body. There are three main components that need to be considered: a uniaxial anisotropy energy UK, an easy-plane anisotropy UP, and an external ﬁeld interaction UH. These are written as follows: UK¼/C0KVð^nK/C1mÞ2; UP¼KPVð^nD/C1mÞ2; UH¼/C0MSVm/C1Hext: In these equations, MSis the saturation magnetization, KPis the easy-plane anisotropy, K¼ð1=2ÞMSHK, and HKis the Stoner-Wohlfarth switching ﬁeld (in units of Teslas). In what follows, we will consider a simpliﬁed model in which theeffects of easy-plane anisotropy are ignored and all external magnetic ﬁelds are absent. However, we retain contributions due to magnetic ﬁelds in our derivations for reasons whichwill be apparent in Sec. III. The full energy landscape then becomes UðmÞ¼U KþUHand reads UðmÞ¼/C0 KV½ð^nK/C1mÞ2þ2h/C1m/C138; (2) where h¼Hext=HK;^nKis the unit vector pointing in the ori- entation of the uniaxial anisotropy axis. Such an energy land- scape generally selects stable magnetic conﬁgurations parallel and anti-parallel to ^nK. The effective interaction ﬁeldHeffis then given by Heff¼/C01 MSVrmUðmÞ¼HK½ð^nK/C1mÞ^nKþh/C138:(3)The symmetries of the problem lead to slightly simpliﬁed equations and the deterministic LLG dynamics can then be expressed as _m¼/C0m/C2½ ð ^nK/C1mÞ^nKþh/C138/C0am /C2½m/C2ð ð ^nK/C1mÞ^nKþhÞ/C138 /C0aIm/C2ðm/C2^kÞþa2Im/C2^k; (4) where we have deﬁned I¼j=ðaHKÞ, conveniently chose ^np as the orientation of our z-axis ( ^k) and introduced the natural timescale s¼c0HKt: III. THERMAL EFFECTS Thermal effects are included by considering uncorre- lated ﬂuctuations in the effective interaction ﬁeld: Heff !HeffþHth. These transform the LLG equation into its Langevin form upon performing the substitution h!Hthin (4). We model the stochastic contribution Hthby specifying its correlation properties, namely hHthi¼0; hHth;iðtÞHth;kðt0Þi ¼ 2Ddi;kdðt/C0t0Þ: (5) The effect of the random torque Hthis to produce a diffusive ran- dom walk on the surface of the M-sphere. An associated Focker- Planck equation describing such dynamics was constructed by Brown.5At long times, the system attains thermal equilibrium; and by the ﬂuctuation-dissipation theorem, we have D¼akBT 2KVð1þa2Þ¼a 2ð1þa2Þn: (6) It is convenient to introduce the notation KV=kBT¼n, repre- senting the natural barrier height of the uniaxial anisotropy energy. Considering only thermal ﬂuctuations, the stochastic LLG equation reads _mi¼AiðmÞþBikðmÞ/H17034Hth;k; (7) where AðmÞ¼aI½am/C2^k/C0m/C2ðm/C2^kÞ/C138 /C0ð^nK/C1mÞ½m/C2^nK/C0að^nK/C0ð^nK/C1mÞmÞ/C138; BikðmÞ¼/C0 /C15ijkmj/C0aðmimk/C0dikÞ; and “/C14Hth;k” means to interpret our stochastic dynamics in the sense of Stratonovich33calculus in treating the multipli- cative noise terms.19 We numerically solve the above Langevin equations by using a standard second order Heun scheme to ensure proper convergence to the Stratonovich calculus. At each time step, the strength of the random kicks is given by the ﬂuctuation-dissipation theorem. Statistics were gathered from an ensem- ble of 5000 events with a natural integration stepsize of 0.01. For concreteness, we set the Landau damping constanta¼0:04. A magnetic ensemble was considered “switched” when half the members of the ensemble have reversed their033901-2 Pinna, Kent, and Stein J. Appl. Phys. 114, 033901 (2013) Downloaded 22 Jul 2013 to 143.88.66.66. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsdirection of magnetization. Different barrier heights were explored although the main results in this paper are shown for a barrier height of n¼80. To explore the simulations for long time regimes, the necessary events were simulated in parallel on an NVidia Tesla C2050 graphics card. To gener- ate the large number of necessary random numbers, we chosea proven combination 20of the three-component combined Tausworthe “taus88”21and the 32-bit “Quick and Dirty” Linear Congruential Generator (LCG).22The hybrid genera- tor provides an overall period of around 2121. IV. SWITCHING DYNAMICS In experiments, one is generally interested in under- standing how the interplay between thermal noise and spin torque effects switch an initial magnetic orientation from parallel to anti-parallel and vice versa. The role of spin-torque can be clariﬁed by considering how energy is pumped in the system, from an energy landscape point of view. As in Sec. III, the magnetic energy of the monodomain is UðmÞ¼/C0 KVð^n K/C1mÞ2: (8) The change in energy over time can be obtained after some straightforward algebra and is found to be 1 MsVH K_U¼/C0 ½ am/C2Heff/C0Iða^k/C0m/C2^kÞ/C138/C1ðm/C2HeffÞ:(9) This expression shows how current pumps energy into the system. In the absence of current, the damping dissipatesenergy and, as one would expect, the dynamical ﬂow is toward the minimum energy conﬁguration. The sign pre- ceeding the current term allows the expression to becomepositive in certain regions of magnetic conﬁguration space. Furthermore, by averaging over constant energy trajectories, one can construct an equivalent dynamical ﬂow equation inenergy space. This kind of approach has already been used in the literature 9and can lead to the appearance of stable limit cycles at currents less than the critical current as can beintuitively inferred by considering which constant energy trajectories lead to a canceling of the ﬂow in (9). Starting from an initially stable magnetic state, spin- torque effects will tend to drive the magnetization toward the current’s polarization axis. Once the current is turned off, the projection of the magnetization vector along the uniaxialanisotropy axis will almost surely determine which stable energy state (parallel or anti-parallel) the magnetic system will relax to as long as the energy barrier nis large enough. As such, switching dynamics are best studied by projecting Eq.(7)along the uniaxial anisotropy axis ^n K. One then obtains a stochastic differential equation describing thedynamics of such a projection _q¼a½ðn zIþqÞð1/C0q2ÞþnxIqp/C138 þa2Inxmyþﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃa nð1/C0q2Þr /H17034_W: (10) In the above equation, we have deﬁned q/C17m/C1^nK,pthe analogous projection onto an axis perpendicular to ^nK.Furthermore, ^nKis taken to lie in the x-z plane (such that we are projecting mon axes rotated about the y-axis); nzandnx are the projections of the uniaxial anisotropy axis, respec- tively, on the zand xaxes. Furthermore, the multiple stochastic contributions are assumed to be Gaussian random variables with identical average and space dependentvariance. _Wis a standard mean zero, variance 1, Wiener pro- cess, and its prefactor explicitly expresses the strength of the stochastic contribution. 19As will be shown in the following Subsections, (10) is a convenient analytical tool in speciﬁc scenarios. In general, it is not useful as it explicitly depends on the dynamics of both the mzandmycomponents of the magnetization. Numerically, we can solve (7)directly. In all our simula- tions, the initial ensemble of magnetizations was taken to beBoltzmann-distributed along the anti-parallel orientation. We assume that the energy barrier height is so large that, before current effects are activated, thermalization has only beenachieved within the antiparallel energy well and no states have had time to thermally switch to the parallel orientation on their own. A typical histogram of magnetic orientations ata given time is shown in Figure 1. We now turn on a current and allow the system to evolve for a ﬁxed amount of time. Once this time has passed,we use the projection rule expressed above to evaluate what fraction of the ensemble has effectively switched from the anti-parallel to the parallel state. V. COLLINEAR SPIN-TORQUE MODEL Having derived the necessary expressions for our macro- spin model dynamics, it is useful to consider the following simpliﬁcation. Let us take the uniaxial anisotropy and spin-current axes to be collinear, namely, ^n K/C17^np/C17^k. In such a scenario, the stochastic LLG equation simpliﬁes signiﬁ- cantly. In particular, (10)reduces to the simpliﬁed form FIG. 1. Histogram distribution of mzafter letting the magnetic system relax to thermal equilibrium (103natural time units). The overlayed red dashed line is the theoretical equilibrium Boltzmann distribution. In the inset, we show a semilog-plot of the probability vs. m2 zdependency. As expected, the data scale linearly.033901-3 Pinna, Kent, and Stein J. Appl. Phys. 114, 033901 (2013) Downloaded 22 Jul 2013 to 143.88.66.66. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions_q¼aðIþqÞð1/C0q2Þþﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃa nð1/C0q2Þr /H17034_W: (11) In this symmetric scenario, qcoincides with mzand magnet- ization reversal has been reduced to a straightforward 1-D problem. For a general value of I<1, the evolution of qhas two local minima and a saddle. The two stable conﬁgurations are at q¼/C01 and q¼1, while the saddle is located at q¼/C0I. For currents I>Ic¼1, there is only one stable minimum. Above critical current, spin torque pushes all magnetic conﬁgurations toward the parallel q¼1 state. This regime is particularly important not just for its simplicity butalso for its similarity to the pure ﬁeld switching model. The collinear spin-torque model is, in fact, mathematically identi- cal to a ﬁeld switching model with applied ﬁeld of intensityIapplied parallel to the uniaxial anisotropy axis of the magnetic system. 17 A. Collinear high current regime In the high current regime I/C29Ic, we expect the deter- ministic dynamics to dominate over thermal effects. We refer to this also as ballistic evolution interchangeably. Thedeterministic contribution of (11) can then be solved analyti- cally given an initial conﬁguration q/C17m z¼/C0m0. The switching time sswill simply be the time taken to get from some mz¼/C0m0<0t omz¼0 and reads ssðm0Þ¼1 að0 /C0m0dm ðIþmÞð1/C0m2Þ ¼1 2aðI2/C01ÞIlog1þm0 1/C0m0/C20/C21 /C0log½1/C0m2 0/C138/C26 /C02 logI I/C0m0/C20/C21 /C27 : (12) Since the magnetic states are considered to be in thermal equilibrium before the current is turned on, one should aver- age the above result over the equilibrium Boltzmann distri-bution in the starting well to obtain the average switching timehs siB. For nlarge enough, such an initial distribution will be qBðmÞ¼ﬃﬃﬃnpexp½/C0n/C138 F½ﬃﬃﬃnp/C138exp½nm2/C138; (13) where F½x/C138¼expð/C0x2ÞÐx 0expðy2Þdyis Dawson’s integral. This expression can be used to compute the average switch- ing time numerically. As the intensity of spin-currents becomes closer to Ic, thermal effects increasingly contribute. Moreover, diffusion gradients add to the deterministic drift, which can be shown explicitly by writing (11)in its equivalent ^Ito form. Doing so leads us to a ﬁrst correction of the ballistic dynamics due to thermal inﬂuences. The z-component behavior then reads _mz¼aðIþmzÞð1/C0m2 zÞ/C0a 2nmzþﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃa nð1/C0m2 zÞr _W:(14) The ﬁrst term on the right hand side is still the ballistic ﬂow that we have just discussed. The second term is the desireddiffusion-gradient drift term. The contribution of such a term generates a net motion away from the stable minima of the ballistic equations as one expects to see under the inﬂuenceof thermal effects. Again, we can solve the drift dominated ﬂow analytically to compute the switching time. Considering diffusion-gradient drift, this reads s sðm0Þ¼1 að0 /C0m0dm ðIþmÞð1/C0m2Þ/C0ð m=2nÞ ¼1 aX jlog½1þðm0=wjÞ/C138 3w2 jþ2Iwj/C01/C01 2n/C18/C19 ; (15) where the wjare the three zeros of the cubic equation w3þIw2/C0ð1/C01 2nÞw/C0I¼0. As before, the average switching hssiBtime will simply be given by averaging numerically over the Boltzmann distribution qB. In Figure 2, the reader can see how well these two limiting results ﬁt thesimulation data. As expected, both expressions coincide in the limit of high currents. VI. UNIAXIAL TILT In the high current regime ( I/C29Ic), where ^nK¼^k(i.e., the uniaxial axis is aligned with the z-axis), the ballistic equa-tion for m zwas shown to decouple from the other components, and the dynamics became one dimensional and deterministic. For the more general case where the uniaxial anisotropy axismay have any tilt with respect to the z-axis, such a critical cur- rent is not as intuitively deﬁned. Unlike the collinear limit, a critical current, above which all magnetic states perceive a netﬂow towards an increasing projection, does not exist. One can plot _q/C17_m/C1^n Kover the unit sphere to see what regions allow for an increasing and decreasing projection as the current ischanged. An example of this is shown in Figure 3. Unfortunately, regions characterizing negative projec- tion ﬂow can be shown to persist at all currents. The FIG. 2. Blue line shows the ﬁt of the ballistic limit to the numerical data (in blue crosses). Red line shows the improvement obtained by including diffu- sion gradient terms. Times are shown in units of ( T/C1s) where Tstands for Tesla: real time is obtained upon division by HK.033901-4 Pinna, Kent, and Stein J. Appl. Phys. 114, 033901 (2013) Downloaded 22 Jul 2013 to 143.88.66.66. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsapproach is reﬁned by requiring that on average, over constant-energy precessional trajectories, the ﬂow is towardthe positive uniaxial anisotropy axis: h_m/C1^n Ki>0.8Such trajectories are found by solving the ﬂow equations with I¼a¼0. Solutions correspond to circular libations about the uniaxial anisotropy axis. The critical current is then rede- ﬁned to be the minimum current at which the average projec- tional ﬂow is positive at all possible precessional energies.This is easily done and results in I/C21max /C15/C0/C15 cosx/C20/C21 ¼1 cosðxÞ¼Icrit; (16) thus allowing for a direct comparison of dynamical switch- ing results between different angular conﬁgurations of uniax- ial tilt. In our discussion of (10), we mentioned how, in the general case, presenting uniaxial tilt, there is no way to reduce the dimensionality of the full dynamical equations. In fact, in the presence of tilt, precessional trajectories mightallow for a magnetization state to temporarily transit through aq>0, “switched” conﬁguration, even though it might spend the majority of its orbit in a q<0, “unswitched” con- ﬁguration. This allows for a much richer mean switching time behavior, especially for currents greater than the critical current, as shown in Figure 4and discussed more in depth later. VII. THERMALLYACTIVATED REGIME For currents I<Ic, the switching relies on thermal effects to stochastically push the magnetization from oneenergy minimum to the other. It is of interest to understand how switching probabilities and switching times depend on temperature and applied current. This is easily done in sto-chastic systems with gradient ﬂow. In such cases, an energy landscape exists and a steady state probability distributioncan be constructed to be used via Kramer’s theory in deriv- ing approximate low-noise switching probabilities. Unfortunately, spin-torque effects introduce a non- gradient term, and the resulting LLG equation does not admit an energy landscape in the presence of applied current. Thecollinear simpliﬁcation, however, is an exception. As already described, in the absence of uniaxial tilt, the dynamics become effectively one dimensional since the m zcomponent decouples from the other magnetization components. Consider then (11): because it is decoupled from the other degrees of freedom, we can construct a corresponding one-component Focker-Planck equation. The evolution in time of the distribution of qis then @ tqðq;tÞ¼ ^L½q/C138ðq;tÞ; (17) where ^L½f/C138¼/C0 a@qðqþIÞð1/C0q2Þ/C01 2nð1/C0q2Þ@q/C20/C21 f:(18) For high energy barriers and low currents, the switching events from one basin to the other are expected to be rare.The probability of a double reversal should be even smaller. We therefore model the magnetization reversal as a mean ﬁrst passage time (MFPT) problem with absorbing bounda-ries at the saddle point. The MFPT will then be given by the solution of the adjoint equation ( ^L †hsiðqÞ¼/C0 1)23 a 2nexpð/C0nðqþIÞ2Þ@q½ð1/C0q2ÞexpðnðqþIÞ2Þ/C138@qsðqÞ¼/C0 1 (19) subject to the boundary condition hsið0Þ¼0. This can be solved to give hsiðqÞ¼2n að0 qduexpð/C0nðuþIÞ2Þ 1/C0u2ðu /C01dsexpðnðsþIÞ2Þ:(20) FIG. 3. _q: green >0, red <0 for applied current I¼5. The plane dissecting the sphere is perpendicular to the uniaxial anistropy axis. Its intersection with the sphere selects the regions with highest uniaxial anisotropy energy. FIG. 4. Mean switching time behavior for various angular tilts above critical current obtained by numerically solving (7). Each set of data is rescaled by its critical current such that all data plotted has Ic¼1. Angular tilts are shown in the legend in units of p=36 such that the smallest angular tilt is 0 and the largest is p=4. Times are shown in units of ( T/C1s) where Tstands for Tesla: real time is obtained upon division by HK.033901-5 Pinna, Kent, and Stein J. Appl. Phys. 114, 033901 (2013) Downloaded 22 Jul 2013 to 143.88.66.66. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsThe rightmost integral can be computed explicitly. Retaining only dominant terms, the ﬁnal integral can be computed by saddlepoint approximation to give hsi’2ﬃﬃﬃpp aexpðnð1/C0IÞ2ÞFðﬃﬃﬃnpð1/C0IÞÞ 1/C0I2: (21) Such a square exponential dependence has recently been derived by Taniguchi and Imamura15,24as well as Butler et al. ,16although a s/expðnð1/C0IÞÞdependence, proposed elsewhere in the literature,7,9,10has also been successfully used to ﬁt experimental data.13,34 To decide between these experimental dependences, we ﬁt the scaling behaviors in Figure 5, along with the theoretical prediction from (24). The square exponential dependence ﬁts the data better, conﬁrming analytical results. Furthermore, comparison of the asymptotic expression (21)to the full theo- retical prediction obtained by solving (20) numerically dem- onstrates that even for mean switching times of the order 10/C01T/C1ms, asymptotically still is not fully achieved. All that remains is to consider the effects of angular tilt on the switching properties in the thermally activated regime. Insight into this problem can be obtained by invok- ing(12) again. For small values of a, the term in square brackets is of leading order over the second ballistic term depending on my. This allows us, in the small aregime, to neglect the second ballistic term altogether. We now concentrate on the behavior of the term in square brackets. For low sub-critical currents, switching will depend on thermal activation for the most part. We expect aninitially anti-parallel conﬁguration to not diffuse very far away from its local energy minima. It will remain that way until a strong enough thermal kick manages to drive it acrossthe energy barrier. Because of this, the second term appear- ing in the square brackets will generally be close to zero as the particle awaits thermal switching. To make the statementmore precise, one can imagine the magnetic state precessing many times before actually making it over the saddle. The second term can then be averaged over a constant energytrajectory where pwill vanish identically. Hence, in the subcritical regime, (12) can be rewritten in the following approximate form: _q’aðn zIþqÞð1/C0q2Þþﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃa nð1/C0q2Þr /H17034_W: (22) This is reminiscent of the 1D projectional dynamics dis- cussed in relation to the collinear limit and shown explicitlyin(11). The only difference between the two is the substitu- tion I!n zI; recall because nz¼cosðxÞ;nzI¼I=Ic.I n other words, the thermally activated dynamics are the samefor all angular separations up to a rescaling by the critical current. We then expect that the mean switching time dependences remain functionally identical to the collinearcase for all uniaxial tilts. We have conﬁrmed this by compar- ison with data from our simulations, and the results are shown in Figure 6. As predicted, all mean switching time data from different uniaxial tilts collapses on top of each other after a rescaling by each tilt’s proper critical current. As already hinted in the introduction to the collinear model, a strong mathematical analogy exists with the ﬁeld switching model studied in the literature. 25–31By consider- ing a macrospin model with external magnetic ﬁeld appliedin the direction of the uniaxial anisotropy axis, one obtains a dynamical equation for the magnetization vector analogous to(11) with the ﬁeld strength in place of the applied current. In fact, one can think of writing an effective energy land- scape Uðm zÞ¼/C0 Kðm2 z/C02ImzÞ, in terms of which the equi- librium Boltzmann distribution has precisely the same formas that given by Brown. 5The thermally activated behavior discussed in the literature also reproduces an exponent 2 FIG. 5. Mean switching time behavior in the sub-critical low current regime obtained by numerically solving (7). Times are shown in units of ( T/C1s)w h e r e Tstands for Tesla: real time is obtained upon division by HK. The red and green lines are born by ﬁtting to the data the functional form hsi¼Cexpð/C0nð1/C0IÞlÞ,w h e r e lis the debated exponent (either 1 or 2) and Cis deduced numerically. The red curve ﬁts the numerical data asymptoti- cally better the green curve. The difference between the red line and (21) is that our theoretical prediction includes a current dependent prefactor, which was not ﬁtted numerically. The differences between numerical data and (20) are due to numerical inaccuracies out to such long time regimes. The differen- ces between (20)and(21)quantify the reach of the crossover regime. FIG. 6. Mean switching time behavior in the sub-critical low current regime obtained by numerically solving (7). Various uniaxial tilts are compared by rescaling all data by the appropriate critical current value. Times are shown in units of ( T/C1s) where Tstands for Tesla: real time is obtained upon division by HK.033901-6 Pinna, Kent, and Stein J. Appl. Phys. 114, 033901 (2013) Downloaded 22 Jul 2013 to 143.88.66.66. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsscaling dependence such as what we have shown here. One must understand, that spin torque effects are generally non conservative and it is only in this collinear scenario that theymay be interpreted in this way. Upon introducing an angular tilt between uniaxial and spin-polarization axes, the analogy with the ﬁeld switching model will generally break (see(10)). It is interesting, to quantify the crossover between the spin torque and ﬁeld switched macrospin model. Coffey 25,26 has already discussed the effect of angular tilt between ani- sotropy and applied switching ﬁeld axes. We introduced noise in the macrospin model by considering a random applied magnetic ﬁeld. In (7), we showed the full form of the dynamical equations. To write the dynamical equations for the ﬁeld switched model, we need to suppress current effects and simply introduce a term identical to the stochastic contri-bution with the exception that now the applied ﬁeld will not be random but ﬁxed at a speciﬁc angular separation from the uniaxial anisotropy axis. Writing the dynamical equationsfor the ﬁeld switched model is straightforward _q¼a½ðn zhþqÞð1/C0q2Þþnxhðmy/C0aqpÞ/C138þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃa nð1/C0q2Þr /H17034_W; (23) where instead of an applied current I, the dynamics depend on an applied ﬁeld with strength h. Comparison with (10) shows how, in general, the two evolution equations are verydifferent from each other. In the thermally activated regime, however, where one is able to average over constant energy trajectories due to their timescales being much smaller thanthose required for actual diffusion or magnetic torque (h/C28h crit), the second term in square brackets will be aver- aged away and one is left with a stochastic evolution equa-tion identical to (22). All thermally activated switching will then again be functionally identical for all angular tilts up to a trivial rescaling of the applied ﬁeld. In comparing our scaling relationships between current and mean switching time with the previous literature, a subtle issue must be addressed. Results obtained by Apalkov andVisscher 9rely upon an initial averaging of the dynamics in energy space over constant energy trajectories (limit for small damping) and only subsequently applying weak noise meth-ods to extrapolate switching time dependences. The small damping and weak noise limits are singular and the order in which they are taken is important. Both limits radically alterthe form of the equations: whereas both limits suppress ther- mal effects, the ﬁrst also severely restrains the deterministic evolution of the magnetic system. Our approach considers theweak noise limit and, only in discussing the effects of an angular tilt between polarization and easy axes do we employ the small damping averaging technique to obtain functionalforms for the mean switching time. The switching time data shown seem to justify, in this particular case, an interchange- ability between these two limits. More generally, however,one should not expect the two limits to commute. VIII. SWITCHING TIME PROBABILITY CURVES Up until now, we have analyzed the main properties of spin-torque induced switching dynamics by concentratingsolely on the mean switching times. In experiments, one gen- erally constructs full probability curves. The probability that a given magnetic particle has a switching time ss/C20scan be explicitely written as P½ss/C20s/C138¼ðmðsÞ 0dxqBðxÞ ¼exp½/C0nð1/C0mðsÞ2Þ/C138F½ﬃﬃﬃnpmðsÞ/C138 F½ﬃﬃﬃnp/C138; (24) where mðsÞis the initial magnetization that is switched deter- ministically in time s. Once one has evaluated mðsÞ, the prob- ability curve follows. Ideally, in the ballistic regime, one would like to invert the ballistic equations. Unfortunately, thesolutions of such ballistic equations are generally transcenden- tal and cannot be inverted analytically. Even in the simpler collinear case, as can be seen from Eqs. (12)and(15), no ana- lytical inversion is possible. One must instead compute the inversion numerically. 35Nonetheless, one can construct appropriate analytical approximations by inverting the domi-nant terms in the expressions. In the case of (12),f o re x a m p l e , one has that for currents much larger than the critical current, sðmÞ’I 2aðI2/C01Þlog1þm 1/C0m/C20/C21 ; (25) which can be inverted to give mðsÞ¼tanh asI2/C01 I/C20/C21 : (26) Plugging into expression (24)for the sprobability curve, one has P½ss/C20s/C138¼exp/C0n cosh2asI2/C01 I/C20/C212 643 75Fﬃﬃﬃnptanh asI2/C01 I/C20/C21 /C20/C21 F½ﬃﬃﬃnp/C138: (27) This expression can be truncated to a simpler form by noting that the leading exponential term dominates over the ratio ofDawson functions. Furthermore, if one considers the limit of large values for s(or, analogously, I/C291), the “cosh” can be also approximated by its leading exponential term. We areﬁnally left with P½s s/C20s/C138’exp/C0n cosh2asI2/C01 I/C20/C212 643 75 /C24exp/C04nexp/C02asI2/C01 I/C20/C21 /C20/C21 ; (28) which is very similar in form to what has already been reported and used for ﬁtting in the literature.8,13 In the low current regime, one constructs probability curves by considering the mean switching time and modelinga purely thermal reversal as a decay process with rate given by Eq.(21). The fraction of switched states then vary in time as033901-7 Pinna, Kent, and Stein J. Appl. Phys. 114, 033901 (2013) Downloaded 22 Jul 2013 to 143.88.66.66. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsPðmz>0Þ¼1/C0expð/C0t=hsiÞ: (29) Upon introducing uniaxial tilt, precessional effects can be witnessed directly on the switching probability curves in thesuper-critical regime. One expects that in the initial phases of switching, the fraction of switched states is sensitive to the time at which the current is turned off. One may acciden-tally turn off the current during a moment of transient pas- sage through the switched region along the precessional orbit. This was checked and veriﬁed from our numerical sim-ulations (see Figure 7). More generally, effects similar to the “waviness” seen in the mean switching time curves can be seen in the probability curves as well as the angle of uniaxialtilt is allowed to vary (see Figure 8). Only a numerical solu- tion of the LLG equation can bring such subtleties to light.IX. CONCLUSION We have constructed the theory underlying the dynamics of a uniaxial macrospin subject to both thermal ﬂuctuationand spin-torque effects. We then studied the subtle interplaybetween these two effects in aiding magnetization reversalbetween energy minima in a magnetically bistable system.Two regimes stand out in such a theory: a ballistic regime dominated by the deterministic ﬂow and a thermally activated regime where reversal is dominated by noise. In the ballisticregime, we discussed how to approximate the mean switchingtime behavior and found that corrections due to the diffusion-gradient term, arising from the stochastic equations, allow one to model the dynamics more accurately. In the thermally activated regime, we solved the relevant mean ﬁrst passage time problem and obtained an expressionfor the dependence of mean switching time on appliedcurrent. In doing so, the correct scaling was shown to behsi/expð/C0nð1/C0IÞ 2Þ, in contrast to the prevailing view thathsi/expð/C0nð1/C0IÞÞ. Analytical results were then compared with detailed numerical simulations of the sto-chastic LLG equation. The massively parallel capabilities ofour GPU devices have allowed us to explore the behavior ofmacrospin dynamics over six orders of temporal magnitude.Comparing to our analytical results, we suggest that thethermal asymptotic behavior is achieved fairly slowly incomparison to the switching timescales that have beenprobed experimentally. Different geometrical conﬁgurations of the uniaxial ani- sotropy axis with respect to the spin-current axis were shownto inﬂuence the thermally activated regime very minimally inasmuch as the currents were rescaled by the proper critical current of the angular setup. Only in the super-critical regimewere distinctions shown to exist due to complex precessionaland transient switching behavior. These results have important implications for the analy- sis of experimental data in which measurements of theswitching time versus current pulse amplitude are used todetermine the energy barrier to magnetization reversal.Clearly, use of the correct asymptotic scaling form is essen-tial to properly determine the energy barrier to reversal. Theenergy barrier, in turn, is very important in assessing thethermal stability of magnetic states of thin ﬁlm elements thatare being developed for long term data storage in STT-MRAM. Further work should address how these resultsextend to systems with easy plane anisotropy and situationsin which the nanomagnet has internal degrees of freedom,leading to a break down of the macrospin approximation. We also note that current ﬂow is a source of shot noise, which at low frequencies acts like a white-noise source in much the same way as thermal noise. It is therefore interest-ing to understand when this additional source of noise plays a role. For a magnetic layer coupled to unpolarized leads, the current induced noise on the magnetization dynamics was found to be CL=CR ð1þCL=CRÞ2V,32where Vis the voltage drop across the magnetic layer, while CL=CRis a dimensionless ratio characterizing the coupling strength of the magnetic layer tothe left (L) and right (R) leads. Thus, the noise is maximal FIG. 7. Inﬂuence of precessional orbits on transient switching as seen from the switching time probability curve in the supercritical current regime. The case shown is that of an angular tilt of p=3 subject to a current intensity of 2.0 times the critical current. Data were gathered by numerically solving (7). The non-monotonicity in the probability curve shows the existence of tran- sient switching. Times are shown in units of ( T/C1s) where Tstands for Tesla: real time is obtained upon division by HK. FIG. 8. Spin-torque induced switching time probability curves for various angular conﬁgurations of uniaxial tilt (a sample normalized current of 10 was used) obtained by numerically solving (7). A log-log y-axis is used following (28)to make the tails of the probability distributions visible.033901-8 Pinna, Kent, and Stein J. Appl. Phys. 114, 033901 (2013) Downloaded 22 Jul 2013 to 143.88.66.66. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions(V=4) for perfectly symmetrical couplings, and is smaller in the limit of highly asymmetric contacts. This basic behavior, and the order of magnitude of the effect, is not likely to bemodiﬁed by polarized leads. We argue that the temperatures at which the experiments have been performed current noise effects are not important. The experiments are performed atroom temperature where T¼300 K. For an all-metallic de- vice, such as a spin-valve nanopillar, the couplings are nearly symmetrical; and at the critical current, a typical voltagedrop across the magnetic layer is less than 10 lV or, equiva- lently, 1 K. For a magnetic tunnel junction device, Vcan be /C241 V. However, in this case, the coupling is asymmetric. One lead (L) forms a magnetic tunnel junction with the nanomagnet, while the other (R) a metallic contact. This gives C R=CL>104and a relevant energy /C241 K, again far lower than room temperature. It appears that current induced noise can only be important at room temperature for a nano-magnet coupled symmetrically between two tunnel barriers. ACKNOWLEDGMENTS The authors would like to acknowledge A. MacFadyen, Aditi Mitra, and J. Z. Sun for many useful discussions and comments leading to this paper. This research was supported by NSF-DMR-100657 and PHY0965015. 1J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 2L. Berger, Phys. Rev. B 54, 9353 (1996). 3J. 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Stratonovich calculus is then to be preferred over ^Ito calculus as ascertained by the Wong-Zakai theorem. 34It is important to note that experiments at ﬁxed temperature are by neces- sity performed over a limited range of I=Icand thus cannot truly distin- guish an exponent of 1 or 2. 35Easily achieved thanks to the monotonicity of their form.033901-9 Pinna, Kent, and Stein J. Appl. Phys. 114, 033901 (2013) Downloaded 22 Jul 2013 to 143.88.66.66. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
1.364877.pdf	Microwave permeability of ferromagnetic thin films with stripe domain structure O. Acher, C. Boscher, B. Brulé, G. Perrin, N. Vukadinovic, G. Suran, and H. Joisten Citation: Journal of Applied Physics 81, 4057 (1997); doi: 10.1063/1.364877 View online: http://dx.doi.org/10.1063/1.364877 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/81/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Structure, ferromagnetic resonance, and permeability of nanogranular Fe–Co–B–Ni films J. Appl. Phys. 99, 08M303 (2006); 10.1063/1.2165926 Microwave permeability spectra of ferromagnetic thin films over a wide range of temperatures J. Appl. Phys. 93, 7202 (2003); 10.1063/1.1555902 Ultrahigh frequency permeability of sputtered Fe–Co–B thin films J. Appl. Phys. 87, 830 (2000); 10.1063/1.371949 Analysis of the complex permeability of a ferromagnetic wire J. Appl. Phys. 85, 5456 (1999); 10.1063/1.369974 Control of the resonance frequency of soft ferromagnetic amorphous thin films by strip patterning J. Appl. Phys. 81, 5166 (1997); 10.1063/1.365158 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.189.170.231 On: Mon, 22 Dec 2014 05:15:34Microwave permeability of ferromagnetic thin ﬁlms with stripe domain structure O. Acher, C. Boscher, B. Brule ´, and G. Perrin CEA-DMAT, BP12 F-91680, Bruye `res le Cha ˆtel, France N. Vukadinovic Dassault Aviation, F-92552 St Cloud Cedex, France G. Suran CNRS, Laboratoire Louis Ne ´el, F-38000 Grenoble, France H. Joisten CEA-LETI, F-38054 Grenoble Cedex 9, France We report the observation of multiple permeability peaks for thin ferromagnetic ﬁlms, in the 10 MHz to 6 GHz range. This behavior is correlated with the presence of perpendicular anisotropy andstripe domains. Because the perpendicular anisotropy is much smaller than the saturationmagnetization of the layer, we propose an adaptation of the classical domain mode resonance modelto the conﬁguration with oblique magnetization in the stripes. © 1997 American Institute of Physics. @S0021-8979 ~97!30408-3 # I. INTRODUCTION Extensive ferromagnetic resonance ~FMR !studies have been performed on ferromagnetic thin ﬁlms. The uniformmodes and spin wave modes are quite well understood. Incontrast, the gyromagnetic resonance of ferromagnetic thinﬁlms in the absence of magnetic ﬁeld has seldom been stud-ied. Recently, several observations of multiple narrow reso-nance modes on ferromagnetic thin ﬁlms have beenreported. 1,2The aim of this article is to give new experimen- tal results related to this phenomenon, and to show that thesemodes should be attributed to domain modes and spin waves. II. EXPERIMENT Amorphous ferromagnetic CoFeZr thin ﬁlms have been manufactured by ion beam sputtering. Thickness was in the0.25–0.6 mm range. The measurement of magnetic proper- ties of CoFeZr using a B–H plotter is described elsewhere.2 The ﬁne domain structure was investigated using an atomicforce microscope with magnetic force microscopy ability~AFM-MFM !. The permeability up to several GHz was mea- sured using a permeameter. 3The samples exhibited a hyster- esis loop typical of ﬁlms with biaxial anisotropy. The in-plane component of the anisotropy K iand the out-of-plane component K'were determined using FMR measurements.2 K'was found to be in the 4 3104–2.33105erg cm23range. This corresponds to out of plane anisotropy ﬁelds in the120–500 Oe range, much smaller than 4 pMs. The samples also exhibited rotatable anisotropy. The AFM-MFM pictureshown in Fig. 1 clearly reveals a stripe domain pattern. It iswell known that stripe domain structure and rotatable anisot-ropy are often associated. 4The typical frequency permeabil- ity behavior is illustrated in Fig. 2 for two different samples.In contrast with previous work, 1,2we have investigated the in-plane permeability both along the in-plane hard axis yand along the easy axis x. Some striking features of these spectra are the multiplicity of the peaks, the sharpness of the reso-nances, and the signiﬁcant levels of permeability in the di-rection parallel to the stripes, along the easy axis @Fig. 2 ~b!#. This contrasts with the behavior usually observed on ferro-magnetic ﬁlms, corresponding to conventional uniform FMRmodes. 5In many cases @see Figs. 2 ~a!and 2 ~b!, and also, to some extent, Figs. 4 and 5 in Ref. 1 #, the frequency spacing between the peaks seems to be nearly constant. In Fig. 2 ~a!, the frequency spacing between the three peaks along the hardaxis is nearly 1.4 GHz. In Fig. 2 ~b!, the frequency spacing between the pair of peaks along the easy and hard axes isclose to 1.9 GHz. III. THEORETICAL APPROACH AND DISCUSSION The peaks are observed at frequencies too high to be due to domain wall movements, so they must correspond to gy-romagnetic resonance. The presence of a gyromagnetic re-sponse along the stripe axis can be clearly related to the factthat the local magnetization has an out-of-plane component.The case where the spins are normal to the ﬁlm plane, alter-natively up and down in the stripes, has been investigatedboth theoretically and experimentally. 6,7It has been shown that this stripe conﬁguration led to two so-called domainmodes ~DM!. In particular, the lower frequency DM is ex- FIG. 1. AFM–MFM images of the stripe domains of a 0.5 mm CoFeZr layer ferromagnetic ﬁlm. 4057 J. Appl. Phys. 81(8), 15 April 1997 0021-8979/97/81(8)/4057/3/$10.00 © 1997 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.189.170.231 On: Mon, 22 Dec 2014 05:15:34cited by a ﬁeld along y~Fig. 3 !, and the higher frequency mode by a driving ﬁeld along x, parallel to the stripes. This is in qualitative agreement with our observations. However,quantitative calculations for CoFeZr samples using the val-ues ofK iandK'obtained from FMR measurement suggest that if the resonance frequency for an excitation along yFy has the right order of magnitude, the calculated Fxis much higher than the experimental value. In addition, the presenceof more than two excitation modes seems difﬁcult to accountusing DM theory. The presence of spinwaves ~SW!with deﬁnite boundary conditions can be suggested. In Ref. 1, SWwith magnetization and wave vector normal to the ﬁlm planewere shown to be a plausible but not completely satisfactoryway to ﬁt experimental results. It predicts a variation of F ny orFny2F0yasn2, which is clearly not what we observe. Be- sides, the anisotropy K'seems not large enough to lead to stripes with up and down magnetization, but rather with ob-lique magnetization 2~Fig. 3 !. We suggest that our observations may be accounted by DM in stripes with oblique magnetization, and SW in thestripe structure. To demonstrate this quantitatively we haveto establish the resonance frequencies of the system. This canbe done using the Smit and Beljers method, 6by differentiat- ing the energy function of the system. One has to consider asystem with two populations of stripes, the magnetization ineach population being represented in spherical coordinatesby u1,f1,u2, and f2as sketched in Fig. 3. It has been shown that in the absence of an external ﬁeld, if the ex-change energy is not taken into account, the equilibrium po-sitions correspond to u150 and u25p.6,7Since we want toconsider not only stripes with magnetization normal to the ﬁlm plane, but also weaker stripes, we have to introduce anexchange term. Since no exchange term can be associatedwith a discontinuous magnetization proﬁle, we take the ex-change term reported in Ref. 4 that approximates the mag-netization proﬁle in weak stripes as a sawtooth proﬁle. Then,the energy function may be written G5K ' 2~sin2u11sin2u2!1Ki 2~sin2u1sin2f1 1sin2u2sin2f2!1p 2NzzMs2~cosu12cosu2!2 1p 2NyyMs2~sinu1sinf12sinu2sinf2!2 1p 2Ms2~cosu11cosu2!21AS4 DD2Su22u1 2D2 , ~1! whereDis the stripe period. The equilibrium position is obtained by imposing that the ﬁrst derivative of Gis null. It yields f15f250u15p2u25u, ~2a! sin 2u~H'24pNzzMs!52A MsS4 DD2 ~p22u!. ~2b! FIG. 2. Complex permeability of two ferromagnetic ﬁlms, measured along the hard axis y, and along the easy axis x:~a!0.5mm CoFeZr layer ~similar to layer of Fig. 1 !,~b!CoFeZr layer deposited under different condition. 4058 J. Appl. Phys., Vol. 81, No. 8, 15 April 1997 Acheret al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.189.170.231 On: Mon, 22 Dec 2014 05:15:34The equation of motion for magnetization is linearized for small variations of uiandfi, which excludes the analysis of the case where ui50. One ﬁnds that the resonance frequen- cies are Fy5AV2V82andFx5AV1V81with ~3a! V2 g5H'cos 2u14pMssin2u 24pMsNzzcos2u;V82 g5Hi, ~3b! V1 g5Hi14pMsNyy, ~3c! V81 g52A MsS4 DD2F12p22u tan~p22u!G, ~3d! where 2K'5MsH'and 2Ki5MsHi. Fyis excited by a driving ﬁeld along y, andFxis excited by a driving ﬁeld along x. It is clear that Fxwill be very weakly excited if the magnetization makes an angle p/22ui small with the xaxis. This may account for the fact that, in several cases, the susceptibility along xaxis seems to be near unity through all the frequency range @Fig. 2 ~a!#. One can check that Fyyields the classical value of FMR frequency in the case the magnetization lies in the plane. The precessionof the two magnetization vectors is in phase for the F ymode, and in opposition of phase for the Fxmode. In a further approach, it is possible to linearize the equa- tion of motion including a damping term.7We choose to use the Gilbert damping term. This allows the determination ofthe full width at half maximum ~FWHM !DFof the m9peak. With the approximations a!1 and DF!F, it can be shown that DFx/y'2a~V61V86!'2aV6~4! in the regime with V6@V86. For the conventional uniform resonance mode, this yields DFy'2a(g4pMs). For the DM, DFyis much smaller because V2is smaller thang4pMs. As a consequence, the model can account for the very small FWHM observed on CoFeZr ~typically 100–200 MHz for DFyandDFx!compared to the same kind of sample after annealing, showing a conventional uniformresonance mode with 600 MHz linewidth. In addition, we suggest that SW directed in the ﬁlm plane along ymay be excited. They should match the peri- odicity conditions along y, so the wave vector should take the form k5p2 p/D. ~5! A simpliﬁed analysis of the coupling between the driving ﬁeld and the excitation modes suggests that a ﬁeld along y excites the modes with even p, and a ﬁeld along xexcites the modes with odd p. A detailed study of these spin modes with the stripe magnetization conﬁguration would clearly requireextensive calculations, but here we propose a more intuitiveand qualitative approach. It has been shown that, in manycases, the presence of spin waves could be accounted 9for by adding a term Hexeffto the anisotropy ﬁeld Hi Hexeff52A Ms4p2 D2p21Hex0. ~6! The point is that if Hexeff@Hex01Hi, which will be true for p2 large enough, then it is possible to approximate Fy g'p2p DA2A MsAV2. ~7! This corresponds to a linear variation of Fywithp, when the approximation is valid. It accounts for the fact that the fre-quency spacing between the modes excited by a ﬁeld along y is constant. In addition, it has been checked that, using rea-sonable values for N zzandu, Eqs. ~3!and~7!can account for the observed resonance frequencies. The indexing of the Fx modes is not clear at the moment. It should be mentioned that one cannot rule out the pos- sibility that the high order peaks correspond to SW with awave vector normal to the ﬁlm plane rather than along y.I n this case, the characteristic frequencies would be related notto the stripe period but to the layer thickness. 1Y. Shimada, M. Shimoda, and O. Kitakami, Jpn. J. Appl. Phys. 1 34, 4786 ~1995!. 2G. Suran, H. Niedoba, M. Naili, O. Acher, V. Meyer, C. Boscher, and G. Perrin, J. Magn. Magn. Mater. 157/158, 223 ~1996!. 3J. C. Peuzin and J. C. Gay, Acte des Journe ´es d’Etude sur la Caracte ´risa- tion Microonde, Limoges, 71 ~1990!. 4N. Saito, H. Fujiwara, and Y. Sugita, J. Phys. Soc. Jpn. 19, 1116 ~1964!. 5J. Russat, G. Suran, H. Ouhmane, M. Rivoire, and J. Sztern, J. Appl. Phys. 73, 1386 ~1993!. 6A. Layadi, F. W. Ciarello, and J. O. Artman, IEEE Trans. Magn. MAG- 23, 3642 ~1987!, and references therein. 7N. Vukadinovic, J. Ben Youssef, and H. Le Gall, J. Magn. Magn. Mater. 150, 213 ~1995!, and references therein. 8B. Lax and K. J. Button, Microwave Ferrites and Ferromagnetics ~McGraw-Hill, New York, 1962 !, p. 171. FIG. 3. Sketch of the magnetization conﬁguration in the domain modes resonance model. 4059 J. Appl. Phys., Vol. 81, No. 8, 15 April 1997 Acheret al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.189.170.231 On: Mon, 22 Dec 2014 05:15:34
1.2837665.pdf	Spin-polarized currents in exchange spring systems Matteo Franchin, Giuliano Bordignon, Thomas Fischbacher, Guido Meier, Jürgen Zimmermann et al. Citation: J. Appl. Phys. 103, 07A504 (2008); doi: 10.1063/1.2837665 View online: http://dx.doi.org/10.1063/1.2837665 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v103/i7 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 28 Apr 2013 to 142.51.1.212. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsSpin-polarized currents in exchange spring systems Matteo Franchin,1,2,a/H20850Giuliano Bordignon,1,2Thomas Fischbacher,2Guido Meier,3 Jürgen Zimmermann,2Peter de Groot,1and Hans Fangohr2 1School of Physics and Astronomy, University of Southampton, SO17 1BJ Southampton, United Kingdom 2School of Engineering Sciences, University of Southampton, SO17 1BJ Southampton, United Kingdom 3Institut für Angewandte Physik und Zentrum für Mikrostrukturforschung, Universität Hamburg, Jungiusstrasse 11, 20355 Hamburg, Germany /H20849Presented on 9 November 2007; received 12 September 2007; accepted 15 November 2007; published online 6 March 2008 /H20850 We present a computational study of the magnetization dynamics of a trilayer exchange spring system in the form of a cylindrical nanopillar in the presence of an electric current. Athree-dimensional micromagnetic model is used, where the interaction between the current and thelocal magnetization is taken into account following a recent model by Zhang and Li /H20851Phys. Rev. Lett. 93, 127204 /H208492004 /H20850/H20852We obtain a stationary rotation of the magnetization of the system around its axis, accompanied by a compression of the artiﬁcial domain wall in the direction of the electronﬂow. © 2008 American Institute of Physics ./H20851DOI: 10.1063/1.2837665 /H20852 INTRODUCTION The effects of spin-polarized currents on the magnetiza- tion of a ferromagnet have received considerable interest inrecent times 1–3after being proposed and studied in earlier works.4Spin-polarized currents may be used to generate mi- crowave oscillations in the magnetization of a ferromagnet5 or to switch the magnetization of a memory element.6The limit to these applications is currently found in the magni-tude of the current density required to obtain signiﬁcant ef-fects, which is of the order of 10 10–1012A/m2. Conse- quently, there is a strong interest in ﬁnding novel deviceswhere spin-torque effects are enhanced and require lowercurrent density. The recent discovery of signiﬁcant giantmagnetoresistance in exchange spring multilayers 7suggests that spin-transfer torque may play a role in these systems. Afurther reason, which makes exchange spring systems attrac-tive, is the possibility of creating “artiﬁcial” domain walls.Their length and shape—whose importance has been re-cently emphasized 8—can be controlled, ﬁrst during the engi- neering phase, then by applying a suitable magnetic ﬁeld.7 THE SYSTEM Consider a system whose ground state energy is degen- erate. It has inﬁnitely many different equilibrium conﬁgura-tions, which all have the same minimal energy and form acontinuous curve in the phase space. This system can be“dragged” through this curve, changing its conﬁgurationfrom one equilibrium state to another and this can beachieved “easily,” because there is no energy barrier betweenthem. In such a system, an electric current may ﬁnd a veryfavorable situation to fully manifest its effects. The idea is very simple but can serve as a guideline to develop micromagnetic systems where spin-transfer-torqueeffects are maximized. In this paper, we discuss a possibleexample of such a system. We study a trilayer exchange spring system in the form of a cylindrical nanopillar, where acentral magnetically soft layer is sandwiched between twomagnetically hard layers, as shown in Fig. 1. The system materials are chosen in the following way: YFe 2for the soft layer and DyFe 2for the two hard layers. This choice allows us to study the system with a model similar to the one usedin our previous work. 9Regarding the geometry, the diameter of the cylindrical nanopillar is 10 nm, while the thicknessesof the hard and soft layers are 5 and 40 nm, respectively. Yttrium has negligible magnetic moment and only two species of atoms contribute to the magnetization of the sys-tem. The ﬁrst one, iron /H20849Fe/H20850, is present in all the three layers and the second one, dysprosium /H20849Dy/H20850, is present only in the two hard layers. Neighboring iron moments are exchangecoupled, throughout all the hard and soft layers and acrossthe hard-soft interfaces. This coupling favors the alignmentof the magnetization of iron throughout the entire nanopillar.This alignment is, however, broken, because the magnetiza-tion of iron in the two hard layers is pinned along oppositedirections, as shown in Fig. 1. The pinning of the iron mo- ments is the result of the joint actions of two strong interac-tions: the cubic anisotropy of DyFe 2, which pins the dyspro- sium moments along an easy axis direction, and theantiferromagnetic coupling Dy–Fe, which transmits this pin-ning to the iron moments of the hard layers. a/H20850Electronic mail: franchin@soton.ac.uk. FIG. 1. /H20849Color online /H20850A sketch of the nanopillar which is studied in the paper /H20849not to scale /H20850. Dysprosium moments /H20849white arrows /H20850pin the iron mo- ments /H20849black arrows /H20850at the borders of the soft layer.JOURNAL OF APPLIED PHYSICS 103, 07A504 /H208492008 /H20850 0021-8979/2008/103 /H208497/H20850/07A504/3/$23.00 © 2008 American Institute of Physics 103, 07A504-1 Downloaded 28 Apr 2013 to 142.51.1.212. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsAmong all the interactions which we take into account, the cubic anisotropy of DyFe 2is the only one which is not symmetric under rotations around the axis of the nanopillar.However, in the case we are considering, where there is noapplied ﬁeld and the soft layer is relatively thick, the dyspro-sium moments keep their direction well aligned with the onethey initially have and the degeneracy of the ground state iswell preserved, as we will see from the results of the com-puter simulations. This means that conﬁgurations which dif-fer by a rotation around the xaxis have almost the same energy. Then if the applied current wants to rotate the wholemagnetization around the xaxis, nothing will oppose to its action. THE MODEL Since the density of iron atoms and their position in the lattice structure is the same for DyFe 2and YFe 2/H20849they both crystallize in laves phase structures /H20850, we use a single magne- tization vector MFeto describe the magnetic conﬁguration of iron in all the three layers. The conﬁguration of dysprosiumis modeled by another magnetization ﬁeld M Dywhich is de- ﬁned over the hard layers only. The model is similar to theone-dimensional model used in Ref. 9, extended to three dimensions /H20849the stray ﬁeld is calculated using the hybrid ﬁ- nite element method /H20849FEM /H20850/boundary element method 10,11/H20850. We also consider the same temperature /H20849100 K /H20850and the same material parameters. The moment densities of iron /H20849in both DyFe 2and YFe 2/H20850and dysprosium are /H20648MFe/H20648=0.55 /H11003106A/m and /H20648MDy/H20648=1.73/H11003106A/m, respectively. The easy axes for the anisotropy are u1=/H208490,1,1 /H20850//H208812,u2=/H208490, −1,1 /H20850//H208812, and u3=/H208491,0,0 /H20850. The coefﬁcients are K1=33.9 /H11003106J/m3, K2=−16.2 /H11003106J/m3, and K3=16.4 /H11003106J/m3. The effects of the electric current are modelled using the Zhang–Li correction to the Landau–Lifshitz–Gilbert equation. 12We assume that only the iron moments interact with the spin of the conduction electrons. The mag-netic electrons in the 4 forbitals of dysprosium are strongly localized at the ion core and their interaction with the con-duction electrons should be negligible. In the simulation thedamping parameter is chosen to be /H9251=0.02, the current den- sity is assumed to be fully polarized /H20849P=1/H20850and/H9264, the ratio between the exchange relaxation time, and the spin-ﬂip re- laxation time, is taken to be /H9264=0.01. The Oersted ﬁeld and the effects of Joule heating are ignored in the present study. RESULTS The simulations are performed by NMAG ,13a FEM-based micromagnetic simulation package. The cylindrical nanopil-lar is modeled by a three-dimensional unstructured mesh andﬁrst order FEM is used to discretize the space. In this case,FEM is preferable with respect to ﬁnite differences becauseit allows a better representation of the cylindrical geometry.Finite differences would introduce artifacts in the discretiza-tion of the rounded surface of the nanopillar. The initial magnetizations M FeandMDyare obtained by letting the system relax to one of its degenerate equilibriumconﬁgurations. The system then evolves from this conﬁgura-tion /H20849t=0/H20850up to t=10.5 ns. The dynamics of /H20855M Fe/H20856, the ironmagnetization averaged over all the nanopillar, is studied in Fig. 2. For simplicity we identify four points on the time axis: A at 0 ns, B at 3.5 ns, C at 7 ns, and D at 10.5 ns. Thetime axis is then subdivided into three regions AB, BC, andCD. The applied current density jis uniform and constant in each of these three time intervals. In particular, it is directedalong the axis of the cylinder: j=jx, with j=10 11A/m2in AB, j=0 in BC, and j=−1011A/m2in CD. We remind the reader that the applied ﬁeld is always zero, throughout all thesimulation. The graph in Fig. 2shows the behavior of the compo- nents of /H20855M Fe/H20856. In region AB, the current produces a preces- sion of the whole magnetization of the system around the x axis. This precession is accompanied by a movement—andconsequent compression—of the artiﬁcial domain wall in thedirection of the electron ﬂow /H20849negative xdirection /H20850, which reﬂects in an increase of the xcomponent of /H20855M Fe/H20856. Such an effect may be explained with a current-induced motion of the artiﬁcial domain wall. Current-induced motion is a wellknown effect for domain walls in nano-wires: it has beenobserved experimentally and has been provedanalytically. 14–16 In the time interval AB, the current pumps energy into the system, which is stored in the compression of the domainwall. In the time interval BC, the current is switched off andthis energy is gradually released. The domain wall decom-presses, restoring the conﬁguration it had at time t=0. Fi- nally, during the time interval CD the system behaves in away which is symmetrical to the one observed in AB. /H20855M Fe,x/H20856 rotates in the opposite direction and the wall is compressed in the positive xdirection, leading to negative values for /H20855MFe,x/H20856. Expressing /H20855MFe/H20856in spherical coordinates with xchosen as the polar axis, we obtained the precession angle /H9278/H20849t/H20850of /H20855MFe/H20856around the xaxis as a function of time t. We computed the time derivative /H9275/H20849t/H20850=/H9278/H11032/H20849t/H20850to obtain the precession fre- quency as a function of time. The result is shown in Fig. 3. FIG. 2. /H20849Color online /H20850The evolution in time of the three components of /H20855mFe/H20856=/H20855MFe//H20648MFe/H20648/H20856, the normalized magnetization of iron averaged over all the nano-pillar. The three boxes above the graph show the conﬁguration ofMFeatt=0,t=3.5 ns and t=10.5 ns.07A504-2 Franchin et al. J. Appl. Phys. 103, 07A504 /H208492008 /H20850 Downloaded 28 Apr 2013 to 142.51.1.212. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsThe sign of /H9275/H20849t/H20850depends on the sense of rotation around the xaxis. This graph shows that the applied current j =/H110061011A/m2, produces a precession motion with fre- quency around 14 GHz, in the microwave frequency range.The frequency seems to be related to the compression of thedomain wall. It increases rapidly when /H20855M Fe,x/H20856increases and stabilizes when also /H20855MFe,x/H20856does. The accuracy of the discretization of space has been veriﬁed by increasing the number of mesh elements /H20849from 4129 to 19251 /H20850, obtaining differences in the precession fre- quency at 3.5 ns lower than 1.2%. DISCUSSION The model we presented does not take into account a number of effects which complicate the physics of real sys-tems. The imperfections of the geometry and the impuritiesin the materials can break the cylindrical symmetry. The ef-fect of such imperfections is difﬁcult to predict. The size of the sample was chosen to speed up the simu- lation. However, we expect a similar precessional dynamicsin nanopillars with greater radii. Also, the materials couldhave been chosen differently and the DyFe 2anisotropy could have been well approximated by an inﬁnite pinning on theiron moments, resulting in a simpliﬁcation of the model.However, this approximation would have removed the onlysource of symmetry breaking, besides the irregularity of theunstructured mesh.Other simulations should be carried out to understand how the precession frequency depends on the current densityand what the role of the system geometry is. To conclude, we remark that a symmetry breaking could be introduced on purpose to obtain bistable systems, wherethe current may be used to switch the magnetization betweentwo states. ACKNOWLEDGMENTS This work has been funded by the Engineering and Physical Sciences Research Council /H20849EPSRC /H20850in the United Kingdom /H20849GR/T09156 and GR/S95824 /H20850. G.M. acknowledges ﬁnancial support by the Deutsche Forschungsgemeinschaft via SFB 668 and GK 1286. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 /H208491998 /H20850. 3L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 4L. Berger, J. Appl. Phys. 49, 2156 /H208491978 /H20850. 5J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 /H208492000 /H20850. 6Y. Acremann, J. P. Strachan, V. Chembrolu, S. D. Andrews, T. Tyliszczak, J. A. Katine, M. J. Carey, B. M. Clemens, H. C. Siegmann, and J. Stöhr,Phys. Rev. Lett. 96, 217202 /H208492006 /H20850. 7S. N. Gordeev, J.-M. L. Beaujour, G. J. Bowden, P. A. J. de Groot, B. D. Rainford, R. C. C. Ward, M. R. Wells, and A. G. M. Jansen, Phys. Rev. Lett. 87, 186808 /H208492001 /H20850. 8M. Kläui, M. Laufenberg, L. Heyne, U. Rüdiger, C. A. F. Vaz, J. A. C. Bland, L. J. Heyderman, S. Cheriﬁ, A. Locatelli, T. O. Mentes, and L.Aballe, Appl. Phys. Lett. 88, 232507 /H208492006 /H20850. 9M. Franchin, J. Zimmermann, T. Fischbacher, G. Bordignon, P. de Groot, and H. Fangohr, IEEE Trans. Magn. 43, 2887 /H208492007 /H20850. 10D. R. Fredkin and T. R. Koehler, IEEE Trans. Magn. 26,4 1 5 /H208491990 /H20850. 11D. A. Lindholm, IEEE Trans. Magn. 20, 2025 /H208491984 /H20850. 12S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 /H208492004 /H20850. 13NMAG , a micromagnetic simulation environment, 2007 /H20849http:// www.soton.ac.uk/~fangohr/nsim/nmag /H20850. 14B. Krüger, D. Pfannkuche, M. Bolte, G. Meier, and U. Merkt, Phys. Rev. B75, 054421 /H208492007 /H20850. 15G. Meier, M. Bolte, R. Eiselt, B. Krüger, D.-H. Kim, and P. Fischer, Phys. Rev. Lett. 98, 187202 /H208492007 /H20850. 16M. Kläui, P.-O. Jubert, R. Allenspach, A. Bischof, J. A. C. Bland, G. Faini, U. Rüdiger, C. A. F. Vaz, L. Villa, and C. Vouille, Phys. Rev. Lett. 95, 026601 /H208492005 /H20850. FIG. 3. /H20849Color online /H20850The time dependence of the frequency /H9275for the precession of /H20855MFe/H20856around the xaxis. The sign of /H9275is related to the sense of rotation.07A504-3 Franchin et al. J. Appl. Phys. 103, 07A504 /H208492008 /H20850 Downloaded 28 Apr 2013 to 142.51.1.212. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
1.1560703.pdf	Influence of domain wall structure on pinning characteristics with self-induced anisotropy H. Asada, Y. Wasada, J. Yamasaki, M. Takezawa, and T. Koyanagi Citation: Journal of Applied Physics 93, 7447 (2003); doi: 10.1063/1.1560703 View online: http://dx.doi.org/10.1063/1.1560703 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/93/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Controlled domain wall pinning in nanowires with perpendicular magnetic anisotropy by localized fringing fields J. Appl. Phys. 115, 17D506 (2014); 10.1063/1.4864737 Micromagnetic study of domain-wall pinning characteristics with grooves in thin films J. Appl. Phys. 97, 10E317 (2005); 10.1063/1.1857652 Field-controlled domain-wall resistance in magnetic nanojunctions Appl. Phys. Lett. 85, 251 (2004); 10.1063/1.1771455 Static properties and nonlinear dynamics of domain walls with a vortexlike internal structure in magnetic films (Review) Low Temp. Phys. 28, 707 (2002); 10.1063/1.1521291 Characteristics of 360°-domain walls observed by magnetic force microscope in exchange-biased NiFe films J. Appl. Phys. 85, 5160 (1999); 10.1063/1.369110 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.59.226.54 On: Wed, 10 Dec 2014 15:04:43Micromagnetics and Field Computation Michael Donahue, Chairman Inﬂuence of domain wall structure on pinning characteristics with self-induced anisotropy H. Asadaa)and Y. Wasada Department of Symbiotic Environmental Systems Engineering, Graduate School of Science and Engineering, Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, Japan J. Yamasaki and M. Takezawa Department of Electrical Engineering, Faculty of Engineering, Kyushu Institute of Technology,1-1 Sensui-cho, Tobata-ku, Kitakyushu 804-8550, Japan T. Koyanagi Department of Symbiotic Environmental Systems Engineering, Graduate School of Science and Engineering,Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, Japan ~Presented on 13 November 2002 ! Wall pinning effects with self-induced spatially varying uniaxial anisotropy in various thick ﬁlms have been studied using micromagnetic simulation based on the Landau–Lifshitz–Gilbert equation.In the simulation, the discretization region is in the cross section normal to the ﬁlm plane. It isclariﬁed that the wall structure is strongly related to pinning characteristics. Depinning ﬁelds of thewall having a ﬂux-closure asymmetric vortex ~C-shaped wall !are different in the wall movement directions due to the asymmetric wall structure. On the other hand, depinning ﬁelds of the wall withtwovortices ~S-shapedwall !whichhaveasymmetricstructuredonotdependonthewallmovement direction. Depinning ﬁelds for the S-shaped wall are different from both depinning ﬁelds for theC-shaped wall. © 2003 American Institute of Physics. @DOI: 10.1063/1.1560703 # I. INTRODUCTION It is well known that amorphous ribbons annealed in a demagnetized state exhibit magnetization reversal with largeBarkhausen discontinuities due to the domain wall pinning.The mechanism for the wall pinning is self-induced anisot-ropy during annealing by the domain wall itself. 1Kerr mi- croscope observation revealed the pinned wall broadeningand magnetization reversal process in a Perminvar-type loop. 2,3Theoretical analysis and micromagnetic simulation on self-induced anisotropy effects on domain wall within aone-dimensional approximation was also performed. 4,5How- ever, the domain wall behaviors with self-induced aniso-tropy, which plays an important role for magnetic properties,has not been clariﬁed well since the domain wall containsNe´el caps and Bloch wall in thin ﬁlms. 6Magnetization within the wall, therefore, rotates along the ﬁlm thicknessdirection as well as the direction normal to the wall plane.We have done the micromagnetic simulation based on theLandau–Lifshitz–Gilbert ~LLG!equation assuming the cross section normal to the ﬁlm plane and studied on domain wallbehaviors such as wall broadening and wall pinning withspatially varying uniaxial anisotropy. 7In this article, we in-vestigate the inﬂuence of domain wall structures on wall pinning characteristics with spatially varying uniaxial anisot-ropy in various thick ﬁlms. II. SIMULATION MODEL Numerical simulations were carried out by integrating the LLG equation.8The cross section normal to the ﬁlm plane was discretized into a two-dimensional array. Self-induced anisotropy was modeled as follows: ﬁrst, with theuniform easy axis set normal to the calculation region, thedomain wall proﬁle was calculated. Next, after relaxation,with the easy axis direction set to be the same as the mag-netization direction, the domain wall proﬁle was recalcu-lated. This procedure was iterated when wall broadening wasinvestigated. The material parameters used in the simulationwere as follows: saturation induction 4 pMs58000 Gauss, uniaxial anisotropy constant Ku53800 erg/cm3, exchange constant A51026erg/cm, and gyromagnetic ratio g51.76 3107erg/(sOe). The damping constant a51.0 was chosen to speed up the computation.The grid element spacings were50 Å for the ﬁlm thickness h50.15 mm, 100 Å for h50.3 and 0.5 mm, and 150 Å for h50.8mm, respectively. III. RESULTS AND DISCUSSION Figure 1 shows magnetic conﬁguration and energy curves of a domain wall part of the calculation region, hav-a!Author to whom correspondence should be addressed; electronic mail: asada@aem.eee.yamaguchi-u.ac.jpJOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 10 15 MAY 2003 7447 0021-8979/2003/93(10)/7447/3/$20.00 © 2003 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.59.226.54 On: Wed, 10 Dec 2014 15:04:43ing a ﬂux-closure asymmetric vortex ~C-shaped wall !,i na 0.8mm thick ﬁlm ~a!with the uniform easy axis set normal to the calculation region ( x-direction !;~b!with the easy axis proﬁle set to the domain wall proﬁle. The arrows in the ﬁg-ures represent the magnetization directions for every fourth(434) grid element. Energies are averaged through the ﬁlm thickness and normalized by the peak of the wall energy inFig. 1 ~a!. The magnetization rotation in Fig. 1 ~b!becomes more gradual not only along the direction normal to the wallplane (y-direction !but also along the ﬁlm thickness direction (z-direction !. Reﬂecting the magnetization conﬁguration, the wall energy curves show an asymmetric shape. The slope ofthe energy curves are steeper at the left side of the Blochwall in the center of ﬁlm thickness, that is, the vortex side.Comparing the wall energy components in Figs. 1 ~a!and 1~b!, the anisotropy energy drastically dropped, which oc- curred at the ﬁrst iteration of an easy axis proﬁle set to thedomain wall proﬁle. The exchange energy also decreasedmonotonically with the repeated iteration, while the demag-netization energy variation was small. 7Positive and negative magnetic ﬁelds were applied along the magnetic domain toinvestigate the pinning characteristics. When the positivemagnetic ﬁelds were applied, the wall moved to the right-hand side of Fig. 1. The time transient of the orthogonalcomponent of an effective ﬁeld is used for determining thedepinning ﬁeld. 9Depinning ﬁelds as a function of ﬁlm thick- ness are shown in Fig. 2. It was conﬁrmed that, in the 0.15mm thick ﬁlm, depinning ﬁelds for a51.0 were the same as those for a50.1. As shown in Fig. 2, depinning ﬁelds are different in the wall movement direction for various thickﬁlms due to the asymmetric wall structure, which causes theasymmetric energy proﬁle as shown in Fig. 1. The depinningﬁelds for both the positive and negative applied ﬁelds de-crease with increasing ﬁlm thickness and tend to saturate.The depinning ﬁelds for h50.8 mm are 0.54 Hkand 0.57 Hk (Hk52Ku/Ms) for the positive and negative applied ﬁelds, respectively. These values are similar to the numerically ob-tained depinning ﬁeld of 0.55 H kwithin the one-dimensional approximation.5On the other hand, the difference of depin- ning ﬁelds in wall movement directions are almost the samefor the various thick ﬁlms. Next, we investigated the depinning ﬁeld for the differ- ent types of wall which has two vortices ~S-shaped wall !. Figure 3 shows the magnetic conﬁguration ~every 4 34 grid elements !and normalized energy curves of a domain wall part of calculation region, having an S-shaped wall, in a 0.15 mm thick ﬁlm with the easy axis proﬁle set to the domain wall proﬁle. As shown in the ﬁgure, the wall energy curvefor the S-shaped wall shows the symmetric shape having twopeaks near each vortex where the magnetization rapidly ro-tates along the yandzdirections. Simulated wall energies of the S-shaped wall (1.7 erg/cm 2forh50.15mm and 0.45 erg/cm2forh50.8mm) is higher than those for the C-shaped wall (1.3 erg/cm2forh50.15mm and 0.32 erg/cm2forh50.8mm). The depinning ﬁelds are also examined by applying positive and negative magnetic ﬁelds.Figure 4 shows depinning ﬁelds of the S-shaped wall as afunction of ﬁlm thickness. The depinning ﬁeld decreases FIG. 1. Simulation results of magnetization conﬁguration and wall energy curve ~solid line !for an asymmetric Bloch wall ~C-shaped wall !in a 0.8 mm thick ﬁlm ~a!easy axis along x;~b!with the easy axis proﬁle set to the domain wall proﬁle. The wall energy components of anisotropy ~dotted !, demagnetization ~dashed !, and exchange ~dotted–dashed !are also indicated. FIG. 2. Depinning ﬁelds of the C-shaped wall as a function of ﬁlm thickness for positive and negative magnetic ﬁelds. FIG. 3. Magnetization conﬁguration and wall energy curve for an S-shapedwall in a 0.15 mm thick ﬁlm. The easy axis proﬁle is set to the wall proﬁle. The wall energy components of anisotropy ~dotted !, demagnetization ~dashed !, and exchange ~dotted–dashed !are also indicated.7448 J. Appl. Phys., Vol. 93, No. 10, Part s2&3 ,1 5M a y 2003 Asada et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.59.226.54 On: Wed, 10 Dec 2014 15:04:43with increasing ﬁlm thickness. In contrast to the C-shaped wall, the depinning ﬁelds for the S-shaped wall do not de-pend on the wall movement direction due to the symmetricstructure. It is also found that the depinning ﬁeld for theS-shaped wall is different from both depinning ﬁelds for theC-shaped wall. Finally, pinning effects of spatially varying uniaxial an- isotropy on domain walls having different kinds of proﬁlesfrom an easy axis proﬁle as shown in Fig. 5 are investigated.The easy axis direction of spatially varying uniaxial anisot-ropy @Fig. 5 ~a!#is set to the same kind of C-shaped wall proﬁle as Fig. 1. The assumed ﬁlm thickness is 0.15 mm. In this simulation, the domain walls having the Bloch wall inwhich the magnetization rotates in the same direction of theeasy axis proﬁle are chosen. First, the interaction for theS-shaped wall is examined. The simulated magnetic conﬁgu- ration ~every 4 34 grid elements !of the domain wall part without the applied ﬁeld is indicated in Fig. 5 ~b!. This type of domain wall would correspond to the experimentally ob-served ‘‘unstable wall’’ having the black-and-white contrastusing the Kerr magneto-optical effect, 2which means that the domain wall at the ﬁlm surface consists of the two magneti-zation regions having 1yand2ycomponents. Obviously, the wall energy for the S-shaped wall is higher than that forthe C-shaped wall energy with the same spatially varyinguniaxial anisotropy. The depinning ﬁelds are 5.0 Oe for boththe positive and negative applied ﬁelds and the dependenceof the wall movement directions on depinning ﬁelds is notobserved. Second, the interaction for different types ofC-shaped walls having the Ne ´el caps where magnetization rotates in the opposite direction of the easy axis proﬁle asdepicted schematically in Fig. 6 is investigated. In this case,there are two pinning sites as shown in Figs. 5 ~c!and 5 ~d!. The pinning site as Fig. 5 ~d!is more stable compared to Fig. 5~c!. The obtained depinning ﬁelds of 3.3 Oe for the positive applied ﬁeld and 1.1 Oe for the negative applied ﬁelds areconsiderably smaller. IV. CONCLUSIONS Numerical simulation shows that the wall structure is strongly related to pinning characteristics with self-inducedspatially varying uniaxial anisotropy. Different wall struc-tures yield different pinning characteristics due to the differ-ent self-induced anisotropy. Depinning ﬁelds of the C-shapedwall are different in the wall movement directions due tothe asymmetric wall structure, while depinning ﬁelds of theS-shaped wall do not depend on the wall movementdirection. 1H. Fujimori, H. Yoshimoto, T. Masumoto, and T. Mitera, J. Appl. Phys. 52, 1893 ~1981!. 2R. Schafer, W. K. Ho, J. Yamasaki,A. Hubert, and F. B. Humphrey, IEEE Trans. Magn. 27, 3678 ~1991!. 3J. Yamasaki, T. Chuman, M. Yagi, and M. Yamaoka, IEEE Trans. Magn. 33, 3775 ~1997!. 4C. Aroca, P. Sanchez, and E. Lopez, Phys. Rev. B 34, 490 ~1986!. 5B. B. Pant, K. Matsuyama, J.Yamasaki, and F. B. Humphrey, Jpn. J.Appl. Phys., Part 1 34, 4779 ~1995!. 6A. Hubert, Phys. Status Solidi 32,1 5 9 ~1969!. 7H. Asada, Y. Wasada, J. Yamasaki, M. Takezawa, and T. Koyanagi, J. Magn. Soc. Jpn. 26,3 9 2 ~2002!. 8S. Konishi, K. Matsuyama, N. Yoshimatsu, and K. Sakai, IEEE Trans. Magn.24, 3036 ~1988!. 9H. Asada, K. Matsuyama, M. Gamachi, and K. Taniguchi, J. Appl. Phys. 75, 6089 ~1994!. FIG. 4. Depinning ﬁelds of the S-shaped wall as a function of ﬁlm thickness for positive and negative magnetic ﬁelds. FIG. 6. Schematic drawing of easy axis @Fig. 5 ~a!#and magnetization of the C-shaped wall @Figs. 5 ~c!and 5 ~d!#at the top of the ﬁlm surface. FIG. 5. ~a!Easy axis directions and magnetization conﬁgurations for ~b! S-shaped and ~c!and~d!C-shaped walls having different kinds of proﬁles from an easy axis proﬁle.7449 J. Appl. Phys., Vol. 93, No. 10, Part s2&3 ,1 5M a y 2003 Asada et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.59.226.54 On: Wed, 10 Dec 2014 15:04:43
5.0041233.pdf	J. Chem. Phys. 154, 124106 (2021); https://doi.org/10.1063/5.0041233 154, 124106 © 2021 Author(s).Reliable transition properties from excited- state mean-field calculations Cite as: J. Chem. Phys. 154, 124106 (2021); https://doi.org/10.1063/5.0041233 Submitted: 21 December 2020 . Accepted: 11 February 2021 . Published Online: 22 March 2021 Susannah Bourne Worster , Oliver Feighan , and Frederick R. Manby COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185 Modeling nonadiabatic dynamics with degenerate electronic states, intersystem crossing, and spin separation: A key goal for chemical physics The Journal of Chemical Physics 154, 110901 (2021); https://doi.org/10.1063/5.0039371 Spin contamination in MP2 and CC2, a surprising issue The Journal of Chemical Physics 154, 131101 (2021); https://doi.org/10.1063/5.0044362The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Reliable transition properties from excited-state mean-field calculations Cite as: J. Chem. Phys. 154, 124106 (2021); doi: 10.1063/5.0041233 Submitted: 21 December 2020 •Accepted: 11 February 2021 • Published Online: 22 March 2021 Susannah Bourne Worster,a) Oliver Feighan, and Frederick R. Manbyb) AFFILIATIONS Centre for Computational Chemistry, School of Chemistry, University of Bristol, Bristol BS8 1TS, United Kingdom a)susannah.bourne-worster@bristol.ac.uk b)Author to whom correspondence should be addressed: fred.manby@bristol.ac.uk. Present address: Entos, Inc., 4470 W Sunset Blvd, Suite 107 PMB 94758, Los Angeles, CA 90027, USA. ABSTRACT Delta-self-consistent field ( ΔSCF) theory is a conceptually simple and computationally inexpensive method for finding excited states. Using the maximum overlap method to guide optimization of the excited state, ΔSCF has been shown to predict excitation energies with a level of accuracy that is competitive with, and sometimes better than, that of time-dependent density functional theory. Here, we benchmark ΔSCF on a larger set of molecules than has previously been considered, and, in particular, we examine the performance of ΔSCF in predicting transition dipole moments, the essential quantity for spectral intensities. A potential downfall for ΔSCF transition dipoles is origin dependence induced by the nonorthogonality of ΔSCF ground and excited states. We propose and test a simple correction for this problem, based on symmetric orthogonalization of the states, and demonstrate its use on bacteriochlorophyll structures sampled from the photosynthetic antenna in purple bacteria. Published under license by AIP Publishing. https://doi.org/10.1063/5.0041233 .,s I. INTRODUCTION The matrix element of the electric dipole operator ˆμbetween two quantum states, commonly known as a transition dipole moment μ, is a crucial quantity in simulating spectra and describ- ing excited-state dynamics of molecular systems. The magnitude of the transition dipole moment \| μ\| defines the strength with which a transition between the two states can couple to the electro- magnetic field to absorb (or emit) light, while the dipole–dipole interaction between transition dipole moments provides the sim- plest model for the coupling between excited states on different chromophores. An important application of this second property is in describ- ing the transport of excitons through a network of chromophores, as is seen in the early stages of photosynthesis, as well as synthetic analogs, such as organic polymer light-emitting diodes1and chro- mophores hosted on DNA scaffolds.2–4These systems are often simulated using a Frenkel exciton Hamiltonian,5–8 ˆH=∑ iEi∣i⟩⟨i∣+∑ i≠jVij∣i⟩⟨j∣, (1)whose off-diagonal elements, Vij, are the coulomb interaction between the transition dipole moments of the relevant excitation on each chromophore. The light-harvesting antenna in photosynthetic organisms typically contains large numbers of chromophores, which are, themselves, relatively large conjugated organic molecules. For example, the antenna in purple photosynthetic bacteria consists of 3–10 light-harvesting II (LHII) complexes (and one LHI complex) per reaction center,9,10each containing 27 (32) bacteriochlorophyll- a (BChla) chromophores of around 140 atoms.11,12Furthermore, the transition dipole moment of each chromophore, and hence the coupling elements of the Hamiltonian, fluctuates constantly with the vibrations of the molecules. To capture the full time- dependent Hamiltonian, even approximately, calculation of the transition dipole moments should, therefore, ideally be computa- tionally cheap as well as reasonably accurate. Current models of exciton dynamics in these systems rely on parameterizing the cou- pling elements Vijfrom experiment5,13–17or use time-dependent density functional theory (TDDFT) to generate representative tran- sition dipole moments from a small handful of chromophores.18,19 On-the-fly TDDFT has been used in this context for a single LHII J. Chem. Phys. 154, 124106 (2021); doi: 10.1063/5.0041233 154, 124106-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp complex,20and the present work forms part of a wider effort to scale and refine the approach reported there. TDDFT21–23is a widely popular method for obtaining the properties, including transition dipole moments, of excited states.24,25With the right choice of exchange–correlation functional and basis set, it yields good accuracy compared to correlated wave- function methods, such as second-order coupled cluster (CC2)26 or equation of motion coupled cluster singles and doubles (EOM- CCSD),25,27at a much lower computational cost. In TDDFT, excitation energies emerge as the eigenvalues of the Casida equations.22,28The transition vectors (which arise as eigen- vectors) are expressed in a basis of excitations i→aand corre- sponding de-excitations. Each one can be reshaped into a transition density matrix, with columns iand rows a, from which the transition properties of the excited state can easily be calculated. The transi- tion dipole moment, for example, is found by tracing the transition density matrix with the dipole operator ˆμ. However, TDDFT is still too costly to perform dynamics cal- culations involving large numbers of BChla chromophores, and this paper amounts to an investigation into how feasible it would be to use the cheaper delta-self-consistent field ( ΔSCF) method. Crucially, ΔSCF is not only simpler for the energy evaluation; the excited-state gradient is also available very cheaply because it can be computed using standard ground-state mean-field gradient theory. ΔSCF is conceptually simple. Excited states are found by pro- moting an electron from an occupied orbital in the ground state to one of the unoccupied virtual orbitals. The orbitals are then reop- timized for the excited electron configuration using a normal SCF iterative procedure.29–33Unlike TDDFT, therefore, ΔSCF produces a distinct set of molecular orbitals for the excited state. The transi- tion dipole moment can be calculated as a matrix element between the ground-state and excited determinants. Initial attempts to locate excited states via an SCF procedure rigidly maintained the orthogonality of the ground and excited states by relaxing the excited state particle (and hole) orbitals within the ground-state virtual space29,30(or in the virtual and occupied spaces, respectively32). In addition to the convenience of dealing with orthogonal states, these procedures also ensure that relaxing the orbitals does not collapse the excited state wavefunction back down to the ground state. Gilbert et al.33later argued that imposing orthogonality in this way led to wavefunctions that were no longer solutions of the full SCF equations and propagated errors and approximations in the ground state. They relaxed the orthogonal- ity condition and searched for high energy solutions to the SCF equations by minimizing the energy of the excited state with the added condition that the occupied orbitals at each step of the itera- tive cycle should overlap as much as possible with their counterparts in the previous iteration. This is known as the maximum overlap method (MOM) and has been shown to be highly successful in find- ing excited state energies.33–36The sizable test set that we consider in this paper adds to this body of evidence, as well as benchmarking the technique for transition dipole moments. However, as Gilbert et al. acknowledge in their original paper, allowing non-zero overlap between the ground and excited states can artificially enhance the size of the transition dipole moment (and other transition properties). Nonorthogonality of the states introduces a non-zero transition charge, equal to the size of the overlap. Transition dipole moments calculated from the chargedtransition density are origin-dependent and, therefore, have a com- pletely arbitrary magnitude. When the state overlap is very small and the molecule is positioned with its center of mass on, or close to, the origin, the error associated with the charged transition density is small, or even negligible. Conversely, if the molecule is positioned far away from the origin, as might be the case for a chromophore located within a larger complex or aggregate centered collectively on the origin, the error associated with this additional charge can quickly escalate. Here, we propose and test a simple correction that can be applied to the transition density matrix after the SCF cycle, to restore the orthogonality of the ground and excited states. II. THEORY The transition dipole for an excitation from an initial state \| Ψ1⟩ to a final state \| Ψ2⟩is defined in the standard length gauge as μ1→2=⟨Ψ2∣ˆμ∣Ψ1⟩, (2) where ˆμis the three-component dipole operator. InΔSCF, the states \| Ψn⟩are Slater determinants constructed from spin orbitals {∣ϕ(n) j⟩}with n= 1, 2. The orbitals are orthonor- mal within each state, but, in general, nonorthogonal between states, with inner products S21 jk=⟨ϕ(2) j∣ϕ(1) k⟩. The inner product of the two determinants is the determinant of the orbital inner products, ⟨Ψ2∣Ψ1⟩=∣S21∣. (3) Following the normal rules for nonorthogonal determinants laid down by Löwdin,37the transition dipole can be written as ⟨Ψ2∣ˆμ∣Ψ1⟩=∑ jkμ21 jkadj(S21)jk, (4) where μ21 jk=⟨ϕ(2) j∣ˆμ∣ϕ(1) k⟩and adj denotes matrix adjugate. Alternatively, the value of the transition dipole can be com- puted from the reduced one-particle transition density matrix, ⟨Ψ2∣ˆμ∣Ψ1⟩=tr(ˆμ∣Ψ1⟩⟨Ψ2∣)=tr(μD21). (5) Here, D21is the one-particle reduced transition density matrix in the atomic-orbital basis, given by D21=C(2)adj(S21)C(1)†, (6) where C(n)are the molecular-orbital coefficients for state n. For unrestricted calculations, the spin summation for the reduced den- sity matrix has additional factors that would be 1 or 0 if a common set of orthonormal orbitals were being used, but here have to be considered explicitly, D21=D21,α∣S21,β∣+D21,β∣S21,α∣, (7) where D21,σis the analog of D21for theσspin channel. J. Chem. Phys. 154, 124106 (2021); doi: 10.1063/5.0041233 154, 124106-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp As noted above, in ΔSCF, the sets of orbitals {ϕ(1) j}and{ϕ(2) k} for the ground and excited states are optimized independently, so that the resulting states \| Ψ1⟩and \|Ψ2⟩are not necessarily orthogo- nal. As previously recognized in the literature,33non-zero overlap between these two states leads to errors in the calculated transition dipole moment. In particular, when states are not exactly orthog- onal, there is a non-zero transition charge equal to the value of the overlap: q21=⟨Ψ2\|Ψ1⟩. This breaks the origin-independence of the transition dipole moment, making the calculated values virtually meaningless. While the transition charge is sometimes exactly zero (when the ground and excited states are of different symmetries) or very small, any violation of translational invariance is certain to pre- vent widespread use of transition properties from ΔSCF and needs to be fixed. ForΔSCF calculations using Hartree–Fock theory, one can clearly proceed by performing nonorthogonal configuration interac- tion,38,39not only fixing the transition dipoles but also (presumably) generally improving the quality of the description. On the other hand, for ΔSCF based on DFT, such a procedure is not well defined because the underlying Slater determinants are understood not to be “the” wavefunctions, nor the Hamiltonian to be “the” Hamil- tonian.40It would be possible to build on the approach developed by Wu et al. in the context of constrained DFT,40but that also introduces other choices and approximations. Another possibility is to correct the transition dipole moment by adding in the dipole of the nuclear charges, weighted by the over- lap of the ground and excited states. This is equivalent to reposition- ing the molecule before calculating the transition dipole moment, such that its center of charge sits at the origin. This approach has been used successfully in simulations of absorption spectra,41,42and we will briefly comment on its effectiveness for calculating the abso- lute magnitudes of transition dipoles. However, while this correc- tion does, by definition, ensure translational invariance of the calcu- lated transition dipole moment, it does not address the underlying cause of the problem and neither does it eliminate the transition charge. Here, we instead focus on the simple expedient of using sym- metric orthogonalization to ensure exact orthogonality. Recall that symmetric orthogonalization mixes the two states to make a pair of states that are orthogonal while being as close as possible to the original states and is defined by the transformation ∣Ψ˜ν⟩=∑ ν∣Ψν⟩[S−1/2]ν˜ν, (8) where S=(1S S1)andS=⟨Ψ2\|Ψ1⟩. Based on this transformation, the transition density between the orthogonalized states is given by ˜D21=1 4(1 +S)[(1−a2)(D11+D22)+(1 +a)2D21+(1−a)2D12], (9) where a=√ 1 +S/√ 1−S; this parameter is equal to 1 when S= 0, recovering the expected result ˜D21=D21in this limit. In this work, we explore the quality of ΔSCF transition dipoles based on the symmetrically orthogonalized transition density.III. COMPUTATIONAL DETAILS Calculations were performed on a set of 109 small closed-shell molecules containing H, C, N, O, and F. These structures are a subset of the benchmark set used in Ref. 43, with molecules of 12 atoms or fewer. Reference energies and transition dipole moments (reported in atomic units) were calculated for the three lowest energy singlet excited states of each molecule using EOM-CCSD with an aug-cc- pVTZ basis set.44–46The same quantities were also calculated for the six lowest energy singlet excited states using TDDFT with the CAM- B3LYP functional47and aug-cc-pVTZ basis set. CAM-B3LYP has consistently been shown to perform well for the prediction of the optical properties of both small molecules26,27and a large number of conjugated chromophores of various sizes.48–50 Both EOM-CCSD and TDDFT calculations were performed using Gaussian 16.51Excited states were cross-referenced between the two methods using the symmetry labels provided by Gaussian. In a small number of cases, the symmetry labeling was unsuccess- ful or defaulted to a different choice (non-Abelian or highest order Abelian) of point group between the two methods. In these cases, the excited states were matched by hand based on descent in symmetry and their composition, energy, and transition dipole moment. A full list of symmetries and indices of the selected transitions can be found in the supplementary material. These data were used to benchmark the performance of ΔSCF in predicting transition properties, both with and without the sym- metric orthogonalization correction proposed in Eq. (9). ΔSCF cal- culations were performed in the Entos Qcore package,52with the CAM-B3LYP functional and aug-cc-pVTZ basis set. We investigated only the HOMO–LUMO singlet transition. Using ΔSCF, we calculated the properties of the state correspond- ing to the spin-conserving excitation of a HOMO electron into the lowest energy virtual orbital. This does not correspond to a true singlet excitation, which would contain a superposition of αandβ excitations. The spin-purification formula, ΔES=2ΔEi,α→a,α−ΔEi,α→a,β, (10) was applied to more accurately estimate the true singlet excitation energy.53–55However, this correction is applied at the end of the SCF cycle and does not affect the composition of the molecular orbitals, which are used to calculate the transition dipole moment.55 SinceΔSCF uses a variational principle to optimize the excited state orbitals, a known weakness is that the calculation can converge on the ground state rather than the desired excited state. In most cases, this can be prevented using MOM,33which selects orbitals to be occupied based on maximum overlap with each occupied molec- ular orbital in the previous iteration. This stops the orbitals from changing significantly in any particular step of the optimization and helps stabilize the calculation around the excited state stationary points, rather than the global minimum (ground state). However, in a small number of cases, additional help was needed to converge the SCF cycle to the correct excited state. There are a number of well-established techniques to address this issue. We used a com- bination of Fock-damping, modifying the direct inversion of itera- tive subspace (DIIS) protocol,56–58and starting from an initial guess corresponding to excitation of half an electron. J. Chem. Phys. 154, 124106 (2021); doi: 10.1063/5.0041233 154, 124106-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The properties of the ΔSCF transition were compared to those of the TDDFT transition with the largest coefficient for HOMO– LUMO excitation (based on the orbital indexing in the TDDFT calculation), along with the corresponding EOM-CCSD transition. For a few molecules, this was not an appropriate comparison to make, either because of a reordering of orbitals with very simi- lar energies or because there was no single state dominated by the HOMO–LUMO transition. In these cases, we either selected the cor- rect TDDFT transition by hand or calculated the ΔSCF transition corresponding to the lowest energy TDDFT transition. Full details of these choices can be found in the supplementary material. IV. RESULTS First, we test the effect of applying the symmetric orthogo- nalization correction, proposed above, to overlapping ground and excited states. Figure 1 shows the error relative to EOM-CCSD in the magnitude of the transition dipole moment, as a function of the ground–excited state overlap for each molecule in the test set using ΔSCF with or without the correction. In panel (a), the coordinates of the entire molecule have been translated by 100 Å in each Cartesian direction. Physical properties, such as excitation energy and tran- sition dipole moment, should be invariant under this translation; but when there is non-zero overlap between the ground and excited states, the calculation of the transition dipole moment becomes origin-dependent, and this coordinate shift introduces an error into the calculated values of \| μ\|. Although the ground–excited state overlaps are small ( <0.02) for every molecule in the test set, when the molecule is displaced far away from the origin, it is sufficient to produce highly unphysical transition dipoles. Using the symmetric orthogonalization correc- tion, the origin dependence is completely removed and these errors do not arise. An important consideration is whether applying the correc- tion degrades the accuracy of the ΔSCF calculation in any way. This is difficult to see since the origin-dependence of the uncorrected transition dipole moments means that they cannot be taken as a reliable indication of the “correct” ΔSCF transition dipole. However, we note that, by construction, the amount of ground and excitedstate dipole that are mixed into the transition density (the amount that the correction “changes the answer”) scales roughly linearly with the size of the overlap for small overlaps. When the overlap is zero (and the uncorrected ΔSCF transition dipole is, therefore, already “correct”), the symmetric orthogonalization procedure does not change the states, transition density, or transition dipole at all. At the largest overlaps present in this test set, the change in the transition dipole that comes from applying the symmetric orthog- onalization correction is still very small, as illustrated in panel (b) of Fig. 1. Note that we do not attach any significance to whether the corrected or uncorrected transition dipole magnitude is closer to the reference value since the uncorrected magnitude can be made to have any value by shifting the coordinates of the molecule. The molecules in this test set are small, with average atomic positions (not the center of mass) defining the origin, so we do not expect the uncorrected transition dipole moments to be wildly wrong. How- ever, even shifting the molecule so that its center-of-mass lies on the origin is sufficient to account for the difference in values seen on the right-hand side of Fig. 1. For the larger molecules in the test set, the transition dipole may not span the whole molecule and the con- cept of the “correct” position or transition dipole for the molecule becomes even less clear. Correcting the transition dipole by including a weighted amount of the nuclear dipole produces near-identical results to the symmetric orthogonalization correction. This is illustrated in Fig. S1 of the supplementary material. It would, therefore, be rea- sonable to choose either of these corrections based on convenience or suitability for a particular application. For the remainder of this paper, the symmetric orthogonal- ization correction will be applied for all reported ΔSCF transition dipoles. Figure 2 compares the excitation energy of each molecule cal- culated using TDDFT and ΔSCF with the value predicted by EOM- CCSD. The energies predicted by ΔSCF are at least as accurate as those predicted using TDDFT, if not slightly more so. TDDFT with CAM-B3LYP has a tendency to slightly underpredict the excitation energy, which is slightly less pronounced in ΔSCF. The exception is one very noticeable outlier, highlighted with a circle in Fig. 2. FIG. 1 . Absolute error in the magnitude of the transition dipole moment calculated using ΔSCF vs EOM-CCSD, as a function of the overlap between the ground and excited states. The coordinates of the molecules have been translated by [100, 100, 100] Å in panel (a) compared to (b). Without any correction (red dots), the coordinate shift results in unphysically large transition dipoles. This can be avoided by using the symmetric orthogonalization correction (black dots). All calculations used the aug-cc-pVTZ basis set.ΔSCF calculations were performed using the CAM-B3LYP functional. J. Chem. Phys. 154, 124106 (2021); doi: 10.1063/5.0041233 154, 124106-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Excitation energies calculated using TDDFT (black cross) or ΔSCF (red dot) compared to the EOM-CCSD reference values. The target values are given by the solid black line y=x. The inset in the lower right-hand corner shows the probability distribution for the error in each method compared to the EOM-CCSD reference. The outlier circled in green is excluded from this error analysis (see also Table I). All calculations used the aug-cc-pVTZ basis set.ΔSCF and TDDFT calculations were performed using the CAM-B3LYP functional. This outlier is a perpendicular ethene dimer, and it serves to illustrate a key situation where ΔSCF may not be an appropriate choice of method. The two highest occupied molecular orbitals in the ground state of the ethene dimer are degenerate, representing theπ-bonding orbital on each monomer. The two lowest unoccu- pied molecular orbitals are similarly very close in energy and are in- phase and out-of-phase combinations of the π-antibonding orbitals on each molecule. Both EOM-CCSD and TDDFT predict that the two lowest energy excitations of the ethene dimer are degenerate lin- ear combinations of the local excitations with an excitation energy of 7.5 eV and transition dipole moments in the xandydirections (the principal axis being z).ΔSCF, by construction, cannot capture the mixed nature of these excitations59and instead predicts excitations with energies around 7 and 10 eV (shown) and transition dipoles in thexyplane. Excluding the outlier, the mean error in the ΔSCF excitation energies compared to EOM-CCSD is 0.35 eV, with a standard devi- ation of 0.25 eV (Table I). For TDDFT, the mean error is 0.41 eV, with a standard deviation of 0.27 eV. For excitation energies, ΔSCF is, therefore, clearly worth considering as a cheap and accurate alter- native to TDDFT. This is in good agreement with earlier studies benchmarking ΔSCF excitation energies for large organic dyes.36,60 Figure 3 shows the same comparison for \| μ\|. By eye, ΔSCF produces slightly more scatter around the EOM-CCSD reference than TDDFT but has a broadly similar accuracy. This is borne out in a more detailed numerical analysis. The mean error in \| μ\| forΔSCF compared to EOM-CCSD is 0.07 a.u. (atomic units for transition dipoles = ea0), with a standard deviation of 0.08 a.u. (Table I). For TDDFT, the mean error is 0.03 a.u., with a standard deviation of 0.06 a.u.TABLE I . Error in excitation energies and transition dipole magnitudes calculated usingΔSCF and TDDFT at the CAM-B3LYP/aug-cc-pVTZ level of theory. Errors are calculated relative to an EOM-CCSD reference value. The outliers highlighted in Figs. 2 and 3 are excluded from the error analysis for the energies and dipole moment, respectively. Error in ΔE(eV) Error in \| μ\| (ea0) ΔSCF TDDFT ΔSCF TDDFT Mean 0.35 0.41 0.07 0.03 Standard deviation 0.25 0.27 0.08 0.06 Min. 0.01 0.02 0.00 0.00 Max. 1.63 1.24 0.52 0.34 There is, again, a single obvious outlier where ΔSCF appar- ently performs far worse than TDDFT. This outlier corresponds to a stretched version of the benzene molecule. Like the ethene dimer described above, the lowest energy excitation of this structure is a roughly equal mix of HOMO to LUMO and HOMO −1 to LUMO + 1 transitions. In this case, however, both HOMO and HOMO − 1 and LUMO and LUMO + 1 are exactly degenerate and this cre- ates some flexibility in the definition of the transition and its dipole moment. The transition dipole moment found by ΔSCF agrees very well with that for an excitation that is an equal mix of HOMO to LUMO + 1 and HOMO −1 to LUMO, which, given the degeneracy of the states, is an equally valid choice. This outlier should, there- fore, be viewed not as a failure of ΔSCF but as a reminder that there is no one correct transition dipole moment when degenerate states are involved. FIG. 3 . Transition dipole magnitudes (\| μ\|) calculated using TDDFT (crosses) or ΔSCF (dots) compared to the EOM-CCSD reference values. The target values are given by the solid black line y=x. The inset in the lower right-hand corner shows the probability distribution for the error in each method compared to the EOM- CCSD reference. The outlier circled in green is excluded from this error analysis (see also Table I). All calculations used the aug-cc-pVTZ basis set. ΔSCF and TDDFT calculations were performed using the CAM-B3LYP functional. J. Chem. Phys. 154, 124106 (2021); doi: 10.1063/5.0041233 154, 124106-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp This test set contains two other structures for benzene, with slightly different bond lengths. For these variations, coupled clus- ter and TDDFT find nearly pure, degenerate HOMO to LUMO and HOMO −1 to LUMO + 1 transitions, for which ΔSCF predicts very accurate transition dipoles. SinceΔSCF was unable to capture the true nature of the tran- sition in the ethene dimer discussed above, we might expect this to account for one of the larger errors in Fig. 3. However, it hap- pens that the magnitude of the dipole moment for the combined transition is very similar to that of the single-determinant transi- tion predicted by ΔSCF (although their directions are different). In general, there does not appear to be a strong correlation between error in the ΔSCF excitation energy and the ΔSCF transition dipole magnitudes. Having established the performance of ΔSCF vs TDDFT, we move on to look at the performance of ΔSCF in calculating the tran- sition properties of the 27 BChla in the LHII complex of purple bac- teria. As before, we focus on the HOMO–LUMO transition, which, in this case, is the Q ytransition between two sets of π-bonding molecular orbitals spread over the conjugated ring system. The cor- responding transition dipole lies approximately along an axis con- necting opposing nitrogen atoms on the tetrapyrrole ring. We take the structures of the chromophores from a single snapshot of the molecular dynamics simulation by Stross et al.18This chromophore is too large to treat with EOM-CCSD, so we use TDDFT as our ref- erence, bearing in mind its performance on the test set of smaller molecules. We use the PBE0 functional61,62and Def2-SVP basis set,63 in line with Ref. 18. As shown in Fig. 2, the excitation energies calculated using ΔSCF correlate extremely well with those predicted by TDDFT and lie well within the range of error of TDDFT. This suggests both that theΔSCF excitation energies are accurate and that small variations in the energy between the different chromophores are physically meaningful. By contrast, there is a significant difference between the mag- nitude of the transition dipoles predicted by TDDFT and ΔSCF, withΔSCF predicting magnitudes that are, on average, 0.42 a.u. larger. This is larger than the average error expected for TDDFT and ΔSCF compared to EOM-CCSD but within the full range of errors observed for the test set of small molecules. We note that the differ- ence between TDDFT and ΔSCF will have contributions from the error in both methods and it is not clear from Fig. 4, which will be the largest contribution. However, while it appears that the errorin the ΔSCF transition dipole moment is toward the higher end of what we might expect, it is reassuring that the values remain well- correlated with those from TDDFT. This suggests that ΔSCF could be used to create a valid picture of how the transition dipoles of each chromophore change over the course of a molecular dynamics simulation. One chromophore is missing from Fig. 4, as the ΔSCF calcula- tion collapsed back to the ground state. This is a hazard of the ΔSCF method, and we plan to keep working on robustness, including, for example, by implementing the initial maximum overlap method34 (IMOM) based on orbitals from an initial averaged calculation. V. DISCUSSION We have benchmarked the excitation energies and transition dipole magnitudes predicted by ΔSCF for a large set of small organic molecules. In line with previous work on larger, organic chro- mophores, we have shown that ΔSCF predicts excitation energies with very similar accuracy to TDDFT, compared to a highly accurate EOM-CCSD reference. TDDFT still outperforms ΔSCF in predict- ing the magnitudes of transition dipoles, but the error in the ΔSCF predictions are sufficiently small that it can still be considered a use- ful alternative when TDDFT is too computationally demanding or when speed is of greater importance than higher precision. In con- trast to earlier studies, we have focused on testing a large number of different molecules, rather than a range of functionals and basis sets. A potential downside of many excited state SCF methods, including the MOM, used here, is that the excited state molecular orbitals are optimized independently of the ground state orbitals and there is consequently no guarantee that the ground and excited states will be orthogonal. In their paper first introducing the MOM, Gilbert et al. argue that orthogonality is not an expected property of SCF states, which are approximations of the exact quantum states.33They further demonstrate that the MOM tends to converge on excited states that only overlap with the ground state by a small amount. Nevertheless, even a small overlap can introduce a problematic origin dependence into the calculation of the transition dipole moment, particularly when the relevant part of the molecule is not close to the origin of the coordinate axis. We have demonstrated that performing a symmetric orthogonalization of the ground and excited states produced by the SCF optimization is a simple way to FIG. 4 . (a) Excitation energies and (b) transition dipole magnitudes of the 27 BChla molecules in the LHII complex of purple bacteria, calculated using ΔSCF vs TDDFT. All calculations used the PBE0 functional and Def2-SVP basis set. To highlight the correlation between the methods, we plot the lines y=x+C on each subplot. The interpretation of the intercept Cis discussed in the text. J. Chem. Phys. 154, 124106 (2021); doi: 10.1063/5.0041233 154, 124106-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp remove these small overlaps without introducing error into the calculation of the excitation energy or significantly changing the identity of the states. We have demonstrated the use of this correction in the context of simulating photosynthetic antenna complexes, which consist of multiple chromophores arranged into a larger aggregate structure. In a molecular dynamics simulation, for example, these complexes would typically be centered globally on the origin, with each individual chromophore, therefore, being displaced well away from the origin. Applying our simple correc- tion to the transition density matrix is significantly more straight- forward than recentering every single chromophore (while also keeping track of its original position relative to all the other chro- mophores). We anticipate that this trick will be extremely useful in the application of cheaper excited-state SCF methods to biological systems. We have seen that the greatest potential for ΔSCF to fail occurs when the transition of interest is highly mixed in nature. This is not surprising, since ΔSCF is constructed to deal with transitions between a single occupied ground state orbital and a single (relaxed) virtual orbital. Highly mixed transitions usually occur when there are low-lying virtual orbitals of the same symmetry with very sim- ilar energies. By calculating the energies and symmetries of the molecular orbitals (programs like Gaussian provide an option to do this automatically), a simple inspection would identify molecules with a greater risk of highly mixed transitions, helping to deter- mine whether ΔSCF could be appropriately used. Furthermore, large molecules, for which TDDFT may become prohibitively expen- sive, typically have much lower symmetry than the small molecules considered here, greatly reducing the chances that near-degenerate molecular orbitals of the same symmetry will exist. Looking forward, we suggest that there is potential to further improve the ability of ΔSCF to predict accurate transition dipole moments. Previous work by Kowalczyk et al.36demonstrates that much of the error in ΔSCF excitation energies arise from spin contamination and that this effect is more pronounced for function- als with a smaller amount of exact exchange. While excitation ener- gies can be, at least partially, corrected for spin contamination using the spin purification formula described above, this correction does not extend to the molecular orbitals used to calculate the transition density and related properties. We hypothesize that the performance ofΔSCF for transition dipoles could be improved by incorporating spin purification into the calculation of the molecular orbitals. This could be done, for example, by minimizing the spin-purified energy in the SCF cycle, rather than applying the correction at the end of the energy calculation. Trialing such a procedure is, however, beyond the scope of the current study. SUPPLEMENTARY MATERIAL See the supplementary material for all numerical data presented in this paper, and xyz structure files for the chlorophyll geometries. ACKNOWLEDGMENTS We gratefully acknowledge the funding agencies that supported this work: O.F. was funded by the U.S. Department of Energy(Grant No. DE-FOA-0001912). 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5.0014487.pdf	J. Math. Phys. 61, 103304 (2020); https://doi.org/10.1063/5.0014487 61, 103304 © 2020 Author(s).Statistical mechanics with non-integrable topological constraints: Self-organization in knotted phase space Cite as: J. Math. Phys. 61, 103304 (2020); https://doi.org/10.1063/5.0014487 Submitted: 19 May 2020 . Accepted: 19 September 2020 . Published Online: 08 October 2020 Naoki Sato ARTICLES YOU MAY BE INTERESTED IN Retraction: “Transition between multimode oscillations in a loaded hair bundle” [Chaos 29, 083135 (2019)] Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 049901 (2020); https:// doi.org/10.1063/5.0008545 Dirac structures in nonequilibrium thermodynamics for simple open systems Journal of Mathematical Physics 61, 092701 (2020); https://doi.org/10.1063/1.5120390 COVID-19 in the United States: Trajectories and second surge behavior Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 091102 (2020); https:// doi.org/10.1063/5.0024204Journal of Mathematical PhysicsARTICLE scitation.org/journal/jmp Statistical mechanics with non-integrable topological constraints: Self-organization in knotted phase space Cite as: J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 Submitted: 19 May 2020 •Accepted: 19 September 2020 • Published Online: 8 October 2020 Naoki Satoa) AFFILIATIONS Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan a)Author to whom correspondence should be addressed: sato@ppl.k.u-tokyo.ac.jp ABSTRACT The object of this study is the statistical mechanics of dynamical systems lacking a Hamiltonian structure due to the presence of non-integrable topological constraints that limit the accessible regions of the phase space. Focusing on the simplest three dimen- sional case, we develop a procedure (Poissonization) that assigns to any three dimensional non-Hamiltonian system an equivalent four dimensional Hamiltonian system endowed with a proper time. The statistical distribution is then constructed in the recovered four dimensional canonical phase space. Projecting in the original reference frame, we show that the statistical distribution departs from standard Maxwell–Boltzmann statistics. The deviation is a function of the knottedness of the phase space, which is measured by the helicity density of the topological constraint. The theory is then generalized to a class of non-Hamiltonian systems in higher dimensions. Published under license by AIP Publishing. https://doi.org/10.1063/5.0014487 I. INTRODUCTION Self-organizing phenomena can be divided into two main categories depending on their thermodynamic properties. On one hand, there are systems, often observed in chemistry and biological sciences, which sustain complex internal structures by exchanging energy and matter with the surrounding environment. These are thermodynamically open systems that are described by non-equilibrium thermodynamics.1,2 On the other hand, certain systems occurring in physics and astrophysics, typically involving fluids and plasmas, exhibit ordered structures that persist even at thermodynamic equilibrium and in a thermodynamically isolated environment. The driving principle for this second type of self-organization is the existence of constraints affecting the phase space, the so called topological constraints.3Topological constraints are intrinsically different from those thermodynamic constraints on the inflow/outflow of energy and matter that enable the formation of stationary non-equilibrium structures in open systems. Indeed, they are purely geometrical in nature. Of course, one type of self-organization does not preclude the other. To understand the physical origin of topological constraints, it is useful to consider a simple example, the motion of a rigid body. Among all the degrees of freedom (positions and momenta) of its microscopic constituents, only the three components of the angular velocity of the rigid body are needed for the description of the dynamics. Redundant degrees of freedom effectively behave as topological constraints that force the system to a small portion of the original phase space. In general, on the spatial and time scales where topological constraints hold, macroscopic structures may arise. Indeed, while the realiza- tion of the totality of microstates eventually results in the disappearance of order by the second law of thermodynamics, reduction of degrees of freedom is synonymous with inhomogeneity. The formulation of statistical mechanics in a topologically constrained phase space represents a major mathematical challenge that must reconcile the emergence of organized structures with the thermodynamic principle of entropy growth. The center of the problem is that a J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-1 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp constrained phase space does not possess, in general, an invariant measure. Furthermore, when available, it is usually “hidden” by a non- trivial change of coordinates. While this difficulty shakes the classical construction of statistical mechanics at its foundation, it also enables us to push the limits of the theory and to address open questions pertaining to the notion of entropy4,5and the applicability of the ergodic ansatz6,7in a generalized framework. There are two types of topological constraints: integrable and non-integrable. Here, integrability is defined in the differential geomet- ric sense of the Frobenius theorem of differential forms.8In essence, the effect of an integrable constraint is to restrict the dynamics to a Casimir leaf, i.e., a smooth submanifold embedded in the phase space, whereas the region of phase space allowed by a non-integrable constraint fails to define a smooth surface.9This situation can be visualized in a three dimensional setting. Here, a topological constraint is represented by a vector field to which the tangent direction of any orbit must be orthogonal. For an integrable constraint, the vec- tor field defines the normal of a surface, and we, therefore, speak of a foliated phase space. For a non-integrable constraint, there is no surface because the vector field is locally “knotted” (it has a non-vanishing helicity density). In this case, the phase space is knotted (see Fig. 1). Statistical mechanics in the presence of integrable constraints has been formulated successfully. Here, the key point is that the con- strained dynamics takes the form of a noncanonical Hamiltonian system.9,10The Casimir leaves are identified by solving for the center (kernel) of the Poisson operator that is associated with the Hamiltonian structure. Then, the Lie–Darboux theorem11–13of differential geometry assigns a set of canonically conjugated variables that define an invariant volume element on the Casimir leaf. The proper (ther- modynamically consistent) entropy measure is thus defined with respect to this measure, and self-organization of heterogeneous struc- tures is explained by the mismatch between the metric tensor of the original unconstrained phase space and that of the foliated phase space.14,15 While the Lie–Darboux theorem is a local result that holds in a finite number of dimensions, the construction described above can be generalized to infinite dimensional noncanonical Hamiltonian systems, such as ideal fluids and magnetohydrodynamic plasmas.10,16,17In this case, the invariant measure is obtained by expanding the phase space variables, which are elements of a Hilbert space, with respect to a discrete basis of orthogonal functions, e.g., a Fourier basis. Then, the statistical distribution is defined for the coefficients of the expansion, which represent a countable number of canonically conjugated variables.18,19 The situation is different when the topological constraints are non-integrable. Indeed, it is known that non-integrable constraints destroy the Hamiltonian structure. More precisely, the antisymmetric operator generating the constrained dynamics fails to satisfy the Jacobi iden- tity,20implying that the associated bracket, while consistent with conservation of energy, does not define a Poisson algebra.21Several physical systems possess non-integrable constraints, usually representing special types of rigidity. Examples of this kind of non-Hamiltonian structure are nonholonomic mechanical systems,22certain charged particle motions in plasmas, such as pure ExB drift dynamics,23the Nosé–Hoover thermostat equations of molecular dynamics,24,25or the Landau–Lifshitz equation for the magnetization in a ferromagnet.26,27These systems do not possess, in general, a time-independent invariant measure.28,29Therefore, the construction of statistical mechanics requires a funda- mentally different approach. Indeed, while ergodicity (the property by which the time spent by a particle with a given energy in an accessible region of the phase space is proportional to the volume of that region) cannot hold in the presence of an integrable constraint, in the inte- grable case, the problem can be solved by enforcing the ergodic property on the smaller phase space defined by Casimir leaves. However, non-integrability does not dictate a reduced phase space, and the applicability of the ergodic hypothesis becomes a major challenge. In this regard, in Ref. 30, it is shown that if a system subject to non-integrable constraints is perturbed with homogeneous fluctuations, the result- ing equilibrium distribution function departs from standard Maxwell–Boltzmann statistics. The discrepancy is measured by the field charge, a quantity that corresponds to the divergence of the Lorentz force in the electromagnetic analogy and that mathematically expresses the FIG. 1. (a) Foliated phase space: the motion of a point particle is restricted to a smooth surface (Casimir leaf). (b) Knotted phase space: the constraint does not define a surface. The orbits tend to diverge due to the missing invariant measure. J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-2 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp departure of the antisymmetric operator from a Beltrami field (in three dimensions, Beltrami fields are vector fields aligned with their own curl that arise as steady solutions of fluid and magnetofluid equations). The connection between the equilibrium properties of statistical ensembles with non-integrable constraints under the effect of homogeneous fluctuations and the geometric properties of stationary solutions of the ideal Euler equations is investigated in Ref. 31. In this paper, we are concerned with the construction of statistical mechanics in the presence of non-integrable constraints. Our strategy is to recover a Hamiltonian phase space structure by introducing fictitious degrees of freedom that compensate the phase space compressibility of the system and to rescale the time variable so that the Jacobi identity is satisfied again. The possibility of reconstruct- ing an invariant measure by increasing the number of variables was already observed in Ref. 30, while the definition of a proper time to elevate an antisymmetric operator to a Poisson operator finds its roots in the work of Chaplygin32concerning the rolling of a symmetri- cal sphere on a horizontal plane and is discussed in the context of the Hamiltonization of nonholonomic systems through reduction by symmetry.33,34 Here, we show that the “Poissonization” procedure described above always holds in three dimensions. In higher dimensional spaces, the method works for a specific class of non-Hamiltonian systems spanned by coupled variables that define two dimensional invariant volume elements. Thanks to the recovered Hamiltonian structure, the standard construction of statistical mechanics is readily applicable. In partic- ular, the relaxation process driving the system toward thermodynamic equilibrium can now be derived in a systematic fashion,35–37and the equilibrium distribution function can be calculated by means of an H-theorem.38We find that once projected into the original phase space, the resulting equilibrium distribution function departs from the Maxwell–Boltzmann distribution. The deviation is a function of the helicity density of the constraint, which measures the knottedness of the phase space. The Maxwell–Boltzmann distribution is recovered in the limit of vanishing helicity density. This paper is organized as follows: In Sec. II, we introduce the mathematical notation and the dynamical systems of interest. In Sec. III, we discuss two examples of non-Hamiltonian systems subject to non-integrable constraints in three dimensions. In Sec. IV, we show how to assign a Poisson structure to an arbitrary three dimensional non-Hamiltonian system. In Sec. V, we apply the Poissonization procedure to an example pertaining to plasma physics. In Sec. VI, we calculate the equilibrium distribution function and determine its dependence on the helicity density of the constraint. The generalization of the theory to higher dimensional spaces is discussed in Sec. VII. Conclusions are drawn in Sec. VIII. II. PRELIMINARIES For the purpose of this paper, we are concerned with a finite number of dimensions n. Let(x1,...,xn)denote a coordinate system in a domain U⊂Rn. In the following, Einstein’s summation convention on repeated indices is used. Lower indices applied to functions denote partial derivatives, e.g., Hj=∂H/∂xj, and we assume that all quantities possess the regularity needed for differentiations. The equations of motion of a system with topological constraints take the form ˙xi=JijHj,i=1,...,n. (1) Here,Jij=−Jji,i,j=1,...,n, is an antisymmetric matrix (antisymmetric operator) whose entries are functions of the variables xi.His the Hamiltonian function, which usually represents the energy of the system. The antisymmetry of the operator Jguarantees conservation of energy, ˙H=HiJijHj=0. (2) Equation (1) defines a noncanonical Hamiltonian system whenever the antisymmetric operator Jsatisfies the Jacobi identity JimJjk m+JjmJki m+JkmJij m=0, i,j,k=1,...,n. (3) Recall that for the case of non-integrable constraints, the Jacobi identity (3) is never satisfied. Hence, the corresponding dynamics is not Hamiltonian. Conversely, it can be shown that all constraints in a noncanonical Hamiltonian system are integrable (Lie–Darboux theorem11–13). Topological constraints are encapsulated in the kernel of the operator J. More precisely, we say that a covector ξ=(ξ1,...,ξn)Tis a topological constraint whenever Jijξj=0, i=1,...,n. (4) Equation (4) implies that the velocity ˙xis always orthogonal to the constraint ξ, regardless of the choice of H, ˙x⋅ξ=JijHjξi=0,∀H. (5) The covector ξdefines a direction in phase space that is not accessible to the dynamics. Furthermore, this direction does not depend on the properties of matter (the energy H), but it represents an intrinsic property of the phase space (which is characterized by J). J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-3 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp Since the number mof topological constraints corresponds to the dimension of the kernel ker (J)of the operator J, we always have m=n−2r≤n. Here, 2 ris the rank of J, which is always even due to antisymmetry. The constraints (ξ1,...,ξm)are integrable if one can findmfunctions (C1,...,Cm)whose gradients ∇C=Ci∇xispan ker (J), i.e., ker(J)=span{ξ1,...,ξm}=span{∇C1,...,∇Cm}. (6) The functions Ciare the anticipated Casimir invariants, which define madditional conservation laws, ˙Ci=Ci jJjkHk=0, i=1,...,m. (7) The integrability conditions on the constraints ξfor Eq. (6) to hold are provided by the Frobenius theorem.8Since in this paper we are concerned with non-integrable constraints, these conditions are always violated, meaning that there is no Casimir foliation of the phase space and the resulting dynamics is not Hamiltonian. Finally, we remark that while, for physical reasons, our focus is oriented toward the role of constraints in self-organization, the theory applies to the limiting case m=0 of an empty kernel. Indeed, the mathematical discriminant for the validity of theory is the violation of the Jacobi identity (3). III. SYSTEMS WITH NON-INTEGRABLE TOPOLOGICAL CONSTRAINTS In this section, we consider two examples of dynamical systems with non-integrable constraints in R3. Taking Cartesian coordinates (x1,x2,x3)=(x,y,z), the action of the antisymmetric operator Jon the Hamiltonian Hcan be represented by the cross product with a vector fieldw, ˙x=J∇H=w×∇H. (8) Here,w=(wx,wy,wz)T=(J32,J13,J21)T. Now, the Jacobi identity (3) reads w⋅∇×w=0. (9) Hence, system (8) is Hamiltonian whenever whas a vanishing helicity density. In the following, we shall denote the helicity density of the vector field wash=w⋅∇×w. The topological constraint is given by ξ=w. Indeed, ˙x⋅w=0,∀H, (10) implying that the velocity ˙xis always orthogonal to the constraining vector field w. The Frobenius integrability condition for the constraint wis again h=0. Intuitively, this is because the vector field wdefines the normal of a surface C=constant whenever w=λ∇Cfor some appropriate choice of functions λandCso that h=λ∇C⋅∇λ×∇C=0. The Frobenius theorem guarantees that the functions λandCalways exist locally, provided that h=0. It is also useful to elucidate the conditions under which system (8) admits an invariant measure μdV=μdxdydz for any choice of the Hamiltonian H. In formulas, one must find a non-vanishing function μsuch that ∇⋅(μ˙x)=∇H⋅∇×(μw)=0,∀H. (11) This condition is satisfied whenever w=μ−1∇νfor some potential ν. For an integrable constraint, μ−1=λandν=C. We, therefore, conclude that, in three dimensions, the validity of the Jacobi identity, the integrability of the constraint w, and the existence of an invariant measure for any choice of the Hamiltonian function are locally equivalent. However, note that when n>3, this equivalence does not hold anymore because the Frobenius integrability condition and the existence of an invariant measure independent of Honly represent the necessary conditions for the validity of the Jacobi identity. For further details on the cases n≠3, we refer the reader to Ref. 30. A. The non-Hamiltonian plasma particle A prototypical example of three dimensional dynamics with a non-integrable constraint is the so called E×Bdrift motion of charged particles. Consider a charged particle of mass mand charge Zmoving in a static magnetic field Band a static electric field E=−∇ϕ. The equation of motion is J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-4 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp m¨x=Z(˙x×B+E). (12) Suppose that the mass mis sufficiently small so that the left-hand side of Eq. (12) can be neglected. Taking the cross product of both sides withB, we obtain ˙x/⊙◇⊞=E×B B2, (13) where ˙x/⊙◇⊞is the component of the velocity in the direction perpendicular to the magnetic field. We further demand that the particle does not move along the magnetic field, i.e., ˙x∥=˙x−˙x/⊙◇⊞=0. Then, we obtain the reduced system of equations of motion ˙x=E×B B2=B B2×∇ϕ. (14) Equation (14) has the form of Eq. (8) with w=B/B2andH=ϕ. Furthermore, Eq. (14) is Hamiltonian if and only if the Jacobi identity (9) is satisfied. In light of the previous discussion, this occurs when one can find two locally defined functions λandCsuch that B=B2λ∇C. An example is the case in which the magnetic field is a harmonic vector field, i.e., B=∇ξwith∇⋅B=0 so thatλ=B−2andC=ξ. However, the Jacobi identity does not hold in the presence of a magnetic field with a non-vanishing helicity density because h=B−4B⋅∇× B. Therefore, E×Bdrift motion (14) is not a Hamiltonian system in general. Below, we compare two examples that highlight the difference between the Hamiltonian and the non-Hamiltonian dynamical settings. First, consider the motion of a rigid body with angular momentum xand momenta of inertia Ix,Iy, and Iz. We have w=x, (15a) H=1 2(x2 Ix+y2 Iy+z2 Iz), (15b) This system is Hamiltonian because the Jacobi identity is satisfied: h=x⋅∇× x=0. The constraining vector field wdefines the normal to the spherical surface x2/2=constant. Hence, C=x2/2 is a Casimir invariant physically representing conservation of total angular momentum. The equations of motion can be expressed as ˙x=∇x2 2×∇H=yz(1 Iz−1 Iy)∂x+ xz(1 Ix−1 Iz)∂y+xy(1 Iy−1 Ix)∂z.(16) Here,(∂x,∂y,∂z)denotes the standard vector basis of R3. Next, consider the E×Bdrift motion of a charged particle with w=(cosz+ sin z)∂x+(cosz−sinz)∂y, (17a) H=1 2(x2+y2+z2). (17b) Note that the magnetic field and the electric field are given by B=w/w2andE=−∇H, respectively. This system is not Hamiltonian because the Jacobi identity is violated: h=2≠0. The equation of motion is ˙x=(cosz−sinz)z∂x−(cosz+ sin z)z∂y +[(cosz+ sin z)y−(cosz−sinz)x]∂z.(18) Figure 2(a) shows the trajectory of the rigid body, and Fig. 2(b) shows the trajectory of the charged particle. Both trajectories lie on the surface of constant energy H=constant. However, while the orbit of the rigid body is a closed curve resulting from the intersection of the level sets of energy Hand Casimir invariant C, the charged particle spirals toward a “sink” and delineates an open path characterized by the non-vanishing divergence of the velocity (18). This example shows that there is a relationship between the existence of an invariant measure and the Hamiltonian nature of the system. The absence of an invariant measure may be interpreted as the consequence of missing degrees of freedom that would compensate the compressibility of the system. This is why we will need to “extend” the system in order to recover a Hamiltonian structure. J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-5 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp FIG. 2. (a) Numerical integration of Eq. (16). (b) Numerical integration of Eq. (18). B. The Landau–Lifshitz equation of ferromagnetism A second example of three dimensional dynamics subject to a non-integrable constraint is the motion of the magnetization xof a ferromagnetic material as described by the Landau–Lifshitz equation,30,39 ˙x=−γx×H+σx×(H×x) x2. (19) Here,γis a physical constant, σis the damping parameter, and His the effective magnetic field, which includes any external magnetic field and the magnetic field caused by the magnetization. The first term of (19) describes the precessional motion of the magnetization around the effective magnetic field H. The second term of (19) is a damping (dissipative) effect that tends to align the magnetization with the effective magnetic field H. Equation (19) conserves the total magnetization H=x2/2, which we identify with the Hamiltonian function. Then, the constraining vector field whas the expression w=γH+σ x2x×H. (20) In general, the effective magnetic field His a function of the magnetization x. For example, we may choose H=(c, 0,z)T, with cbeing a constant external magnetic field. One can verify that the corresponding Jacobi identity is violated. Indeed, h=cγσ x4[σyz γ−2cx2−2z4 c+x(x2+y2−3z2)]. (21) J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-6 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp FIG. 3. Superpositions of solutions of the equation ˙x=w×(∇H+Γ), wherewis given by (20), H=x2/2, and Γis a Gaussian white noise process mimicking fluctuations in the magnetization of the ferromagnetic material. Observe how the magnetization describes a damped precessional motion around the zaxis, which represents the direction of easiest magnetization of the ferromagnet. Figure 3 shows the qualitative evolution of magnetization. IV. RECONSTRUCTION OF PHASE SPACE The purpose of this section is to develop a systematic procedure (Poissonization) to “repair” an arbitrary three dimensional antisymmet- ric operator and obtain an equivalent Hamiltonian system describing the same dynamics. As anticipated in the Introduction, the procedure consists of two steps: an extension to four dimensions through the introduction of a fictitious degree of freedom sand a time reparameteriza- tiont→τ. The physical interpretation of the variable sand the proper time τwill be discussed in Sec. V when we apply the procedure to the E×Bdrift motion of charged particles. To embed the system in four dimensions n=4 with(x1,x2,x3,x4)=(x,y,z,s), it is convenient to return to the matrix representation J of the three dimensional antisymmetric operator. The embedding is performed by extending Jto a four dimensional antisymmetric operator Jin the following way: J=⎡⎢⎢⎢⎢⎢⎢⎢⎣0−wzwya wz 0−wxb −wywx 0 c −a−b−c0⎤⎥⎥⎥⎥⎥⎥⎥⎦. (22) Here, the coefficients a,b, and chave to be determined by demanding that the extended equations of motion ˙xi=JijHj,i=1,...,n, (23) are a Hamiltonian system up to a time reparameterization. Note that the Hamiltonian His unchanged, implying that ∂H/∂x4=0. Therefore, the new terms a,b, and cdo not alter the original equations of motion, which are given by ˙x1,˙x2, and ˙x3. The time reparameterization is defined by the differential equation dτ dt=r, (24) where r=r(x1,...,x4)is a non-vanishing function called conformal factor that will be determined when enforcing the Jacobi identity on the time-reparameterized extended system. In proper time, the equations of motion read dxi dτ=r−1JijHj,i=1,...,n. (25) Now our task is to choose a,b,c, and rso that the antisymmetric operator r−1Jsatisfies the Jacobi identity (3). Suppose that the antisymmetric matrix with components r−1Jijis invertible with inverse Ωij. Then, the Jacobi identity (3) is equivalent to the condition that ∂Ωij ∂xk+∂Ωjk ∂xi+∂Ωki ∂xj=0, i,j,k=1,...,n. (26) J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-7 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp Mathematically, Eq. (26) expresses the closure of Ωijwhen intended as a differential two form. The equivalence can be verified by multiplying each side of the equation by r−3JliJmjJnkand by summing over i,j,k. The inverse matrix can be calculated to be Ω=r awx+bwy+cwz⎡⎢⎢⎢⎢⎢⎢⎢⎣0 c−b−wx −c 0 a−wy b−a 0−wz wxwywz 0⎤⎥⎥⎥⎥⎥⎥⎥⎦. (27) Equation (26) can be satisfied by assuming that r=awx+bwy+cwz, (28a) a=∂ ∂y(Az+swz)−∂ ∂z(Ay+swy), (28b) b=∂ ∂z(Ax+swx)−∂ ∂x(Az+swz), (28c) c=∂ ∂x(Ay+swy)−∂ ∂y(Ax+swx), (28d) where A=(Ax,Ay,Az)Tis an arbitrary vector field whose components Ax,Ay, and Azare functions of x,y, and zonly. We conclude that the antisymmetric operator r−1Jresulting from substitution of (28) is a four dimensional Poisson operator satisfying the Jacobi identity (3). The vector field Amust be specified by requiring that the conformal factor rhas a definite sign and that it converges to unity when the helicity density htends to zero. The first condition ensures that the proper time τis monotonic with respect to changes in time t, while the second condition implies that proper time τcoincides with twhen the original three dimensional dynamics is a Hamiltonian system. Observe that, from (28), r=w⋅∇× (A+sw)=w⋅∇× A+sh. (29) To enforce the conditions above, we choose Ato be a solution of the first order partial differential equation w⋅∇× A=1 (30) and restrict the domain of the fictitious degree of freedom sso that sh≠−1. For example, we may demand that sis a periodic variable, s∈[0,1 M),M=2 sup Ω∣h∣. (31) Note that with this choice, r=1 +sh≥1 2>0. (32) We have thus shown that an arbitrary three dimensional non-Hamiltonian system can be transformed to an equivalent four dimensional Hamiltonian system with a proper time. The effect of the extension to four dimensions is to restore an Hamiltonian independent invariant measure. Indeed, the extended equations of motion in time tcan be written in vector notation as ˙x=w×∇H, (33a) ˙s=−∇× (A+sw)⋅∇H. (33b) Hence, the divergence of the vector field V=(˙x,˙y,˙z,˙s)Tis given by div(V)=∇⋅(w×∇H)−∇×w⋅∇H=0,∀H. (34) Once the incompressibility is recovered, a Hamiltonian structure can be obtained through a time reparameterization (32) that absorbs the effect of a non-vanishing helicity density hon the dynamics. J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-8 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp V. POISSONIZATION of E×BDYNAMICS Here, we apply the procedure developed in Sec. IV to Poissonize the E×Bdrift motion studied in Sec. III A. This exercise will help us understand the physical meaning of the new degree of freedom sand the proper time τ. First, recall that the constraining vector field wis given by Eq. (17a). Furthermore, wis related to the magnetic field Bbyw=B/B2[see Eq. (14)]. Since w⋅B=1, Eq. (30) can be satisfied by identifying Awith the vector potential associated with the magnetic field, i.e., B=∇×A. Then, the equation of motion (33b) for shas the expression ˙s=−(B+sB B2)⋅∇H=−(1√ 2+s√ 2)∂H ∂ℓ. (35) Here, we used the fact that ∇×w=wandB2=1/2. Furthermore, we introduced the variable ℓthat measures the length along a field line and has the tangent vector ∂ℓ=B/B. Next, performing the change of variables ˜v∥=log(1 + 2 s) m√ 2, (36) we have md˜v∥ dt=−∂H ∂ℓ. (37) From this equation, we see that the variable ˜ v∥can be interpreted as a velocity in the direction parallel to the magnetic field B, i.e., ˜v∥=˙ℓ. Thus, the missing degree of freedom sdescribes a fictitious dynamics along the magnetic field. We remark again that, however, the dynamics associated with sdoes not correspond to a real orbit in R3because the parallel component of the velocity was removed in the derivation of Eq. (14). Since the helicity density takes the value h=2, the conformal factor is dτ dt=r=1 + 2 s=e√ 2m˜v∥. (38) Note that dτ/dt=1 when s=˜v∥=0. Recalling that, by hypothesis, the mass of the particle is small, and the exponential on the right-hand side of (38) can be expanded in powers of√ 2m˜v∥, dτ dt=1 +√ 2m˜v∥+o(2m2˜v2 ∥). (39) Neglecting the second order terms and using ˜ v∥=˙ℓ, one obtains τ≃t+√ 2mℓ. (40) Hence, the discrepancy between proper time τand time tis proportional to the fictitious length traveled by the particle along the magnetic field. This shows that we can think of log (1 + 2 s)/√ 2 and (τ−t)/√ 2mas coupled momentum and position coordinates. The Poisson operator r−1Jnow has the expression r−1J=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣0 02By 1 + 2 sBx 0 0 −2Bx 1 + 2 sBy −2By 1 + 2 s2Bx 1 + 2 s0 0 −Bx−By 0 0⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (41) with Bx=(cosz+ sin z)/2 and By=(cosz−sinz)/2. In order to write the equations of motion in Hamilton’s canonical form, we must find a change of coordinates that transforms (41) into a symplectic matrix. This can be accomplished by the transformation qx=(1 + 2 s)Bx, (42a) px=−x, (42b) qy=(1 + 2 s)By, (42c) J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-9 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp py=−y. (42d) The transformation above can be derived in a straightforward manner by invoking the exactness of the inverse matrix Ωwhen intended as a differential two-form. These technical aspects will be discussed in Sec. VII. In terms of new variables, the following relationships hold: z=arcsin⎡⎢⎢⎢⎢⎢⎣qx−qy√ 2(q2x+q2y)⎤⎥⎥⎥⎥⎥⎦, (43a) s=−1 2+√ q2x+q2y, (43b) H=1 2⎛ ⎜ ⎝p2 x+p2 y+ arcsin2⎡⎢⎢⎢⎢⎢⎣qx−qy√ 2(q2x+q2y)⎤⎥⎥⎥⎥⎥⎦⎞ ⎟ ⎠. (43c) Hamilton’s canonical equations have the expressions dqx dτ=∂H ∂px=px, (44a) dpx dτ=−∂H ∂qx=−qy q2x+q2yarcsin⎡⎢⎢⎢⎢⎢⎣qx−qy√ 2(q2x+q2y)⎤⎥⎥⎥⎥⎥⎦, (44b) FIG. 4. Numerical integration of system (44). (a) Evolution of px,qx/τ,py, and qy/τwith respect to the proper time τ. (b) Evolution of s+ 1/2 and zwith respect to the proper timeτ. J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-10 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp dqy dτ=∂H ∂py=py, (44c) dpy dτ=−∂H ∂qy=qx q2x+q2yarcsin⎡⎢⎢⎢⎢⎢⎣qx−qy√ 2(q2x+q2y)⎤⎥⎥⎥⎥⎥⎦. (44d) Figure 4 shows a numerical integration of the canonical Hamiltonian system (44). Note that the solution progressively approaches a two dimensional uniform rectilinear motion. This orbit should be compared with the original orbit in R3[Fig. 2(b)]. Finally, observe that in time t, the equations of motion for the variables px,qx,py, and qytake the form ˙qx=r−1Hpx, (45a) ˙px=−r−1Hqx, (45b) ˙qy=r−1Hpy, (45c) ˙py=−r−1Hqy. (45d) These equations, which are not Hamiltonian, imply that the “force” acting on the particle is only proportional to the gradient of the Hamil- tonian with the proportionality factor r−1. Therefore, the same energy gradient produces different forces depending on the position in space. Such behavior departs from the standard laws of physics and signals the importance of the Jacobi identity in determining the struc- ture of the equations of motion. This inhomogeneity is also the reason why Hamiltonian equations can be restored by rescaling the time variable. VI. STATISTICAL MECHANICS IN KNOTTED PHASE SPACE: THERMODYNAMIC EQUILIBRIUM The purpose of this section is to exploit the reconstructed canonical phase space to derive the distribution function of thermody- namic equilibrium for an ensemble of particles with equations of motion (8) by following the classical formulation of statistical mechan- ics. To this end, we must identify the invariant measure of the system. As already shown in Eq. (34), in time t, the preserved volume element is dV=dxdydzds . (46) However, this measure is different from the preserved volume element associated with proper time τ. Indeed, by Liouville’s theorem, the canonical phase space measure is dΠ=dpxdqxdpydqy, (47) where px,qx,py, and qyare the canonical variables obtained by application of the Poissonization procedure (recall that, once a Poisson operator is obtained, canonical variables can always be constructed locally in accordance with the Lie–Darboux theorem). In general, given two vector fields dx/dtanddy/dτ, with x=(x1,...,xn)Tandy=(y1,...,yn)Tand such that ∂ ∂xi(dxi dt)=∂ ∂yi(∂yi ∂τ)=0, (48) it can be shown that the Jacobian gof the coordinate change dx1...dxn=gdy1...dynis given by g=dt dτ. (49) Indeed, using (48), we have ∂ ∂xi(dxi dt)=1 g∂ ∂yi(gdτ dtdyi dτ) =1 gdyi dτ∂ ∂yi(gdτ dt) =1 gd dτ(gdτ dt).(50) J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-11 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp This quantity vanishes when gdτ/dtis constant in proper time τ. By a rescaling of t, we may assume this constant to be unity and therefore obtain (49) (for the sake of simplicity, we do not discuss the case in which non-integrable constraints and Casimir invariants coexist; in such a case, both the invariant measure and distribution function will depend on the Casimir invariants, but the derivations are essentially the same). Applying this result to the four dimensional case x=(x,y,z,s)Tandy=(px,qx,py,qy)T, we conclude that the Jacobian gof the coordinate change is g=dt dτ=r−1=1 1 +sh. (51) This formula for the Jacobian can be verified explicitly for the example studied in Sec. V [Eq. (42)]. LetP=P(y) be the distribution function of an ensemble of particles in the canonical phase space with the volume element dΠat ther- modynamic equilibrium τ→∞. We want to know how the distribution function Pis seen in the coordinates with the volume element dV. Using Eq. (51), we have PdΠ=PrdV , (52) which implies that the distribution function f(x)ondVis related to Pas f=Pr=P(1 +sh). (53) From the result above, we see that the discrepancy between Pand fis controlled by the helicity density h. Furthermore, by integrating over the variable s, we can calculate the shape of the distribution F(x,y,z)in the original coordinates ( x,y,z), F=∫s1 s0f ds=∫s1 s0Pds+h∫s1 s0Psds . (54) Here,[s0,s1]is the domain of the variable s. Let us now calculate the form of the distributions at thermodynamic equilibrium. Since dΠis the preserved volume element of a symplectic manifold spanned by canonical variables, we can exploit the usual formulation of statistical mechanics and define the differential entropy Σof the distribution function Pas follows: Σ=−∫ΠPlogP dΠ. (55) Here, the integral is performed on the whole phase space Π. The total number of particles and the total energy Eof the ensemble are given by N=∫ΠP dΠandE=∫ΠHP dΠ, respectively. The form of the distribution function at equilibrium is calculated my maximizing the entropy Σ under the constraints NandEaccording to the variational principle, δ(Σ−αN−βE)=0. (56) Here,αandβare the Lagrange multipliers associated with NandE. The result of the variation is P=1 Ze−βH. (57) In the above equation, Z=e1+αis the normalization constant. Thus, recalling Eqs. (53) and (54), we arrive at the following formulas for fand Fat thermodynamic equilibrium: f=1 Z(1 +sh)e−βH, (58a) F=s1−s0 Z(1 +s1+s0 2h)e−βH. (58b) The conclusion is that the thermodynamic equilibrium of a three dimensional ensemble governed by an antisymmetric operator departs from the standard Boltzmann distribution of homogeneous probability density on constant energy surfaces. The distortion is controlled by the helicity density h, i.e., by the failure of the Jacobi identity. As an example, consider an ensemble of magnetized particles moving by E×Bdrift according to Eq. (14). The magnetic field Bis assumed to be of the form J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-12 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp FIG. 5. Thermal equilibrium F(x,y)byE×Bdrift in the magnetic field (59). The inhomogeneous distribution is caused by the knottedness of the phase space, which is quantified by the helicity density hof the topological constraint. B=∂x+(y−sinycosy 2−sinx)∂z. (59) Note that∇⋅B=0. Recalling that the constraining vector field is w=B/B2, we have w=∂x+(y−sinycosy 2−sinx)∂z 1 +(y−sinycosy 2−sinx)2(60) and also h=sin2y [1 +(y−sinycosy 2−sinx)2]2. (61) A typical scenario encountered in magnetized plasmas is quasi-neutrality. In such a situation, the electric potential ϕis, on average, zero. Therefore, the Hamiltonian of each “massless” particle is itself zero, H=ϕ=0. However, electrostatic fluctuations δϕgenerated by random interactions among charged particles drive the ensemble toward equilibrium, which according to (58b) is F=s1−s0 Z⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩1 +(s1+s0)sin2y 2[1 +(y−sinycosy 2−sinx)2]2⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭. (62) Here, we used Eq. (61). Figure 5 shows a plot of the predicted thermal equilibrium. The shape of the distribution departs from the flat profile one would expect by a naïve application of the entropy principle in the initial (non-Hamiltonian) coordinates. This discrepancy is a consequence of the failure of the Jacobi identity, i.e., the knottedness of the phase space. VII. GENERALIZATION TO HIGHER DIMENSIONS In this section, we consider the case in which the dimension of the starting space is n>3. The discussion unavoidably requires the wedge notation of differential geometry, but it provides insight into the mechanism allowing a non-Hamiltonian system to be transformed in a Hamiltonian one by the procedure developed in this study. This section is therefore not essential to convey the message of this paper and is given for the interested reader who is familiar with differential geometry. The wedge product is convenient to represent antisymmetric tensors. Denoting by (∂1,...,∂n)the basis of tangent vectors associated with a coordinate system (x1,...,xn)in a domain U⊂Rn, and with (dx1,...,dxn)the basis of cotangent vectors for the dual space, the wedge product on pairs of tangent or cotangent vectors consists of the tensor products J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-13 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp ∂i∧∂j=∂i⊗∂j−∂j⊗∂i, (63a) dxi∧dxj=dxi⊗dxj−dxj⊗dxi. (63b) Using this notation, a twice contravariant antisymmetric tensor Jand a twice covariant antisymmetric tensor Ωcan be expressed as J=1 2Jij∂i∧∂j, (64a) Ω=1 2Ωijdxi∧dxj. (64b) The closure condition (26) ensuring that the equations of motion define a Hamiltonian system now reads dΩ=1 2∂Ωij ∂xkdxi∧dxj∧dxk=0. (65) Note that, here, the wedge product is used to represent a three times covariant antisymmetric tensor. Equation (27) is equivalent to Ω=r awx+bwy+cwz {[a−s(∂w z ∂y−∂w y ∂z)]dy∧dz +[b−s(∂w x ∂z−∂w z ∂x)]dz∧dx +[c−s(∂w y ∂x−∂w x ∂y)]dx∧dy +d(wxs)∧dx+d(wys)∧dy+d(wzs)∧dz}.(66) From this expression, it is now clear why by enforcing the conditions of Eq. (28), the two-form Ωsatisfies Eq. (26). Indeed, if we associate with the constraining vector field wand the vector field Athe differential one forms θandAdefined by θ=wxdx+wydy+wzdz, (67a) A=Axdx+Aydy+Azdz, (67b) one can verify that the two-form Ωbecomes Ω=d(A+sθ). (68) It immediately follows that dΩ=0. Consider again the plasma particle of Sec. III A. In this case, θ=2A=(sinz+ cos z)dx+(cosz−sinz)dyso that Ω=d[(s+1 2)θ] =−d{xd[(s+1 2)(cosz+ sin z)] +yd[(s+1 2)(cosz−sinz)]} =dpx∧dqx+dpy∧dqy.(69) From this equation one sees that the canonically conjugated variables are those of Eq. (42). Suppose that we are given a dynamical system in the form (1), with n>3. By adding the new degrees of freedom, we can always make the dimension of the system even. Furthermore, as shown in Ref. 30, the extension can be carried out so that the extended system pos- sesses an invariant measure for any choice of the Hamiltonian function: given an antisymmetric operator J, the extended antisymmetric operator is J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-14 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp J=J+xn+1∂Jij ∂xi∂j∧∂n+1, (70) where xn+1is the new variable. The existence of the Hamiltonian independent invariant measure is expressed by the property ∂Jij ∂xi=0, j=1,...,n+ 1. (71) If the starting system has an even number of dimensions, but it does not possess an Hamiltonian independent invariant measure, the invari- ant measure can be recovered by first extending the system so that it has an odd number of dimensions and then by extending it again according to Eq. (70). Therefore, from this point on, we shall assume that n=2mfor some positive integer m≥2, and that the opera- torJhas been obtained from an extension such that the volume element dx1∧⋅⋅⋅∧ dx2mis an invariant measure for any choice of the Hamiltonian H. Now consider a 2 mdimensional antisymmetric matrix Mwith constant entries Mij=−Mji∈R. By application of an orthogonal transformation, the matrix Mcan be cast in the block diagonal form M=m ∑ i=1λi∂i∧∂m+i,λi∈R, (72) where±iλiare the complex eigenvalues of M. Provided that the antisymmetric operator Jis sufficiently regular, we expect a representation analogous to (72) to hold upon performing a suitable change of coordinates x=(x1,...,x2m)T→y=(y1,...,y2m)T, J=m ∑ i=1λi˜∂i∧˜∂m+i, (73) where ( ˜∂1,...,˜∂2m) is the basis of tangent vectors associated with the new coordinate system. Note that the λi=λi(y)are now functions of y. IfJadmits an inverse ˜Ω, it, therefore, has the expression ˜Ω=m ∑ i=1λ−1 idym+i∧dyi. (74) Note that d˜Ω≠0, in general, since the λiare not constants. Hence, the equations of motion generated by Jdo not define a Hamiltonian system, although they possess an invariant measure. We want to show that if the functions λican be factorized as λ−1 i=α1...αi−1αi+1...αm, (75) for some functions αi=αi(yi,ym+i)≠0,i=1,...,m, then the equations of motion ˙xi=JijHj,i=1,..., 2m (76) can be Poissonized by introducing the time reparameterization dt=r−1dτ=α1...α2mdτ. (77) To see this, it is sufficient to show that the two-form Ω=r˜Ω, which represents the inverse of the antisymmetric operator r−1Jarising from the time reparameterization, is closed. By construction, we have dΩ=dm ∑ i=1dym+i∧dyi αi(yi,ym+i)=0. (78) The class of dynamical systems that admit the factorization (75) can be characterized by the property ∂˙yi ∂yi+∂˙ym+i ∂ym+i=λi(∂2H ∂yi∂ym+i−∂2H ∂ym+i∂yi)=0, (79) i.e., each pair (yi,ym+i)defines a two dimensional incompressible flow. J. Math. Phys. 61, 103304 (2020); doi: 10.1063/5.0014487 61, 103304-15 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp VIII. CONCLUDING REMARKS In this paper, we investigated the formulation of statistical mechanics for dynamical systems that are governed by non-Hamiltonian equations of motion due to the presence of non-integrable topological constraints. These systems occur, for example, in plasma physics, fer- romagnetism, molecular dynamics, and nonholonomic dynamics. We showed that, given a three dimensional non-Hamiltonian system, it is always possible to construct an equivalent Hamiltonian system by introducing a new fictitious degree of freedom that compensates for the compressibility of the system and a proper time that absorbs the knottedness of the phase space (the helicity density of the constraining vector field). This procedure applies to non-Hamiltonian systems in higher dimensions when the antisymmetric operator satisfies the conditions discussed in Sec. VII. Once the Hamiltonian structure is recovered, the statistical distribution can be defined in the classical way on the invari- ant measure assigned by Liouville’s theorem. We found that, at thermodynamic equilibrium, the system self-organizes into a heterogeneous state and that the statistical distribution seen in the original reference system departs from the standard Maxwell–Boltzmann distribution, the discrepancy being measured by the helicity density of the constraining vector field. ACKNOWLEDGMENTS N.S. was partially supported by JSPS KAKENHI Grant Nos. 18J01729 and 17H01177. The author is grateful to Professor Z. Yoshida for useful discussion on the statistical mechanics of constrained systems. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. 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1.2390668.pdf	Quasistatic nonlinear characteristics of double-reed instruments André Almeidaa/H20850 IRCAM–Centre Georges Pompidou–CNRS UMR9912, 1 Place Igor Stravinsky, 75004 Paris, France Christophe Vergezb/H20850 Laboratoire de Mécanique et Acoustique–CNRS UPR7051, 31 Ch. Joseph Aiguier, 13402 Marseille Cedex 20, France René Causséc/H20850 IRCAM–Centre Georges Pompidou–CNRS UMR9912, 1 Place Igor Stravinsky, 75004 Paris, France /H20849Received 15 June 2006; revised 12 October 2006; accepted 15 October 2006 /H20850 This article proposes a characterization of the double reed in quasistatic regimes. The nonlinear relation between the pressure drop, /H9004p, in the double reed and the volume ﬂow crossing it, q,i s measured for slow variations of these variables. The volume ﬂow is determined from the pressuredrop in a diaphragm replacing the instrument’s bore. Measurements are compared to otherexperimental results on reed instrument exciters and to physical models, revealing that clarinet,oboe, and bassoon quasistatic behavior relies on similar working principles. Differences in theexperimental results are interpreted in terms of pressure recovery due to the conical diffuser role ofthe downstream part of double-reed mouthpieces /H20849the staple /H20850.© 2007 Acoustical Society of America. /H20851DOI: 10.1121/1.2390668 /H20852 PACS number /H20849s/H20850: 43.75.Ef /H20851NHF /H20852 Pages: 536–546 I. INTRODUCTION A. Context The usual method for studying and simulating the be- havior of self-sustained instruments is to separate them intotwo functional parts that interact through a set of linked vari-ables: the resonator, typically described by linear acoustics,and the exciter, a nonlinear element. Although this separationmay be artiﬁcial because of the difﬁculty in establishing aprecise boundary between the two systems, it is usually asimpliﬁed view that allows one to describe the basic func-tioning principles of the instrument. In reed instruments, forinstance, the resonator is assimilated to an air column insidethe bore, and the exciter to the reed, which acts as a valve. In the resonator of reed instruments, the relation be- tween the acoustic variables, pressure /H20849p/H20850and volume ﬂow /H20849q/H20850, can be described by a linear approximation to the acous- tic propagation which has no perceptive consequences in sound simulations /H20851Gilbert et al. /H208492005 /H20850/H20852. On the other hand, the exciter is necessarily a nonlinear component, so that thecontinuous source of energy supplied by the pressure insidethe musician’s mouth can be transformed into an oscillatingone /H20851Helmholtz /H208491954 /H20850; Fletcher and Rossing /H208491998 /H20850/H20852. The characterization of the exciter thus requires the knowledge ofthe relation between variables pandqat the reed output /H20849the coupling region /H20850. In principle this relation is noninstanta- neous, because of inertial effects in the reed oscillation andthe ﬂuid dynamics. Nevertheless, a ﬁrst insight /H20849and com- parison to theoretical models /H20850can be achieved by restricting the measurement of the characteristics to a case where de-layed dependencies /H20849or, equivalently, time derivatives in the mathematical description of the exciter /H20850can be neglected. This paper aims at measuring the relation between the pressure drop across the reed and volume ﬂow at the double-reed output in a quasistatic case, that is, when the time varia-tions of pandqare sufﬁciently small so that all time deriva- tives can be neglected in the nonlinear characteristic relation,and proposing a model to explain the measured relation. B. Elementary reed model In quasistatic conditions, a simple model can be used to describe the reed behavior /H20851Wilson and Beavers /H208491974 /H20850; Backus /H208491963 /H20850/H20852. The reed opening area /H20849S/H20850is controlled by the difference between the pressure inside the reed /H20849pr/H20850and the pressure inside the mouth /H20849pm/H20850. In the simplest model, the relation between pressure and reed opening area is consid- ered to be linear and related through a stiffness constant /H20849ks/H20850, /H20849/H9004p/H20850r=pm−pr=kS/H20849S0−S/H20850. /H208491/H20850 In this formula, S0is the reed opening area at rest, when the pressure is the same on both sides of the reed. In most in-struments /H20849such as clarinets, oboes, or bassoons /H20850the reed is said to be blown-closed /H20849orinward-striking /H20850/H20851Helmholtz /H208491954 /H20850/H20852, because when the mouth pressure /H20849p m/H20850is increased, the reed opening area decreases. The role of the reed is to control and modulate the vol- ume ﬂow /H20849q/H20850entering the instrument. The Bernoulli theorem applied between the mouth and the reed duct determines the velocity of the ﬂow inside the reed /H20849ur/H20850independently of the reed opening area,a/H20850Electronic mail: andre.almeida@ircam.fr b/H20850Electronic mail: vergez@lma.cnrs-mrs.fr c/H20850Electronic mail: rene.causse@ircam.fr 536 J. Acoust. Soc. Am. 121 /H208491/H20850, January 2007 © 2007 Acoustical Society of America 0001-4966/2007/121 /H208491/H20850/536/11/$23.00 Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termspm+1 2/H9267um2=pr+1 2/H9267ur2. /H208492/H20850 In this equation, /H9267is the air density. Usually, the flow veloc- ityumis neglected inside the mouth, because of volume flow conservation: inside the mouth the flow is distributed along amuch wider cross section than inside the reed duct. The volume ﬂow /H20849q/H20850is the integrated ﬂow velocity /H20849u r/H20850 over a cross section of the reed duct. For the sake of sim- plicity, the ﬂow velocity is considered to be constant over thewhole opening area, so that q=Su r. Using Eq. /H208492/H20850, the ﬂow is given by q=S/H208812/H20849pm−pr/H20850 /H9267. /H208493/H20850 Combining Eq. /H208493/H20850and Eq. /H208491/H20850, it is possible to ﬁnd the relation between the variables that establish the couplingwith the resonator /H20849p randq/H20850, q=pM−/H20849/H9004p/H20850r ks/H9267/H208812/H20849/H9004p/H20850r /H9267. /H208494/H20850 The relation deﬁned by Eq. /H208494/H20850is plotted in Fig. 1, con- stituting what will be called in this article the elementarymodel for the reed. The static reed beating pressure /H20851Dalmont et al. /H208492005 /H20850/H20852 p M=ksS0/H20849minimum pressure for which the reed channel is closed /H20850is an alternative parameter to S0, and can be used as a magnitude for proposing a dimensionless pressure, p˜=/H20849/H9004p/H20850r/pM. /H208495/H20850 Similarly a magnitude can be found for q, leading to the definition of the dimensionless volume flow, q˜=ks pM3/2/H20881/H9267 2q. /H208496/H20850 Equation /H208494/H20850can then be rewritten in terms of these dimen- sionless quantities, q˜=/H208491−p˜/H20850p˜1/2. /H208497/H20850 This formula shows that the shape of the nonlinear char- acteristic curve of the elementary model is independent ofthe reed and blowing parameters, although the curve isscaled along the pressure pand volume ﬂow qaxis both by the stiffness k sand the beating pressure pM=ksS0. C. Generalization to double reeds For reed instruments, the quasistatic nonlinear character- istic curve has been measured in a clarinet mouthpiece/H20851Backus /H208491963 /H20850; Dalmont et al. /H208492003 /H20850/H20852, and the elementary mathematical model described above can explain the ob-tained curve remarkably well almost until the reed beatingpressure /H20849p M/H20850. For double-reed instruments it was not veriﬁed that the same model can be applied. In fact, there are some geometri-cal differences in the ﬂow path that can considerably changethe theoretical relation of Eq. /H208497/H20850. Local minima of the reed duct cross section may cause the separation of the ﬂow fromthe walls and an additional loss of head of the ﬂow/H20851Wijnands and Hirschberg /H208491995 /H20850/H20852, and in that case the char- acteristics curve could change from single-valued to multi-valued in a limited pressure range. This kind of change couldhave signiﬁcant consequences on the reed oscillations. However, the nonlinear characteristic relation was never measured before for double reeds, justifying the work that ispresented below. II. PRINCIPLES OF MEASUREMENT AND PRACTICAL ISSUES The characteristic curve requires the synchronized mea- surement of two quantities: the pressure drop across the reed/H20849/H9004p/H20850 rand the induced volume ﬂow q. A. Volume ﬂow measurements One of the main difﬁculties in the measurement of the reed characteristics lies in the measurement of the volumeﬂow. There are instruments which can accurately measurethe ﬂow velocity in an isolated point /H20849LDA, hot-wire probes /H20850 or in a region of a plane /H20849PIV /H20850, but it can be difﬁcult to calculate the corresponding ﬂow by integrating the velocityﬁeld. In fact, it is difﬁcult to do a sampling of a completecross section of the reed because a large number of pointswould have to be registered. Supposing that the ﬂow is axi-symmetric at the reed output /H20851which is conﬁrmed by experi- mental results in Almeida /H208492006 /H20850/H20852, the measurement along a diameter of the reed would be sufﬁcient, but regions close tothe wall are inaccessible. On the other hand, commercial ﬂow meters usually have the disadvantage of requiring a direct reading, which wouldhave been impractical for a complete characteristic measure-ment /H20849large number of readings in a short time interval /H20850. An indirect way of measuring the ﬂow was then pre- ferred to the above-mentioned methods. It consists of intro-ducing a ﬂow resistance in series with the reed, for which thepressure can be accurately related to the ﬂow runningthrough it /H20849see Fig. 2 /H20850. The diaphragm method, used successfully by Ollivier /H208492002 /H20850to measure the nonlinear characteristic of single reeds, is based on this principle. The resistance is simply aperforated metal disk which covers the reed output. FIG. 1. A theoretical nonlinear characteristic curve for a reed of dimensions similar to an oboe reed, given by Eq. /H208494/H20850using pM=20 kPa and ks=5 /H11003109kg m−3s−2. J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Almeida et al. : Quasistatic double-reed characteristics 537 Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsFor such a resistance, and assuming laminar, inviscid ﬂow, the pressure drop /H20849/H9004p/H20850d=pr−patmacross the diaphragm can be approximated by the Bernoulli law, because the ﬂow velocity at the reed output is neglected when compared to thevelocity inside the diaphragm /H20849S d/lessmuchSoutput /H20850, /H20849/H9004p/H20850d=pr−patm=1 2/H9267/H20873q Sd/H208742 , /H208498/H20850 where qis the flow crossing the diaphragm, Sdthe cross section of the hole, and /H9267the density of air. In our experi- ment, pressure patmis the pressure downstream of the dia- phragm /H20849usually the atmospheric pressure, because the flow opens directly into free air /H20850. The volume flow qis then determined using a single pressure measurement pr. B. Practical issues and solutions 1. Issues The realization of the characteristic measurement ex- periments encountered two main problems. a. Diaphragm reduces the range of /H20849/H9004p/H20850rfor which the measurement is possible . The addition of a resistance to the air ﬂow circuit of the reed changes the overall nonlinearcharacteristic of the reed plus diaphragm system /H20851corre- sponding to /H20849/H9004p/H20850 sin Fig. 2 and to the dashed line in Fig. 3 /H20852. The solid line plots the ﬂow against /H20849/H9004p/H20850r, the pressure dropneeded to plot the nonlinear characteristics. When the resis- tance is increased, the maximum value of the system’s char-acteristic is displaced towards higher pressures /H20851Wijnands and Hirschberg /H208491995 /H20850/H20852, whereas the static beating pressure /H20849p M/H20850value does not change /H20851because when the reed closes there is no ﬂow and the pressure drop in the diaphragm /H20849/H20849/H9004p/H20850d/H20850is zero /H20852. Therefore, if the diaphragm is too small /H20849i.e., the resis- tance is too high /H20850, part of the decreasing region /H20849B’C/H20850of the system’s characteristics becomes vertical, or even multival- ued, so that there is a quick transition between two distantﬂow values, preventing the measurement of this part of thecharacteristic curve /H20851Dalmont et al. /H208492003 /H20850/H20852as illustrated in Fig. 3. A critical diaphragm size /H20849S d,crit=0.58 S0/H20850can be found below which the characteristic curve becomes multivalued /H20849see the Appendix /H20850. b. Reed auto-oscillations . Auto-oscillations have to be prevented here to stay consistent with the quasistatic mea-surement /H20849slow variations of pressure and ﬂow /H20850. This proved to be difﬁcult to achieve in practice. In fact, auto-oscillationsbecome possible when the reed ceases to act as a passiveresistance /H20849a positive /H11509q//H11509p, which absorbs energy from the standing wave inside the reed channel /H20850to become an active supply of energy /H20849/H11509q//H11509p/H110210/H20850. All real acoustic resonators are slightly resistive /H20849the input admittance Yinhas a positive real part /H20850. This can compensate in part the negative resistance of the reed in its active region, but only below a threshold pres-sure, where the slope of the characteristic curve is smallerthan the real part of Y infor the resonator as shown by Debut and Kergomard /H208492004 /H20850. One way to avoid auto-oscillations is thus to increase the real part of Yin, which is the acoustic resistance of the reso- nator. It is known that an oriﬁce in an acoustical duct with asteady ﬂow works as an acoustic resistance /H20851Durrieu et al. /H208492001 /H20850/H20852, so that if the diaphragm used to measure the ﬂow /H20849see Sec. II A /H20850is correctly dimensioned, the acoustic admit- tance seen by the reed Y incan become sufﬁciently resistive to avoid oscillations. 2. Solutions proposed to address these issues a. Size of the diaphragm . The volume ﬂow is determined from the pressure drop across the diaphragm placed down-stream of the reed. In practice, there is a trade-off that deter-mines the ideal size of the diaphragm. If it is too wide, thepressure drop is too small to be measured accurately, andreed oscillations are likely to occur. If the diaphragm is toosmall, the system-wide characteristic can become too steep,making part of the /H20849/H20849/H9004p/H20850 r/H20850range inaccessible. The ideal diaphragm cross section is then found empiri- cally, by trying out several resistance values until one com-plete measurement can be done without oscillations or sud-den closings of the reed. The optimal diaphragm diameter issought using a medical ﬂow regulator with continuously ad-justable cross section as a replacement for the diaphragm. b. Finer control of the mouth pressure p m. During the attempts to ﬁnd an optimal diaphragm, it was found thatsudden closures were correlated to sudden increases in themouth pressure. A part of the problem is that the mouthpressure depends both on the reducer setting and on the FIG. 2. /H20849Color online /H20850Use of a diaphragm to measure ﬂow and pressure difference in the reed. Labeled rectangles correspond to the pressure probesused in the measurement. FIG. 3. Comparison of the theoretical reed characteristics /H20849solid line /H20850with the model of the overall characteristics of the reed associated with a dia-phragm /H20849dashed /H20850—mathematical models, based on the Bernoulli theorem: Based on Wijnands and Hirschberg /H208491995 /H20850. 538 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Almeida et al. : Quasistatic double-reed characteristics Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsdownstream resistance. By introducing a leak upstream of the experimental apparatus /H20849thus not altering the experi- ment /H20850, it is possible to improve measurements in the decreas- ing region of the characteristic /H20849BC/H20850, at least when the system-wide characteristic is not multivalued /H20849see Sec. I IB1a /H20850. c. Increase the reed mass . One other way to reduce the oscillations is thus to prevent the appearance of instabilities,or to reduce their effects. An increase in the reed dampingwould certainly be a good method to avoid oscillations, be-cause it cancels out the active role of the reed /H20849which can be seen as a negative damping /H20850, Debut /H208492004 /H20850. It is difﬁcult to increase the damping of the reed without altering its opening or stiffness properties. The simplest wayfound to prevent reed oscillations was thus an increase in thereed mass. This mass increase was implemented by attaching small masses of Blu-Tack /H20849a plastic sticking material usually used to stick paper to a wall /H20850to one or both blades of the reed /H20849Fig. 4 /H20850. During measurements on previously soaked reeds it was difﬁcult to keep the masses attached to the reed, so thatan additional portion of Blu-Tack is used to connect the two masses together, wrapping around the reed. This wrapping isnot expected to have a great effect on the measured elasticproperties, because it does not pull the masses together; itsobjective is to avoid the main masses from falling due to theeffect of gravity. A comparison of the results using differentmasses showed that their effect on the quasistatic character-istics can be neglected /H20849effects are weaker than variations for experiments in the same reed /H20850/H20851Almeida /H208492006 /H20850/H20852.C. Experimental setup and calibrations The experimental device is shown in Fig. 5. An artiﬁcial mouth /H20851Almeida et al. /H208492004 /H20850/H20852was used as a blowing mecha- nism and support for the reed. The window in front of thereed allows the capture of frontal pictures of the reed open-ing. Artiﬁcial lips, allowing adjustment of the initial openingarea of the reed, were not used here, to avoid modiﬁcationsin some of the elastic properties of the reed, possibly in adifferent way from what happens with real lips. As stated before, the plot of the characteristic curve re- quires two coordinated measurements: the pressure differ-ence /H20849/H9004p/H20850 racross the reed and the induced volume ﬂow q, determined from the presure drop /H20849/H9004p/H20850dacross a calibrated diaphragm /H20849Sec. II A /H20850. In practice thus, the experiment requires two pressure measurements pmandpr, as shown in Fig. 5. 1. Pressure measurements The pressure is measured in the mouth and in the reed using Honeywell SCX series, silicon-membrane differentialpressure sensors whose range is from −50 to 50 kPa. These sensors are not mounted directly on the measure- ment points, but one of the terminals in each sensor is con-nected to the measurement point using a short ﬂexible tube/H20849about 20 cm in length /H20850. Therefore, one tube opens in the inside wall of the artiﬁcial mouth, 4 cm upstream from thereed, and the other tube crosses the rubber socket attachingthe diaphragm to the reed output. The use of these tubes doesnot inﬂuence the measured pressures as long as their varia-tions are slow. The signal from these sensors is ampliﬁed before enter- ing the digital acquisition card. The gain is adjusted for eachtype of reed. The system consisting of the sensor connectedto the ampliﬁer is calibrated as a whole in order to ﬁnd thevoltage at the ampliﬁer output corresponding to each pres-sure difference in the probe terminals: the stable pressuredrop applied to the probe is also measured using a digitalmanometer connected to the same volumes, and compared tothe probe tension read using a digital voltmeter. Voltage isfound to vary linearly with the applied pressure within themeasuring range of the sensor. FIG. 4. /H20849Color online /H20850Front view of the reed /H20849sketch /H20850with attached masses, at left in dry conditions, at right in soaked conditions /H20849to prevent the masses from slipping /H20850. FIG. 5. /H20849Color online /H20850Device used for characteristics measurements. J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Almeida et al. : Quasistatic double-reed characteristics 539 Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/terms2. Diaphragm calibration The curve relating volume ﬂow qto the pressure differ- ence through diaphragms /H20849/H9004p/H20850dcan be approximated by the Bernoulli theorem. In fact, diaphragms are constructed so as to minimize friction effects /H20849by reducing the length of the diaphragm channel /H20850and jet contraction—the upstream edges are smoothed by chamfering at 45° /H20849Fig. 6 /H20850. The chamfer height /H20849c/H20850is approximately 0.5 mm. The diaphragm channel is 3 mm long /H20849L/H20850. Nevertheless, this ideal characteristics was checked in stationary conditions for each diaphragm /H20849see Fig. 7 /H20850using a gas volume meter, which would not be usable for variablevolume ﬂows. It was found that the effective cross section isslightly smaller than the actual cross section /H20849about 10% /H20850, which is probably due to some Vena Contracta effect in the entrance of the diaphragm. Moreover, above a given pressuredrop the volume ﬂow increase is lower than what is pre-dicted by Bernoulli’s theorem /H20849corresponding to a lower ex- ponent than 1/2 predicted by Bernoulli /H20850. This difference is probably due to turbulence generated for high Reynoldsnumbers. In Fig. 7 the dashed line corresponding to the criti-cal value of the Reynolds number /H20849Re c=ud//H9263=2000 /H20850is shown. It is calculated using the following formulas for u /H20849the average ﬂow velocity in the diaphragm /H20850andd/H20849the dia- phragm diameter /H20850:d=/H208732 /H9266q u/H208741/2 /H208499/H20850 u=/H208732/H20849/H9004p/H20850d /H9267/H208741/2 , /H2084910/H20850 so that the constant Reynolds relation is given by Q1/2/H9004p1/4=R e c/H9263/H20873/H9266 2/H208741/2/H20873/H9267 2/H208741/4 , /H2084911/H20850 where the right-hand side should be a constant based on the diaphragm geometry. Since a suitable model was not found for the data dis- played in Fig. 7, we chose to interpolate the experimentalcalibrations in order to ﬁnd the ﬂow corresponding to eachpressure drop in the diaphragm. Linear interpolation wasused in the /H20849p,q 2/H20850space. 3. Typical run In a typical run, the mouth pressure pmis balanced with the atmospheric pressure in the room at the beginning of theexperiment. Both p mand prare recorded in the computer through a digital acquisition device at a sampling rate of4000 Hz. Pressure p mis increased until slightly above the pressure at which the reed closes, left for some secondsabove this value, and then decreased back to the atmosphericpressure. The whole procedure lasts for about 3 min, and isdepicted in Fig. 8. D. Double reeds used in this study and operating conditions Among the great variety of double reeds that are used in musical instruments, we chose as a ﬁrst target for these mea-surements a natural cane oboe reed fabricated using standardprocedures /H20849byGlotin /H20850, and sold to the oboist /H20849usually a beginner oboist /H20850as a ﬁnal product /H20849i.e., ready to be played /H20850. The choice of a ready-to-use cane reed was mainly re- tained because it can be considered as an average reed. Thisavoids considering a particular scraping technique amongmany used by musicians and reed makers. Of course, this FIG. 6. Detail of the diaphragm dimensions. FIG. 7. Calibration of diaphragms used in characteristic measurements /H20849dots are experimental data and lines are Bernoulli predictions using the measureddiaphragm diameters /H20850. The dashed black line represents the pressure/ﬂow relation corresponding to the expected transition between laminar and tur-bulent ﬂows /H20849Re c=2000 /H20850, parametrized by the diaphragm diameter d. FIG. 8. Time variation of the mouth pressure /H20849pm/H20850and the pressure inside the reed /H20849pr/H20850during a successful characteristics measurement. 540 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Almeida et al. : Quasistatic double-reed characteristics Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsdoes not greatly facilitate the task of the reed measurement, because natural reeds are very sensitive to environment con-ditions, age, or time of usage. Other reeds were also tested, as a term of comparison with the natural reeds used in most of the experiments. How-ever, none of these reeds was produced by a professionaloboist or reed maker, although it would be an interestingproject to investigate the variations in reeds produced bydifferent professionals. To conclude, the results presented in the next section may depend to a certain extent on the reed chosen for theexperiments, and a larger sample of reeds embracing the bigdiversity of scraping techniques needs to be tested beforeclaiming for the generality of the results that will be pre-sented. Another remark has to be made on the conditions during the experiments. The kind of reeds used in most experimentsare always blown with highly moisturized air. In fact, in reallife, reeds are often soaked before they are used, and con-stantly maintained wet by saliva and water vapor condensa-tion. These conditions were sought throughout most of theexperiments, although the sensitivity of the reed to environ-mental conditions was also investigated. For instance, theadded masses were found to have no practical inﬂuence onthe nonlinear characteristics, whereas the humidity increasesthe hysteresis in the complete measurement cycle /H20849increasing followed by decreasing pressures /H20850, while reducing the reed opening at rest /H20851Almeida /H208492006 /H20850/H20852. In our measurements, humidiﬁcation is achieved by let- ting the air ﬂow through a plastic bottle half-ﬁlled with hotwater at 40° /H20849see Fig. 5 /H20850, recovering it from the top. Air arriving in the artiﬁcial mouth has a lower temperature, be-cause its temperature is approximately 10° when entering thebottle. This causes the temperature and humidity to decreasegradually along the experiments. Future measurementsshould include a thermostat for the water temperature in or-der to ensure stable humidiﬁcation. III. RESULTS AND DISCUSSION A. Typical pressure vs ﬂow characteristics Using the formula of Eq. /H208498/H20850, and the calibrations carried out for the diaphragm used in the measurement, the volumeﬂow /H20849q/H20850is determined from the pressure inside the reed /H20849p r/H20850. The pressure drop in the reed corresponds to the difference between the mouth and reed pressures /H20849/H20849/H9004p/H20850r=pm−pr/H20850. Vol- ume ﬂow is then plotted against the pressure difference /H20849/H20849/H9004p/H20850r/H20850, yielding a curve shown in Fig. 9. In this ﬁgure, the ﬂow is seen to increase until a certain maximum value /H20849/H20849/H9004p/H20850r/H112296 kPa /H20850. When the pressure is in- creased further, ﬂow decreases due to the closing of the reed. Instead of completely vanishing for /H20849/H9004p/H20850r=pM, as predicted by the elementary model shown in Sec. I B, the volume ﬂow ﬁrst stabilizes at a certain minimum value and then slightlyincreases when the pressure is increased further, indicatingthat it is very hard to completely close the reed. The ﬂow remaining after the two blades are in contact suggests that, despite the closed appearance of the doublereed, some narrow channels remaining between the twoblades are impossible to close, behaving like rigid capillary ducts, which is corroborated by the slight increase in theresidual ﬂow for high pressures. Since the logarthmic plot ofthe nonlinear characteristics /H20849Fig. 10 /H20850shows a 1/2 power dependence on the residual volume ﬂow, this suggests thatthe residual ﬂow is controlled by inertia rather than viscosity. When reducing the pressure back to zero, the reed fol- lows a different path in the p/qspace than the path for in- creasing pressures. This hysteresis is due to memory effectsof the reed material which have been investigated experi-mentally for single-reed /H20851Dalmont et al. /H208492003 /H20850/H20852and double- reed instruments /H20851Almeida et al. /H208492006 /H20850/H20852. B. Comparison with other instruments 1. Bassoon Since oboes are not the only double-reed instruments, it is interesting to compare the nonlinear characteristic curvesfrom different instruments. The bassoon is also played usinga double reed, but its dimensions are different: its opening FIG. 9. A typical result for the measurement of the volume ﬂow vs pressure characteristic of a natural cane oboe reed. FIG. 10. Double-logarithmic plot of the characteristic curve of Fig. 9 toshow the 1/2 power dependence when the reed is almost shut. Inset showsthe whole range of data, from which the part corresponding to the closedreed is magniﬁed in the main graph. J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Almeida et al. : Quasistatic double-reed characteristics 541 Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsarea at rest is typically 7 mm2/H20849against around 2 mm2for the oboe /H20850and the cross-section proﬁle varies slightly from oboe reeds. Figure 11 compares two characteristic curves for natural cane oboe and bassoon reeds. Both were measured undersimilar experimental conditions, as far as possible. The reedwas introduced dry in the artiﬁcial mouth, but the suppliedair is moisturized at nearly 100% humidity, and masses wereadded to both reeds to prevent auto-oscillations. The dia-phragms used in each measurement are different, however,and this is because the opening area of the reed at rest ismuch larger in the case of the bassoon, so that a smallerresistance /H20849a larger diaphragm /H20850is needed to avoid the reed closing suddenly in the decreasing side of the characteristiccurve /H20851see Sec. II C and Eq. /H20849A10 /H20850/H20852. This should not have any consequences in the measured characteristic curve. In the qaxis, the bassoon reed reaches higher values, and this is probably a consequence of its larger opening areaat rest, although the surface stiffness is likely to change aswell from the oboe to the bassoon reed. In the paxis, the bassoon reed extends over a smaller range of pressures sothat the reed beating pressure is about 17 kPa in the case ofthe bassoon reed, whereas it is near 33 kPa for the oboe reed. Apart from these scaling considerations, the shapes of the curves are similar and this can be better observed if ﬂowand pressure are normalized using the maximum ﬂow pointof each curve /H20849Fig. 12 /H20850. 2. Clarinet The excitation mechanism of clarinets and saxophones share the same principle of functioning with double reeds.However, there are several geometric and mechanical differ-ences between single reeds and double reeds. For instance,ﬂow in a clarinet mouthpiece encounters an abrupt expansionafter the ﬁrst 2 or 3 mm of the channel between the reed andthe rigid mouthpiece, and the single reed is subject to fewermechanical constraints than any double reed. These differ-ences suggest that the characteristic curve of single-reed in-struments might present some qualitative differences withrespect to the double reed /H20851Vergez et al. /H208492003 /H20850./H20852The nonlinear characteristic curve of clarinet mouth- pieces displayed in Figs. 11 and 12 was measured by Dal-mont et al. /H208492003 /H20850using similar methods as the ones we used for the double reed. A comparison between the curves forboth kinds of exciters /H20849in Fig. 11 /H20850shows that the overall behavior of the excitation mechanism is similar in bothcases. Similarly to when comparing oboe to bassoon reeds,the scalings of the characteristic curves of single reeds aredifferent from those of oboe reeds, although closer to thoseof the bassoon. This is probably a question of the dimensionsof the opening area. A different issue is the relation between reference pres- sure values in the curve /H20849shown in the adimensionalized rep- resentation of Fig. 12 /H20850. As predicted by the elementary model described in Sec. I B, in the single reed the pressure at maxi-mum ﬂow is about 1/3 of the beating pressure of the reed,whereas in double-reed measurements, the relation seems tobe closer to 1/4. This deviation from the model is shown inSec. IV to be linked with the diffuser effect of the conicalstaple in double reeds. Figure 12 also shows that in the clarinet mouthpiece used by Dalmont et al. /H208492003 /H20850the hysteresis is relatively less important than in both kinds of double reeds. In fact,whereas the measurements for double reeds were performedin wet conditions, the PlastiCover® reed used for the clarinetwas especially chosen because of its smaller sensitivity toenvironment conditions. IV. ANALYSIS A. Comparison with the elementary model The measured nonlinear characteristic curve of Fig. 9 can be compared to the model described in Sec. I B. In thismodel, two parameters /H20849k sandS0/H20850control the scaling of the curve along the pandqaxis. They are used to adjust two key points in the theoretical curve to the experimental one: thereed beating pressure p Mand the maximum volume ﬂow qmax. Once qmaxis determined through a direct reading, the stiffness ksis calculated using the following relation: FIG. 11. Comparison of the characteristic curves of different reed exciters for different instruments. Clarinet data were obtained by Dalmont et al. /H208492003 /H20850for a PlastiCover® reed. Oboe and bassoon reeds are blown using moisturized air. FIG. 12. Data from Fig. 11, normalized along qby the maximum ﬂow for increasing pressures, and along pby the corresponding pressure. 542 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Almeida et al. : Quasistatic double-reed characteristics Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsks=qmax−1/H208732 3pM/H208743/2 /H9267−1/2. /H2084912/H20850 This allows adjustment of a theoretical characteristic curve /H20851corresponding to the elementary model of Eq. /H208494/H20850/H20852to each of the branches of the measured characteristic curve, for in-creasing and decreasing pressures /H20849Fig. 13 /H20850. When compared to the elementary model of Sec. I B, the characteristic curve associated with double reeds shows adeviation of the pressure at which the ﬂow reaches its maxi-mum value. In fact, it can be easily shown that for the el-ementary model this value is 1/3 of the reed beating pressure p M, which is also veriﬁed in the clarinet /H20849Sec. III B 2 /H20850.I nt h e measured curves however, this value is usually situated be-tween 1/4 p Mand 1/5 pM. Nevertheless, the shapes of the curves are qualitatively similar to the theoretical ones. B. Conical diffuser The former observations about the displacement of the maximum value can be analyzed in terms of the pressurerecoveries due to ﬂow decelerations inside the reed duct.Variations in the ﬂow velocity are induced by the increasingcross section of the reed towards the reed output /H20849Fig. 14 /H20850. This can be understood simply by considering energy andmass conservation between two different sections of thereed, p in+1 2/H9267/H20873q Sin/H208742 =pout+1 2/H9267/H20873q Sout/H208742 , /H2084913/H20850 where qis the total volume flow that can be calculated either at the input or the output of the conical diffuser by integrat-ing the flow velocity over the cross section S inorSout, re- spectively. In practice, however, energy is not expected to be com- pletely conserved along the ﬂow because of its turbulent na-ture. In fact, for instance at the reed output /H20849diameter d/H20850, the Reynolds number of the ﬂow /H20851Re=ud/ /H9263=4/H20849q//H9266d/H9263/H20850/H20852can be estimated using data from Fig. 9 to reach a maximum value of 5000. Given that this number is inversely proportional tothe diameter of the duct d, the Reynolds number increases upstream, inside the reed duct, so that the ﬂow is expected tobe turbulent also for lower volume ﬂows. For turbulent ﬂows, no theoretical model can be applied to calculate the pressure recovery due to the tapering of thereed duct. However, phenomenological models are availablein engineering literature, where similar duct geometries areknown as “conical diffusers.” Unlike in clarinet mouth-pieces, where the sudden expansion of the proﬁle is likely tocause a turbulent mixing without pressure recovery /H20851Hirsch- berg /H208491995 /H20850/H20852, this effect must be considered in conical dif- fusers. The pressure recovery is usually quantiﬁed in termsof a recovery coefﬁcient C pstating the relation between the pressure difference between both ends of the diffuser and theideal pressure recovery which would be achieved if the ﬂowwas stopped without losses, C P=pout−pin 1 2/H9267uin2. /H2084914/H20850 CPvalues range from 0 /H20849no recovery /H20850to 1 /H20849complete recov- ery, never achieved in practice /H20850. According to Eq. /H2084914/H20850, pressure recovery is proportional to the square of the ﬂow velocity at the entrance of the coni-cal diffuser, and consequently to the squared volume ﬂowinside the reed. The overall pressure difference across thereed /H20849p m−pout/H20850is deduced from the corresponding pressure difference without pressure recovery /H20849pm−pin/H20850according to the formula /H20849pm−pout/H20850=/H20849pm−pin/H20850−/H9251q2, /H2084915/H20850 where /H9251=1 2/H9267/H20849Cp/Sin2/H20850is a constant. This explains why the curve q=f/H20849pm−pout/H20850in Fig. 13 is more shifted to the left compared to the curve q=f/H20849pm−pin/H20850at the top, where qis higher. FIG. 13. Comparison of the experimental nonlinear characteristics curve with the elementary model shown in Fig. 1. Two models are ﬁtted, forincreasing /H20849p M=35 kPa, ks=1.04 /H110031010kg m−3s−2/H20850and decreasing /H20849pM =27 kPa, ks=8.86 /H11003109kg m−3s−2/H20850mouth pressures. FIG. 14. /H20849Color online /H20850Cross-section proﬁles /H20849axis and area /H20850of an oboe reed, measured on a mold of the reed channel, and indexes used in Sec.IV B: mouth, constriction, and reed output. J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Almeida et al. : Quasistatic double-reed characteristics 543 Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/terms1. Reed model with pressure recovery In order to take into account the pressure recovery be- fore the reed output, the ﬂow is divided into two sections, theupstream, until the constriction at 28 mm /H20849index cin Fig. 14 /H20850 and the conical diffuser part from the constriction until thereed output. In the upstream section, no pressure recovery isconsidered, so that the ﬂow velocity can be calculated usingthe pressure difference between the mouth and this pointusing a Bernoulli model, as in Eq. /H208493/H20850, but replacing p rwith pc, q=S/H208812/H20849pm−pc/H20850 /H9267. /H2084916/H20850 Similarly, the reed opening is calculated using the same pressure difference, /H20849/H9004p/H20850c=pm−pc=kS/H20849S0−S/H20850. /H2084917/H20850 The total pressure difference used to plot the character- istic curve, however, is different, because the recovered pres-sure has to be added to /H20849/H9004p/H20850 c, pm−pr=/H20849/H9004p/H20850c−Cp1 2/H9267/H20873q Sc/H208742 , /H2084918/H20850 where Scis the reed duct cross section at the diffuser input, i.e., at the constriction, which is found from Fig. 14, Sc=4 /H1100310−6m2. Using these equations, the modiﬁed model can be ﬁtted to the experimental data. In Fig. 15, the same parameters ks andS0were used as in Fig. 13, leaving only Cpas a free parameter for the ﬁtting. Figure 15 was obtained for a valueofC p=0.8. This value can be compared to typical values of pressure recovery coefﬁcients found in industrial machines/H20851Azad /H208491996 /H20850/H20852. In engineering literature, C Pis found to depend mostly on the ratio between output and input cross sections /H20849AR =Sout/Sin/H20850and the diffuser length to initial diameter ratio /H20849L/din/H20850/H20851White /H208492001 /H20850/H20852. The tapering angle /H9258inﬂuences the growth of the boundary layers, so that above a critical angle/H20849/H9258=8° /H20850the ﬂow is known to detach from the diffuser walls, considerably lowering the recovered pressure. An in-depth study of turbulent ﬂow in conical diffusers can be found inthe literature /H20851Azad /H208491996 /H20850/H20852, usually for diffusers with much larger dimensions than the ones found in the double reed. The geometry of the conical diffuser studied in Azad /H208491996 /H20850can be compared to the one studied in our work: the cross section is circular and the tapering angle /H9258=3.94° is not very far from the tapering angle of the reed staple /H9258 =5.2° /H20851in particular, both are situated in regions of similar ﬂow regimes, Kilne and Abbott /H208491962 /H20850, as a function of the already mentioned ARandL/din/H20852. Reynolds numbers of his ﬂows /H20849Re=6.9 /H11003104/H20850are also close to the maximum ones found at the staple input /H20849Re/H11229104/H20850. The length to input di- ameter ratio of the reed staple L/d=20 is bigger than that found in Azad /H208491996 /H20850; however, the pressure recovery coef- ﬁcient can be extrapolated from his data to ﬁnd the valueC P/H112290.8 /H20851Fig. 2 in Azad /H208491996 /H20850/H20852, or a slightly smaller value ofCP/H112290.7 based on Fig. 6.28b in White /H208492001 /H20850. V. CONCLUSION The quasistatic nonlinear characteristics were measured for double reeds using a similar device as the one used forsingle-reed mouthpieces by Dalmont et al. /H208492003 /H20850. The ob- tained curves are close to the ones found for single reeds, andin particular no evidence of multivalued ﬂows for a samepressure was found, as was suggested by theoretical consid-erations made by Wijnands and Hirschberg /H208491995 /H20850or Vergez et al. /H208492003 /H20850. However, double-reed characteristic curves present sub- stantial quantitative differences for high volume ﬂows whencompared to elementary models for the reed. These differ-ences can be explained using a model of pressure recovery inthe conical staple, proportional to the square of the inputﬂow velocity. In Vergez et al. /H208492003 /H20850a similar model had already been considered with the pressure difference between the jet andthe output of the reed depending on the square of the volumeﬂow q/H20851Eq. 15, p. 969 in Vergez et al. /H208492003 /H20850/H20852, p m=pj+1 2/H9267vj2/H2084919/H20850 pj=pr+1 2/H9267/H90232 Sraq2, /H2084920/H20850 with Srathe cross section of the double reed where the jet reattaches. However, through theoretical considerations, thetypical values /H9023were estimated to be positive /H20849would cor- respond to a negative C p/H20850. The result was a nonlinear char- acteristic q=f/H20849pm−pout/H20850increasingly shifted to the right com- pared to the curve q=f/H20849pm−pin/H20850asqincreases. Based on the experimental results presented in the present paper, it is now possible to explain why the underly-ing conjectures were wrong. In fact, our estimation wasbased on a jet contraction factor /H9251=0.8, whereas our recent experiments have revealed that no jet contraction occurs/H20851Almeida /H208492006 /H20850/H20852. Consequently, in Eq. 16 of Vergez et al. /H208492003 /H20850head losses were the most important terms leading to FIG. 15. Comparison of the experimental nonlinear characteristics curve with a reed model with pressure recovery in the ﬁnal part of the duct. Fittedk sandS0are the same as in Fig. 13 and different for increasing and decreas- ing mouth pressures. In both cases the value Cp=0.8 was used. 544 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Almeida et al. : Quasistatic double-reed characteristics Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsa positive /H9023. On the other hand imposing /H9251=1 would mean Sj=Srain Eq. 16 of Vergez et al. /H208492003 /H20850, leading to /H9023=−1 +/H20849Sra/Sr/H208502+/H9023losses. Given that /H20849Sra/Sr/H208502/H112294/H1100310−2, a nega- tive/H9023, i.e., a positive Cp, is expected. It is worth noting that direct application of the measured characteristic curves to the modeling of the complete oboe isnot that obvious. Indeed, the assumption that the mouthpieceas a whole /H20849reed plus staple /H20850can be modeled as a nonlinear element with characteristics given by the above experimentwould be valid if the size of the mouthpiece was negligiblewith respect to a typical wavelength. Given the 7 cm of themouthpiece and the several tens of centimeters of a typicalwavelength, this is questionable. Should the staple be con-sidered as part of the resonator? In that case, the separationbetween the exciter and the resonator would reveal a longerresonator and an exciter without pressure recovery. Thiscould be investigated through numerical simulations by in-troducing the pressure recovery coefﬁcient /H20849C P/H20850as a free parameter. Moreover, the underlying assumption of the models is that nonstationary effects are negligible /H20849all ﬂow models are quasistatic /H20850. Some clues indicate that this could also be put into question. /H20849i/H20850 First of all, experimental observations of the ﬂow at the output of the staple /H20851through hot-wire measure- ments, Almeida /H208492006 /H20850/H20852revealed signiﬁcant differ- ences in the ﬂow patterns when considering static andauto-oscillating reeds. /H20849ii/H20850Moreover, nondimensional analysis revealed Strouhal numbers much larger than for simple reed instruments/H20851Vergez et al. /H208492003 /H20850, Almeida /H208492006 /H20850/H20852. ACKNOWLEDGMENTS The authors would like to thank J.-P. Dalmont and J. Gilbert for experimental data on the clarinet reed character-istics and fruitful discussions and suggestions about the ex-periments and analysis of data, and A. Terrier and G. Ber-trand for technical support. APPENDIX: CALCULATION OF THE MINIMUM DIAPHRAGM CROSS SECTION The total pressure drop in the reed-diaphragm system /H20849Fig. 2 /H20850is /H20849/H9004p/H20850s=/H20849/H9004p/H20850r+/H20849/H9004p/H20850d. /H20849A1/H20850 The system’s characteristics become multivalued when there is at least one point on the curve where the slope isinﬁnite, /H11509 /H11509q/H20849/H9004p/H20850s=/H11509 /H11509q/H20849/H9004p/H20850r+/H11509 /H11509q/H20849/H9004p/H20850d=0 . /H20849A2/H20850 Because of simplicity, the derivatives in Eq. /H20849A2/H20850are replaced by their inverse, /H11509 /H11509q/H20849/H9004p/H20850s=/H20873/H11509q /H11509/H20849/H9004p/H20850r/H20874−1 +/H20873/H11509q /H11509/H20849/H9004p/H20850d/H20874−1 =0 , /H20849A3/H20850 yielding/H20873S /H9267/H208732/H20849/H9004p/H20850r /H9267/H20874−1/2 −1 ks/H208732/H20849/H9004p/H20850r /H9267/H208741/2/H20874−1 +/H9267 Sd/H208732/H20849/H9004p/H20850d /H9267/H208741/2 =0 . /H20849A4/H20850 Solving for Sd, Sd=−/H208732/H20849/H9004p/H20850d /H9267/H208741/2 /H11003/H20873S /H9267/H208732/H20849/H9004p/H20850r /H9267/H20874−1/2 −1 ks/H208732/H20849/H9004p/H20850r /H9267/H208741/2/H20874. /H20849A5/H20850 Simplifying, Sd=−S/H20873/H20849/H9004p/H20850d /H20849/H9004p/H20850r/H208741/2 +2 ks/H20849/H20849/H9004p/H20850d/H20849/H9004p/H20850r/H208501/2. /H20849A6/H20850 From Eqs. /H208493/H20850and /H208498/H20850, we can ﬁnd /H20849/H9004p/H20850r=/H20873Sd S/H208742 /H20849/H9004p/H20850d, /H20849A7/H20850 and Eq. /H20849A6/H20850can be written Sd=−SS Sd+2 ksS Sd/H20849/H9004p/H20850r. /H20849A8/H20850 Now we can replace S=S0−/H20851/H20849/H9004p/H20850r/ks/H20852to ﬁnd Sd2=/H20873−S0+3/H20849/H9004p/H20850r ks/H20874/H20873S0−/H20849/H9004p/H20850r ks/H20874. /H20849A9/H20850 It is clear that the right-hand side of this equation must be positive. Moreover, it is a parabolic function of /H20849/H9004S/H20850 =/H20849/H9004p/H20850r/ks, with its concavity facing downwards. The maximum value of Sd2/H20849S/H20850, max /H20849Sd2/H20849S/H20850/H20850=S02 3, /H20849A10 /H20850 is thus the value for which there is only a single point where the characteristic curve has an infinite slope. We thus conclude that Sd=S0//H208813=0.58 S0is the mini- mum value of the diaphragm cross section that should beused for ﬂow measurements. Almeida, A. /H208492006 /H20850. “Physics of double-reeds and applications to sound synthesis,” Ph.D. thesis, Univ. Paris VI. Almeida, A., Vergez, C., and Caussé, R. /H208492004 /H20850. “Experimental investiga- tions on double reed quasi-static behavior,” in Proceedings of ICA 2004 , Vol.II, pp. 1229–1232. Almeida, A., Vergez, C., and Caussé, R. /H208492006 /H20850. “Experimental investigation of reed instrument functioning through image analysis of reed opening,”Acustica /H20849accepted /H20850. Azad, R. S. /H208491996 /H20850. “Turbulent ﬂow in a conical diffuser: a review,” Exp. Therm. Fluid Sci. 13, 318–337. Backus, J. /H208491963 /H20850. “Small-vibration theory of the clarinet,” J. Acoust. Soc. Am. 35/H208493/H20850, 305–313. Dalmont, J. P., Gilbert, J., and Ollivier, S. /H208492003 /H20850. “Nonlinear characteristics of single-reed instruments: Quasi-static volume ﬂow and reed openingmeasurements,” J. Acoust. Soc. Am. 114 /H208494/H20850, 2253–2262. Dalmont, J.-P., Gilbert, J., Kergomard, J., and Ollivier, S. /H208492005 /H20850. “An ana- lytical prediction of the oscillation and extinction thresholds of a clarinet,”J. Acoust. Soc. Am. 118 /H208495/H20850, 3294–3305. Debut, V. /H208492004 /H20850.Deux études d’un instrument de musique de type clari- nette: Analyse des fréquences propres du résonateur et calcul des auto-oscillations par décomposition modale /H20849Two studies of a clarinet-like mu- sical instrument: Analysis of the eigen frequencies and calculation of theself-sustained oscillations by modal decomposition /H20850. Ph.D. thesis, Univer- sité de la Mediterranée Aix Marseille II. J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Almeida et al. : Quasistatic double-reed characteristics 545 Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsDebut, V., and Kergomard, J. /H208492004 /H20850. “Analysis of the self-sustained oscil- lations of a clarinet as a Van-der-pol oscillator,” in Proceedings of ICA 2004, Vol. II, pp. 1425–1428. Durrieu, P., Hofmans, G., Ajello, G., Boot, R., Aurégan, Y., Hirschberg, A., and Peters, M. C. A. M. /H208492001 /H20850. “Quasisteady aero-acoustic response of oriﬁces,” J. Acoust. Soc. Am. 110 /H208494/H20850, 1859–1872. Fletcher, N. H., and Rossing, T. D. /H208491998 /H20850.The Physics of Musical Instru- ments , 2nd ed. /H20849Springer, Berlin /H20850. Gilbert, J., Dalmont, J.-P., and Guimezanes, T. /H208492005 /H20850. “Nonlinear propaga- tion in wood-winds,” in Forum Acusticum 2005 , pp. 1369–1372. Helmholtz, H. /H208491954 /H20850.On the Sensations of Tone as a Physiological Basis for the Theory of Music /H20849Dover, New York /H20850/H20849English translation by Alex- ander Ellis /H20850. Hirschberg, A. /H208491995 /H20850.Mechanics of Musical Instruments /H20849Springer, Berlin /H20850, Chap. 7, pp. 229–290. Kilne, S. J., and Abbott, D. E. /H208491962 /H20850. “Flow regimes in curved subsonicdiffusers,” J. Basic Eng. 84, 303–312. Ollivier, S. /H208492002 /H20850.Contribution á l’étude des Oscillations des Instruments à Vent à Anche Simple /H20849“Contribution to the study of oscillations in single- reed wind instruments /H20850.” Ph.D. thesis, Université du Maine, Laboratoire d’Acoustique de l’Université du Maine–UMR CNRS 6613. Vergez, C., Almeida, A., Causse, R., and Rodet, X. /H208492003 /H20850. “Toward a simple physical model of double-reed musical instruments: Inﬂuence ofaero-dynamical losses in the embouchure on the coupling between thereed and the bore of the resonator,” Acta. Acust. Acust. 89, 964–973. White, F. M. /H208492001 /H20850.Fluid Mechanics , 4th ed. /H20849McGraw-Hill, New York /H20850. Wijnands, A. P. J., and Hirschberg, A. /H208491995 /H20850. “Effect of a pipe neck down- stream of a double reed,” in Proceedings of the International Symposium on Musical Acoustics , pp. 149–152. Societe Française d’Acoustique. Wilson, T. A., and Beavers, G. S. /H208491974 /H20850. “Operating modes of the clarinet,” J. Acoust. Soc. Am. 56/H208492/H20850, 653–658. 546 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Almeida et al. : Quasistatic double-reed characteristics Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/terms
1.2172890.pdf	Ultrafast magnetization dynamics investigated in real space (invited) M. Vomir, L. H. F. Andrade, E. Beaurepaire, M. Albrecht, and J.-Y. Bigot Citation: Journal of Applied Physics 99, 08A501 (2006); doi: 10.1063/1.2172890 View online: http://dx.doi.org/10.1063/1.2172890 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/99/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Investigating the contribution of superdiffusive transport to ultrafast demagnetization of ferromagnetic thin films Appl. Phys. Lett. 102, 252408 (2013); 10.1063/1.4812658 Ultrafast studies of carrier and magnetization dynamics in GaMnAs J. Appl. Phys. 107, 033908 (2010); 10.1063/1.3275347 Vortex-antivortex assisted magnetization dynamics in a semi-continuous thin-film model system studied by micromagnetic simulations Appl. Phys. Lett. 86, 052504 (2005); 10.1063/1.1855413 Investigation of ultrafast spin dynamics in a Ni thin film J. Appl. Phys. 91, 8670 (2002); 10.1063/1.1450833 Subnanosecond magnetization dynamics measured by the second-harmonic magneto-optic Kerr effect Appl. Phys. Lett. 74, 3386 (1999); 10.1063/1.123353 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Tue, 25 Nov 2014 16:17:24Ultrafast magnetization dynamics investigated in real space „invited … M. Vomir, L. H. F . Andrade, E. Beaurepaire, M. Albrecht, and J.-Y . Bigota/H20850 Institute of Physics and Chemistry of Materials at Strasbourg (IPCMS), Louis Pasteur University, CNRS, UMR 7504, 23 rue du Loess, BP 43, 67034 Strasbourg Cedex 2, France /H20849Presented on 31 October 2005; published online 25 April 2006 /H20850 The ultrafast magnetization dynamics induced in ferromagnetic thin ﬁlms by femtosecond optical pulses is investigated in real space. Our experimental method allows us to retrieve thethree-dimensional trajectory of the magnetization vector over a large temporal range, from/H11011100 fs to /H110111 ns. This approach carries important information both on the initial spin dynamics and the magnetization precession. An ultrafast decrease of the magnetization modulus, occurringwithin /H11011100 fs, reveals the initial laser induced demagnetization. It is accompanied by a reorientation of the magnetization vector, taking place during the ﬁrst picosecond, a process whichstrongly depends on the material anisotropy. Finally, the three-dimensional trajectory of themagnetization during its precession and damping undertakes a complex pathway as themagnetization modulus varies until the energy is dissipated to the environment in the nanosecondtime scale. © 2006 American Institute of Physics ./H20851DOI: 10.1063/1.2172890 /H20852 I. INTRODUCTION The use of femtosecond optical pulses to induce and investigate fast magnetic processes in highly correlated spinsystems is quite interesting as it brings an unprecedentedtemporal resolution as compared to techniques using pulsedmagnetic ﬁelds. Typically one can easily explore the tempo-ral range from a few femtoseconds to about 1 ns duringwhich several fundamental interaction processes occur in dy-namical magnetic systems. 1,2In addition, the versatility of the current laser sources, which are available over a widespectral range from the near UV to the midinfrared, allowsus to explore very different materials, including ferromag-netic metals, or semiconductors, semimetals, or dielectrics. Abasic experimental conﬁguration consists of exciting, for ex-ample, a ferromagnetic ﬁlm with a sequence of two delayedfemtosecond pulses. 3One modiﬁes the electronic ground state and the second one probes the excited electronic andspin states at various temporal delays t, via an analysis of the transmission T/H20849t/H20850, the reﬂectivity R/H20849t/H20850, and the Kerr or Fara- day rotations /H9258K,F/H20849t/H20850and ellipticities /H9257K,F/H20849t/H20850, with or without the presence of an external static magnetic ﬁeld H. Naturally this so-called time resolved pump-probe magneto-opticalconﬁguration is not unique and several femtosecond opticaltechniques are now also available. Instead, one can detect,for example, the modiﬁcation of the photoemission yield ofcharges and spins from bulk or surface states, 4,5the ampli- tude and polarization state of the second harmonicgeneration, 6,7or the terahertz emission resulting from the pump pulse.8The common ground to these experiments is the possibility of creating a highly nonequilibrium electrondistribution via interband and intraband optical processes. 9 While the mechanism associated with an ultrafast modiﬁca-tion of the electronic distribution is clearly related to thelarge excess of energy acquired by the electrons above theFermi level, 10leading in some cases to a temperature in-crease of a few thousands of Kelvins, there are still debates concerning the mechanism leading to an ultrafast modiﬁca-tion of the magnetization which has been reported by severalgroups. 1–8,11–18It is indeed an interesting opened problem where several candidate processes are suspected to play asigniﬁcant role such as the spin-orbit interaction, a time de-pendent exchange interaction with the excitation of Stoner pairs, an infrared photon emission accompanied by spin re-versal, spin scattering at surfaces, and the spin-phonon inter-action. Some works have also studied band ﬁlling effectswhich may lead to an apparent demagnetization. 19,20 In spite of the complex many body theoretical approach required to understand the experimental works reporting anultrafast magnetization dynamics, there are many progresseson the experimental side which carry out interesting informa-tion to clarify the overall puzzle. As we shall see in thefollowing sections it appears quite clearly that the delta-function excitation associated with the pump pulse not onlyinduces an ultrafast partial or total demagnetization, local-ized at the focused laser spot, but may also induce a reorien-tation of the magnetization as well as a damped motion ofprecession. The beauty of the femtosecond time resolvedmagneto-optical techniques is that they allow us to observethe overall dynamics of the magnetization vector, not only inreal time but also in the three directions of space. 21It is a direct visualization of the complex magnetization trajectoryoccurring in a system which has been brought quasiinstanta-neously far from its ground state equilibrium. The presentwork focuses on such dynamics which we have been able tocapture in nickel and cobalt ﬁlms. Let us stress that our ap-proach allows us to reveal the role played by important quan-tities such as the magnetocrystalline anisotropy. In addition,it offers a nice playground for analyzing the spin dynamics ina system where the modulus of the magnetization is not con-served which naturally requires to go beyond the usualLandau-Lifshitz-Gilbert equations. 22 a/H20850Electronic mail: bigot@ipcms.u-strasbg.frJOURNAL OF APPLIED PHYSICS 99, 08A501 /H208492006 /H20850 0021-8979/2006/99 /H208498/H20850/08A501/5/$23.00 © 2006 American Institute of Physics 99, 08A501-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Tue, 25 Nov 2014 16:17:24II. METHODOLOGY In order to access the full trajectory of the magnetization vector one can take advantage of the versatility of themagneto-optical polarization techniques which allows us toretrieve the three spatial components in the so-called polar,longitudinal, and transverse directions. 23–25In Fig. 1 they are referred to as the x,y, and zaxes. The sample plane, a thin ferromagnetic metallic ﬁlm in our case, is set in the yzplane. An external static magnetic ﬁeld Hcan be rotated around the zaxis so that it makes an angle /H9278with respect to the xaxis. Correspondingly, the static magnetization points in a direc-tion M /H9278out of the sample plane. This direction is naturally different than H/H9278, as the magnetization results from the total effective ﬁeld including the anisotropy, the exchange, and theexternal ﬁelds. The probe beam is set with either sorp polarization with respect to its plane of incidence which isdeﬁned by the xyplane in Fig. 1. The angle of incidence /H9251of the probe beam with respect to the normal to the sample /H20849x axis/H20850can be modiﬁed. Although this is not required it allows us to double check the consistency of the measurements aswe did in our experiments using two different probes withangles of incidence /H9251=3° and /H9251=52°. The components of the magnetization are retrieved from the reﬂected probebeam /H20849Kerr geometry /H20850by analyzing its polarization for dif- ferent complementary angles /H9278of the external ﬁeld: the cor- responding detected signal is named Shereafter. The polar and longitudinal components are given, respectively, by Pol=1 2/H20851S/H20849/H9278/H20850−S/H20849/H9266−/H9278/H20850/H20852and Long=1 2/H20851S/H20849/H9278/H20850−S/H20849−/H9278/H20850/H20852. The transverse component, whenever it is present, is obtained di- rectly from the reﬂectivity Rof the p-polarized probe beam, i.e., without polarization bridge analysis, via the quantity Trans=1 2/H20851R/H20849/H9278/H20850−R/H20849/H9266−/H9278/H20850/H20852. Simultaneously, we determine the transmission Tand reﬂectivity Rof the probe beam. The temporal variation of the ﬁve quantities of interest /H20849transmis- sion, reﬂectivity, and the three components of the magneti-zation vector /H20850is then obtained from the standard pump-probe conﬁguration by measuring the following quantities for eachpump-probe delay t:/H9004T/T=/H20851T/H20849t/H20850−T 0/H20852/T0,/H9004R/R=/H20851R/H20849t/H20850 −R0/H20852/R0,/H9004Pol/Pol= /H20851Pol/H20849t/H20850−Pol 0/H20852/Pol 0,/H9004Long/Long =/H20851Long /H20849t/H20850−Long 0/H20852/Long 0, and /H9004Trans/Trans= /H20851Trans /H20849t/H20850 −Trans 0/H20852/Trans 0, where the subscript 0 refers to the static values. The laser is a titanium:sapphire oscillator ampliﬁed at2.5 kHz. The fundamental wavelength is 790 nm and the pulse duration is 130 fs. Part of this beam is split into twoweak beams corresponding to the two probes while the pumpbeam is obtained by frequency doubling /H20849395 nm /H20850in a beta barium borate /H20849BBO /H20850crystal in order to avoid spurious noise in the detected signals due to interference effects between thepump and probe around zero delay. The pump pulse durationis/H11011200 fs. The time resolved measurements are made by a synchronous detection scheme using a lock-in ampliﬁer andlow frequency chopping of the 2.5 kHz pump pulse train.The pump-probe delay can be varied up to 1 ns with a mini-mum step of 6 fs. The ratio between the pump and probespot diameters is /H110112. The maximum energy density of the pump pulse is 2 mJ cm −2, while the energy density of the probe beam is kept very low. The maximum amplitude of thestatic magnetic ﬁeld is 4 kOe. We have studied three ferromagnetic thin ﬁlms, two co- balt ﬁlms having a different magnetocrystalline anisotropyaxis, and a nickel ﬁlm. The ﬁrst cobalt sample, referred to asCo/Al 2O3in the following, is a 16-nm-thick Co ﬁlm grown by molecular beam epitaxy on a /H208490001 /H20850oriented sapphire substrate. It has a hexagonal compact crystalline phase withthecaxis along the /H208490001 /H20850direction /H20849perpendicular to the ﬁlm, along the xaxis in Fig. 1 /H20850. The magnetization at satu- ration in the sample plane /H20849yz/H20850occurs for an applied ﬁeld of /H110110.9 kOe. It is slightly anisotropic with coercive ﬁelds vary- ing within /H1101110%. The perpendicular magnetization along the hard axis /H20849xaxis/H20850saturates for an external ﬁeld of /H1101120 kOe. The second cobalt sample, named Co/MgO here- after, is grown along the /H20849110/H20850direction of a MgO substrate. The resulting hexagonal caxis is in the plane of the sample, along the yaxis of Fig. 1. Along this easy axis, the magne- tization saturates for an applied ﬁeld of /H110110.2 kOe. The mag- netization perpendicular to the ﬁlm /H20849xaxis/H20850could not be saturated up to 25 kOe, while in the sample plane along thehard direction /H20849zaxis/H20850the saturation ﬁeld is /H1101110 kOe. This sample is much thicker /H20849/H1101150 nm /H20850than Co/Al 2O3. The nickel sample is a 15-nm-thick Ni ﬁlm with cubic anisotropy deposited on an Al 2O3substrate. III. EXPERIMENTAL RESULTS AND DISCUSSIONS Let us ﬁrst consider the magnetization dynamics of the Co/Al 2O3sample. In Fig. 2 we have represented the differ- ential transmission /H9004T/T/H20849full line /H20850together with the dynam- ics of the longitudinal component /H20849opened circles /H20850, when ap- plying the external ﬁeld H/H9278in the sample along the yaxis /H20849/H9278=90° /H20850. In that case the initial condition is such that the magnetization is saturated along the ydirection. The short /H20849t/H1102120 ps /H20850and long /H20849t/H110211n s /H20850delay behaviors are shown in Figs. 2 /H20849a/H20850and 2 /H20849b/H20850, respectively. These signals are easily interpreted in terms of the electron /H20849/H9004T/T/H20850and spin /H20849/H9004Long/Long /H20850dynamics. The initial decrease of the trans- mission within the pump pulse duration corresponds to an increase of the electronic temperature which then comesback to its initial value in two steps. First, the electron-phonon interaction contributes to an equilibrium between theelectron and lattice temperatures with a time constant of900 fs. The second step is a temperature relaxation, associ- FIG. 1. Schematic of the magneto-optical conﬁguration used to measure the three-dimensional magnetization trajectory.08A501-2 Vomir et al. J. Appl. Phys. 99, 08A501 /H208492006 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Tue, 25 Nov 2014 16:17:24ated with the heat diffusion to the environment, which occurs with a time constant of 220 ps. It is remarkable that duringthis entire time scale the spins follow the same dynamics asthe electronic distribution in agreement with the previousobservations in CoPt thin ﬁlms. 15,16Let us emphasize that here the pulse duration is relatively long /H20849200 fs /H20850so that the thermalization of the electronic distribution and spins, which have been shown to occur with a time constant of /H1101160 fs,16 cannot be resolved. When the external magnetic ﬁeld is set with an angle out of the sample plane, the dynamics is muchricher. Figure 3 represents the temporal evolution of the po-lar, longitudinal, and transverse signals up to 1 ns, as well asthe reconstructed three-dimensional trajectory when /H9278is set to 15° /H20849and to its complementary values of 165° and −15° /H20850. A detailed look at these ﬁgures and their corresponding shortdelay behaviors /H20849not shown in Fig. 3 /H20850allows us to trace back the pathway of the magnetization vector. Initially/H208490/H11021t/H11021300 fs /H20850the magnitude of both the polar and longitu- dinal components decreases while the transverse component remains to zero. It clearly reﬂects the decrease of themagnetization modulus during the ﬁrst hundreds offemtoseconds. 21Simultaneously and up to /H110111 ps, the rela- tive changes of the polar and longitudinal components aredifferent. It corresponds to a rotation of the magnetizationvector which occurs in the xyplane as no transverse compo- nent still shows up. After the electrons and spins are in equi-librium with the lattice /H20849t/H11022900 fs /H20850, one gradually sees the development of a motion of precession which now leads to a transverse component /H9004Trans/Trans. Like the polar and lon- gitudinal components, it oscillates with a period of 100 psfor an applied ﬁeld of 2.3 kOe and damps out with a timeconstant of /H11011300 ps. The phase of these signals is, however, different as the magnetization vector rotates around the ef-fective ﬁeld. Let us emphasize that during the ﬁrst few hun- dreds of femtoseconds, when the magnetization modulus de-creases, the pump pulse acts as a delta-function perturbationto excite the spins. During this time scale a description of themagnetization dynamics in terms of spin waves is not rel-evant. It is only after a few tens of picoseconds that spinwaves with their spatial dispersion become the appropriateconcept to describe the magnetization dynamics. The magnetization dynamics in the Co/MgO sample present very signiﬁcant differences. Let us recall that thissample has an easy hexagonal caxis in the ydirection in the sample plane and that its thickness is much larger than thepenetration depth of the laser beams /H20849/H1101115 nm /H20850. Figure 4 /H20849a/H20850 shows the long delay behaviors of the polar and longitudinal signals for an external ﬁeld of 3.5 kOe and an angle /H9278=5°. The signals have been slightly smoothed to better show abeating in the oscillatory behavior associated with theprecession. This beating corresponds to the excitation oftwo magnon modes. 26Their respective frequencies /H9024P0 =25.4 GHz and /H9024P1=31.3 GHz are obtained from the Fou- rier transform in the inset of Fig. 4 /H20849a/H20850. The corresponding zero and ﬁrst order standing spin waves can be simulta-neously excited in this thick cobalt ﬁlm /H2084950 nm /H20850. Assuming a quadratic dispersion of the ﬁrst order mode /H9024 P1=/H9024P0+Dk2 leads to a constant Dof/H11011620 meV Å2. A more detailed study of the magnon dispersion, using, for example, the Bril-louin light scattering, would allow a comparison of our dy- FIG. 2. Time dependent differential transmission /H20849/H9004T/T/H20850and longitudinal /H20849/H9004Long/Long /H20850signals of the Co/Al2O3sample with an applied magnetic ﬁeld H/H9278set in the sample plane /H20849/H9278=90° /H20850, for short /H20849a/H20850and long /H20849b/H20850pump- probe delays. FIG. 3. Magnetization dynamics of Co/Al2O3obtained with an applied magnetic ﬁeld H/H9278set at the complementary angles /H9278= ±15° ,165°. Polar /H20849a/H20850, longitudinal /H20849b/H20850, and transverse /H20849c/H20850differential signals. Corresponding three-dimensional trajectory /H20849d/H20850.08A501-3 Vomir et al. J. Appl. Phys. 99, 08A501 /H208492006 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Tue, 25 Nov 2014 16:17:24namical determination of Din this particular sample which has a large in-plane anisotropy. Figure 4 /H20849b/H20850shows the pro- jection of the magnetization trajectory on the xyplane for long temporal delays. It is interesting to notice that, in con-trast to Co/Al 2O3, the precession damps out long before the modulus of the magnetization has fully recovered its initialvalue. This is due to the fact that for Co/MgO the thermaldissipation on the MgO substrate lasts longer. Consequently,the lattice temperature of the cobalt ﬁlm being still signiﬁ-cant at 1 ns, the magnetization is lower than its initial value.Additional information can be traced back from the magne-tization orientation, in connection with magnetocrystallineanisotropy. In Fig. 4 /H20849b/H20850one can see that after the precession motion has ended not only the modulus of the magnetizationvector has not recovered but also its direction. This is due toa variation of the magnetocrystalline anisotropy with the lat-tice temperature. In cobalt it is known that the anisotropydecreases with increasing temperature. 27Therefore the effec- tive ﬁeld, which results from the in-plane easy axis, the de-magnetizing ﬁeld, and the applied external ﬁeld tend to pointout of the sample plane when the temperature is larger thanits initial value. This dynamical process is indeed present as soon as the temperature increases, as can be seen in the de-tailed view of the trajectory plotted for short temporal delaysin Fig. 4 /H20849c/H20850. The fact that the trajectory moves upward to- wards the polar direction reﬂects the decrease of the magne-tocrystalline anisotropy ﬁeld in the sample plane. A detailedstudy of this mechanism has been recently reported andmodeled. 22 The versatility of our experimental approach allows us toinvestigate the magnetization dynamics in many different conﬁgurations. For example, Fig. 5 shows the evolution ofthe precession period on the nickel ﬁlm when increasing theangle /H9278of the external ﬁeld H/H9278, i.e., when Hturns towards the plane of the sample. The plotted quantity is now thetemporal variation of the differential Kerr rotation angle/H9004 /H9258K//H9258K. Clearly, the precession frequency /H9024P,/H9278increases as/H9278increases /H20849/H9024P,10°=4.2 GHz, /H9024P,30°=7.7 GHz, and /H9024P,80°=11.1 GHz /H20850. This is due to an increase of the effective ﬁeld related to a decrease of the modulus of the demagnetiz-ing ﬁeld associated with the shape anisotropy pointing in the−xdirection. The amplitude of the signal decreases since the measurements are performed with the probe beam having anangle of incidence of 3°, which is essentially sensitive to thepolar direction. Naturally, when /H9278=90°, the magnetization is initially oriented along the yaxis and therefore no precession occurs and the polar signal vanishes. In conclusion, in the present work we have investigated the magnetization dynamics in the three dimensions of spaceover a broad temporal range covering the ultrafast demagne-tization and reorientation of the magnetization as well as itssubsequent motion of precession. It shows that in order tohave an accurate description of the magnetization pathway, itis important to retrieve the entire spatiotemporal dynamicssince the anisotropies /H20849magnetocrystalline and shape /H20850of the material are not constant during the entire temporal evolutionof the magnetization. It has important consequences bothfrom the applied and theoretical points of view. Indeed, whenmodeling the magnetization dynamics in a consistent wayover such a large temporal range one cannot assume that themodulus of the magnetization is conserved like in theLandau-Lifshitz-Gilbert approach. A recent phenomenologi-cal approach has been developed which predicts most of thepresent observations. 22It considers a magnetization modulus and a magnetocrystalline anisotropy which are time depen-dent via their temperature variations. The model includestwo coupled heat equations, associated with the electron/spinand lattice temperatures /H20849two-temperature model /H20850, simulta- neously solved with the Bloch equations for the three com-ponents of the magnetization taking into account the tem-perature dependent effective ﬁeld. Simulatenously, theconstraint that the magnetization modulus follows adiabati-cally the electron/spin temperature is imposed. An accurate FIG. 4. Magnetization dynamics of Co/MgO. /H20849a/H20850Time dependent differen- tial polar /H20849close circles /H20850and longitudinal /H20849open circles /H20850signals. Fourier transform of the polar signal /H20851inset of /H20849a/H20850/H20852. Projection of the trajectory on the xyplane for long /H20849b/H20850and short /H20849c/H20850temporal delays. The arrows indicate the direction of time. FIG. 5. Time dependent differential Kerr rotation signals of the Ni/Al2O3 sample for several directions of the applied magnetic ﬁeld H/H9278 /H2084910°/H33355/H9278/H3335590°/H20850. For clarity each curve is displayed with an offset of 10−2.08A501-4 Vomir et al. J. Appl. Phys. 99, 08A501 /H208492006 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Tue, 25 Nov 2014 16:17:24determination of the spatiotemporal dynamics is also rel- evant when considering realistic magnetic devices con-strained to important temperature variations. In this contextour approach is interesting for understanding the detaileddynamics of such magnetic devices. ACKNOWLEDGMENTS This work has been carried out with ﬁnancial supports from the European Community program “Dynamics” andfrom the Centre National de la Recherche Scientiﬁque inFrance. 1See, for example, J.-Y. Bigot, C. R. Acad. Sci., Ser IV: Phys., Astrophys. 2,1 4 8 3 /H208492001 /H20850. 2For a review, see in Spin Dynamics in Conﬁned Magnetic Structures I , Topics in Applied Physics Vol. 83, edited by B. Hillebrands and K. Oun-adjela /H20849Springer-Verlag, Berlin, 2002 /H20850, pp. 1–326; Spin Dynamics in Con- ﬁned Magnetic Structures II , Topics in Applied Phyics Vol. 87, edited by B. Hillebrands and K. 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Scholl, L. Baumgarten, R. Jacquemin, and W. Eberhardt, Phys. Rev. Lett. 79, 5146 /H208491997 /H20850. 12G. Ju, A. Vertikov, A. V. Nurmikko, C. Canady, G. Xiao, R. F. C. Farrow, and A. Cebollada, Phys. Rev. B 57, R700 /H208491998 /H20850; G. Ju, A. V. Nurmikko, R. F. C. Farrow, R. F. Marks, M. J. Carey, and B. A. Gurney, Phys. Rev.Lett. 82, 3705 /H208491999 /H20850. 13J. Güdde, U. Conrad, V. Jähnke, J. Hohlfeld, and E. Matthias, Phys. Rev. B59, R6608 /H208491999 /H20850; U. Conrad, J. Güdde, V. Jähnke, and E. Matthias, Appl. Phys. B: Lasers Opt. 68,5 1 1 /H208491999 /H20850. 14B. Koopmans, M. van Kampen, J. T. Kohlhepp, and W. J. M. de Jonge, J. Appl. Phys. 87, 5070 /H208492000 /H20850. 15E. Beaurepaire, M. Maret, V. Halté, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Phys. Rev. B 58, 12134 /H208491998 /H20850. 16L. Guidoni, E. Beaurepaire, and J.-Y. Bigot, Phys. Rev. Lett. 89, 017401 /H208492002 /H20850. 17J.-Y. Bigot, L. Guidoni, E. Beaurepaire, and P. N. Saeta, Phys. Rev. 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1.4944650.pdf	Deterministic switching of a magnetoelastic single-domain nano-ellipse using bending Cheng-Yen Liang , Abdon Sepulveda , Scott Keller , and Gregory P. Carman Citation: J. Appl. Phys. 119, 113903 (2016); doi: 10.1063/1.4944650 View online: http://dx.doi.org/10.1063/1.4944650 View Table of Contents: http://aip.scitation.org/toc/jap/119/11 Published by the American Institute of Physics Deterministic switching of a magnetoelastic single-domain nano-ellipse using bending Cheng-Y en Liang, Abdon Sepulveda, Scott Keller, and Gregory P . Carman Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, California 90095, USA (Received 7 January 2016; accepted 9 March 2016; published online 21 March 2016) In this paper, a fully coupled analytical model between elastodynamics with micromagnetics is used to study the switching energies using voltage induced mechanical bending of a magnetoelastic bit. The bit consists of a single domain magnetoelastic nano-ellipse deposited on a thin ﬁlm piezo-electric thin ﬁlm (500 nm) attached to a thick substrate (0.5 mm) with patterned electrodes under- neath the nano-dot. A voltage applied to the electrodes produces out of plane deformation with bending moments induced in the magnetoelastic bit modifying the magnetic anisotropy. To mini-mize the energy, two design stages are used. In the ﬁrst stage, the geometry and bias ﬁeld (H b)o f the bit are optimized to minimize the strain energy required to rotate between two stable states. In the second stage, the bit’s geometry is ﬁxed, and the electrode position and control mechanism isoptimized. The electrical energy input is about 200 (aJ) which is approximately two orders of mag- nitude lower than spin transfer torque approaches. VC2016 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4944650 ] I. INTRODUCTION The use of multiferroic systems for development of low energy memory applications has received considerable atten-tion in the past few years. The main concept is to use amultilayered composite material system consisting of piezo-electric and magnetoelastic layers and control the magnetiza-tion by induced strain. 1–3The actuation mechanism consists of applying voltage to a piezoelectric substrate creating de-formation which in turns transfers to a ferromagnetic (FM)nano-dot placed on top. However, the multiferroic compositememory elements intrinsically reside on a fairly thick sub-strate system. This thick substrate clamps the piezoelectric/magnetoelastic material limiting the amount of strain that can be generated and poses a signiﬁcant problem for this area of study. 4–8In this paper, we present a mechanism to overcome this limitation, and more importantly, we showthat local strain proﬁles can be used to reorient the magnet-ization vector between two stable equilibrium points. Researchers have demonstrated the feasibility of the magnetization control between stable states in thin ﬁlm mag-netoelastic material deposited on a thick piezoelectric sub-strate. 5–7This voltage induced strain mediation effect to manipulate the magnetization is generally referred to as theconverse magnetoelectric effect. 8There have been extensive studies that contain both theoretical and experimental workson strain-mediated magnetization changes, coercivity changes, and strain-induced anisotropy in continuous mag- netic thin ﬁlms. 9In all of the continuous ﬁlm studies, the strain is appropriately assumed to be fully transferred fromthe ferroelectric to the ferromagnetic layer by treating themagnetoelastic energy as a pure uniaxial anisotropy to beapplied to the magnetic media. For example, Brintlingeret al . 10reported both experimental and analytical predic- tions, using OOMMF and constant strain assumptions thatshow reversible switching in FeGa/BaTiO 3(BTO) thin ﬁlm. In recent years, several additional studies, such as ferroelec- tric/ferromagnetic ﬁlm coupling by Lahtinen et al.2and mag- netic thin ﬁlm stress modeling by Bai et al .,11have demonstrated that this constant strain methodology works reasonably well for continuous thin ﬁlms. The strain-mediated effect in multiferroic nanostruc- tures has been used to alter magnetic domains12,13and to shift the magnetic coercive ﬁeld. Bur et al .14reported strain-induced coercive ﬁeld changes in patterned single-domain nickel nanostructures deposited on a thick Si/SiO 2 substrate using external mechanical loads. Moutis et al .15 reported electric-ﬁeld modulation of coercive ﬁeld H cusing a piezoelectric on a periodic array of FM Co 50Fe50stripes but once again this was on an entire substrate. Out of plane magnetic reorientation has also been achieved with mag- netic BiFeO 3(BFO)/CoFe 2O4(CFO) vertical heterostruc- tures embedded into a ferroelectric described by Zavalicheet al. 16While demonstrating the concept this approach pro- duced excitation in all the magnetic elements simultane- ously and thus, the mechanism does not lend itself toindividual element control nor could it be used for deter- ministic reorientation of the magnetic moment. Brandlmaier et al. 17used the biaxial strain difference pro- duced on the side of a piezoelectric stack actuator to control the magnetic anisotropy of a thin crystalline Fe 3O4ﬁlm on bulk material. As an alternative to using in plane polarized piezoelectric material, some researchers such as Wu used the auxetic piezoelectric strain produced by [011] cut (1-x)[Pb(Mg 1/3Nb2/3)O3]-x[PbTiO 3] (PMN-PT), while others have used (1-x)Pb(Zn 1/3Nb2/3)O3–xPbTiO 3(PZN-PT) single crystals.9These single crystal approaches resulted in a proposed design of a magnetoelectric memory system but once again required bulk piezoelectric material, which is not 0021-8979/2016/119(11)/113903/8/$30.00 VC2016 AIP Publishing LLC 119, 113903-1JOURNAL OF APPLIED PHYSICS 119, 113903 (2016) amenable to memory fabrication processes. In a device pro- posed by Hu et al.,18they suggested individual magnetic ele- ments could be controlled by very small single crystal PMN-PT elements, a conﬁguration that presents signiﬁcant fabrica- tion challenges. The development of a strain mediated multiferroic mem- ory device requires that the magnetization to be individually controllable for each nano-dot and the ferroelectric thin ﬁlm be grown on a substrate (e.g., Si wafer). The main difﬁcultyhere is that the thin ﬁlm piezoelectric is clamped by the thick substrate and prevents strain transfer. Cui et al. 19suggested the use of patterned electrodes to overcome substrate clamp-ing and obtain highly localized strain in a thin ﬁlm piezo- electric and the magnetic material. The concept was demonstrated on bulk ceramic and relatively large magneticelements and did not include detailed analysis (or experi- ments) for thin ﬁlm piezoelectric on a thick substrate control- ling a single magnetic domain element. 20In addition to single-bit multiferroic memory devices, nanomagnetic-based Boolean logic circuit also attracts research attention because of its non-volatile and energy-efﬁcient properties. D’Souzaet al. 21experimentally demonstrated strain-induced switching of single-domain magnetostrictive nanomagnets (lateral dimensions /C24200 nm) fabricated on bulk PMN–PT substrates can implement a nanomagnetic Boolean NOT gate and steer bit information unidirectionally in dipole-coupled nanomagnet chains. From their estimation, the energy dissipation for logicoperations using thin ﬁlm is only about /C241 aJ/bit. The design of single domain switchable magnetoelectric heterostructures requires the use of Landau-Lifshitz-Gilbert(LLG) equation. The micromagnetics tools used today are largely based on phenomenological approaches developed in the 1950s that have been reﬁned considerably in recentyears. 22An important addition to micromagnetics was the inclusion of strain (or stress) for magnetostrictive materials by Zhu et al.23as an extra term in the effective magnetic ﬁeld. This was then used by Hu24to model the effect of stress on hysteresis curves and magnetization dynamics, showing the interaction of stress with coercivity and the easyaxis of magnetoelastic materials. Based on these results, Hu et al. 25,26used stability conditions and proposed an electric ﬁeld read and write magnetoresistance random-access mem-ory (MERAM) device. A balance of shape anisotropy and strain anisotropy was used to describe an elliptical nanomag- net that could be switched under stress by Roy et al . 27 However, in most of these studies, the magnetization and the strain were assumed to be spatially uniform and thus did not consider the clamping issue or the effects of a properly tai-lored strain ﬁeld proﬁle. D’Souza et al. 28proposed and ana- lyzed a low-power 4-state universal logic gate using a linear array of multiferroic nanomagnets but did not consider thesubstrate clamping issue. Tiercelin et al. 30described and an- alyzed a magnetoelectric memory cell that balanced strain anisotropy, shape anisotropy, and a bias ﬁeld. In this laterwork, the elastic contribution was modeled separately, and the piezoelectric ﬁlm was not attached to a substrate. In Liang’s study, 20a design based on four patterned electrodes was introduced. In this work, a fully coupled model was used to analyze the design and was shown that, by applyingan electric ﬁeld through the thickness of the piezoelectric substrate, the clamping effect can be overcome. Also, in thiswork, it was concluded that the effect of shear lag produceslocalized strain proﬁles (70%). Biswas et al . 29proposed a scheme that can ﬂip the magnetization of the soft layer (com- plete 180/C14rotation) in magnetic tunnel junction (MTJ) multi- ferroic memory bit with stress alone and without the need forany feedback circuitry that undermines the energy-efﬁciencyand reliability of the bit writing scheme. In this paper, the system consists of a nanoscale single domain magnetoelastic ellipse deposited on a thin ﬁlm piezo-electric wafer attached to a thick substrate. This composite ismodeled by analytically coupling electrostatics, micromag-netics (LLG), and elastodynamic partial differential equa- tions. The piezoelectric thin ﬁlm (500 nm) is attached and clamped to a thick substrate, which prevents relative in-plane motion of the piezoelectric ﬁlm. In order to inducelocalized strains, two electrodes are placed under the Niellipse with an insulation layer. When a voltage is applied tothese electrodes, bending deformation is exited, producingcompression on the Ni dot. Furthermore, unlike Tiercelin’swork, the piezoelectric clamping effect is fully captured bythe model. The intrinsic coupling of the piezoelectricresponse with the magnetoelastic response through strain ismodeled by coupled partial differential equations (i.e., elec-trostatics, micromagnetics, and elastodynamics). The numer-ical formulation uses tetrahedral ﬁnite elements with a maximum size equal to the exchange length of Ni ( /C248.5 nm) providing both spatially varying strains, electric ﬁelds, andmagnetic spins throughout structure. Therefore, the modelcaptures all the relevant physics required to accurately pre-dict the response of this multiferroic nanoscale structure. II. THEORY In this section, a fully-coupled micromagnetic elastody- namic simulation with piezoelectrics for ﬁnite size 3D struc-tures is described. The coupled partial differential equationsto be solved as well as the numerical method to simulate a wide range of shape and geometries are provided below. For a more detailed derivation, the readers are referred to Ref. 20. The model consists of magnetization dynamics using the LLG equations coupled with the mechanical strains andstresses via the equations of elastodynamics. The piezoelec-tric response of the thin ﬁlm is modeled with linear constitu-tive equations relating strain with the electric ﬁeld using aquasi-static electric ﬁeld approximation. Other modelingassumptions include small elastic deformations, linear elas-ticity, electrostatics, and negligible electrical current contri- butions. The coupled governing equations used in this work are as follows. The elastodynamics governing equation formechanical stresses and displacements is q du2 dt2¼r/C1 r; (1) where qis the mass density, ris the stress tensor, u is the displacement vector, and t is time. The dynamics of magnetization is deﬁned by the phe- nomenological LLG equation113903-2 Liang et al. J. Appl. Phys. 119, 113903 (2016) @m @t¼/C0l0cm/C2Hef f/C0/C1þam/C2@m @t/C18/C19 ; (2) where l0is the permeability of free space, cis the Gilbert gyromagnetic ratio, ais the Gilbert damping constant, and m is the normalized magnetization vector. The effective mag- netic ﬁeld, Heff, includes the external ﬁeld ( Hext), the exchange ﬁeld ( Hex), the demagnetization ﬁeld ( Hd), the magnetocrystalline anisotropy ﬁeld ( Hanis), and the magne- toelastic ﬁeld ( Hme) effects. Detailed expressions for these terms can be found in recent literatures.20,22,23,31–34The demagnetization ﬁeld is calculated by using the quasi-static Ampere’s law. This leads to Hd¼/C0 r u, where Hdis thedemagnetization ﬁeld vector and uis the potential. Combining this equation with the divergence of magnetic induction equal to zero, and the constitutive relation, B¼l0ðHþMÞ, produces the equation for the magnetic potential, u, in terms of the magnetization (see Equation (5)). The magnetization is coupled with the effective mag- netic ﬁeld through this demagnetization term. Substituting the piezoelectric constitutive relations into the elastodynamics equation (1) and LLG equations (Equation (2)) produces a cross-coupled set of non-linear equations containing displacements, magnetization, electri- cal ﬁeld, and magnetic potential as follows (detail derivationin Ref. 20): qdu2 dt2/C0r/C1 C1 2ruþr uðÞT/C16/C17/C20/C21 þr/C1 CkmmmThi þr/C1 CdE/C16/C17 ¼0; (3) @m @t¼/C0l0cm/C2HextþHexmðÞþHduðÞþHanimðÞþHmem;uEðÞðÞ/C0/C1 /C0/C1þam/C2@m @t/C18/C19 ; (4) r2u¼Msðr /C1 mÞ; (5) where Cis the stiffness tensor, and kmis the magnetostric- tion tensor. Eis the electric ﬁeld vector, and dis the piezo- electric coupling tensor. In a similar fashion to magnetic potential, the quasi-static Faraday’s Law implies that E¼/C0 r V,w h e r e Vis the electric potential. This equation coupled with Gauss’s Law and provides the piezoelectric coupling within the model. These coupled sys-tems of partial differential equa tions are solved for the mechani- cal displacement ( u, v, w ), the electric potential ( V), the magnetic potential ( u), and the magnetization ( m x,my,mz). The numerical solution of micro-magneto-electro-me- chanical coupled equations is obtained by using a ﬁnite ele- ment formulation (COMSOL) with an implicit backwarddifferentiation (BDF) time stepping scheme. 35,36In order to decrease solution time, the system of equations is solvedsimultaneously but using a segregated method, which splitsthe solution process into sub-steps using a damped Newton’smethod. This coupled model provides dynamic results forthe full strain and micromagnetic spin distribution in themagnetoelastic component coupled with a piezoelectriclayer. For all numerical problems, convergence studies (i.e.,mesh size and time steps) were evaluated to ensure accuracy. III. SIMULATION SETUP Figure 1shows the conﬁguration studied in this paper. The design variables to be determined are the geometry ofthe ellipse (aaxis, baxis, and caxis), the position of electrodes(dist), and the time history for the applied voltage V(t). Adetailed description on determined these quantities is givenin Section IV.The material properties for the magnetoresistance ran- dom-access memory (MRAM) element system are asfollows. The material properties for the nickel nano-dotare 36Ms¼4:8/C2105ðA=mÞ,Aex¼1:05/C210/C011ðJ=mÞ, k100¼/C046/C210/C06,k111¼/C024/C210/C06,c11¼2:5/C21011 ðN=m2Þ,c12¼1:6/C21011ðN=m2Þ, and c44¼1:18/C21011 ðN=m2Þ. The nickel nano-dot is assumed polycrystalline; therefore, the magnetocrystalline anisotropy is neglected.The Gilbert damping ratio is set as a¼0.5 to improve stabil- ity and process time. Using the high Gilbert damping ratio would cause the overdamped precessional motion in themagnetization response. When using realistic (lower) valueof Gilbert damping, more precessional motion will be shown in the magnetization response (Fig. 6(a)). The magnetization precesses with the amplitude gradually decreasing to equilib-rium. For both Gilbert damping ratios, the ﬁnal equilibriumstate is the same. The PZT-5H material properties ared 33¼5:93/C210/C010ðC=NÞ, d31¼/C02:74/C210/C010ðC=NÞ, c11¼c22¼1:27205 /C21011ðN=m2Þ,c12¼8:02/C21010ðN=m2Þ, c13¼c23¼8:46/C21010ðN=m2Þ, c33¼1:17/C21011ðN=m2Þ, c44¼c55¼2:29885 /C21010ðN=m2Þ, and q¼7500ðkg=m3Þ. The Young’s modulus and the Poisson’s ratio for Au are EAu¼7/C21010ðN=m2Þand /C23Au¼0:44, respectively. The exchange constant for nickel isﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2Aex l0Ms2q /C248:5ðnmÞ. The nickel nano-dot is discretized using tetrahedral elements with a size on the order of nickel’s exchange length. The re- mainder of the structure (i.e., Pb[ZrxTi 1-x]O3(0/C20x/C201) (PZT)-5H thin ﬁlm, Au electrodes) is discretized using tetra-hedral elements with graded element sizes dependent uponlocal geometry. The boundary conditions of the piezoelectric ﬁlm are the four sides, and the bottom surface of the ﬁlm is clamped.The bottom surface is also grounded. The top surface is free113903-3 Liang et al. J. Appl. Phys. 119, 113903 (2016) to deform. The piezoelectric ﬁlm is poling in /C0Z-direction. Two electrodes (electrodes A and B) are underneath thenano-dot. IV. NANO-DOT DESIGN The objective in these memory applications is to design the elliptical nano-dot and actuation mechanism for mini-mum magnetization switching energy. In order to havedeterministic rotation, the s table states are offset by 5 /C14. This condition is not really required, but it was included here to show how this requirement can be incorporated to the design process. Also, the energy barrier between stablestates is constrained to be at least 40 kT for thermal stability(T¼300 K room temperature). Geometric design variables for the ellipse are the major axis (aaxis), the eccentricity (e,b¼e*aaxis), and the thickness (caxis) as well as the magni- tude of the H bto achieve the offset of 5/C14. Control design variables are the position of the electrodes (dist), amplitude,and duration of the voltage pulse. The main difﬁculty insolving this optimization problem is that the numerical so-lution of the system equations (in particular the LLG equa-tion) is extremely heavy computationally, and therefore the approach of using and off the shelf optimizer linked to the ﬁnite element code in impractical as a design tool. Themethod adopted in this paper is to solve the problem in twosteps. First, the plant (ellipse geometry, H b) is optimized with no control system, and then the control system (actua-tor, voltage) is optimized for a ﬁxed plan. The followingsare the details of these two optimization procedures. A. Plant design In this ﬁrst stage, the geometry of the ellipse and bias ﬁeld (H b) are optimized for minimum strain magnetization rotation between stable states. This design can be doneneglecting the magnetization dynamics to avoid the solutionof the LLG equation each time we change a dimension.Thus, we pose the problem as minimizing the average strain(e xx-eyy) required for a 90/C14rotation (the reason to use 90 instead of 180 will become evident in Section IV B) subjectto the conditions that the energy barrier is at least 40 kT and that the stable states are offset by 5/C14relative to the x-axis. The design variables are aaxis, e, caxis, and H b. The resultant optimal dimensions are aaxis ¼130 nm, e ¼0.9, c ¼10 nm, Hb¼492 A/m. The optimum strain, which actually corre- sponds to an estimate of the amplitude required by thedynamic strain, is 1000 le. Also, it is important to mention that with the resultant optimal dimensions a full dynamicanalysis was performed to verify that the nano-dot designsatisﬁes the design requirements. B. Actuation and control system design In this second stage, we keep the geometry of the ellipse ﬁxed and optimized design for the control mecha-nism. There are two parts to it, ﬁrst is the position of the electrodes and second the cont rol law for the applied volt- age (magnitude and duration of the pulse). For this optimi-zation, we run the complete fully coupled dynamic model.Due to the symmetry of the conﬁguration, by applying thesame voltage to both electrodes, the magnetization willrotate 90 /C14at the most. Then, for the full rotation, the sym- metry has to be broken. Saying this ﬁrst step was to deter-mine the distance (dist) and the magnitude of the voltagefor a 90 /C14rotation. This is done for a minimum switching energy criterion. The result of this design phase isdist¼52 nm and V max¼1 V (electrical ﬁeld through the thickness is 2 (MV/m)). With these values, we determinedthe proper duration of the pulses and ﬁnalize the controllaw for the voltage (how it is applied) First, the voltage isapplied on electrodes A and B simultaneously at time¼0. Attime¼4 (that is when the magnetization has rotated 90 /C14), the voltage is removed from electrode B, returning to ground state. Following this, at time¼6, the voltage on electrode A is removed, and since this longer pulse on elec-trode A makes the magnetization cross the energy maxi-mum, the process will make the magnetization to settle atthe second energy well. Total time duration for the cycle is 15. All the simulations for this second optimization phase consider the solution for the fully coupled dynamic model. FIG. 1. Schematic plot and design arrangements.113903-4 Liang et al. J. Appl. Phys. 119, 113903 (2016) Also, prior to application of the bias magnetic ﬁeld and/or voltage, all magnetic spins are uniformly canted out of the x-y plane at 5/C14and allowed to precess toward an equilib- rium state. V. SIMULATION RESULTS AND DISCUSSION Figure 2shows the deformation and strain distribution (exx) results for a bias ﬁeld Hb¼492ðA=mÞ, and the voltage 1 V is applied on both electrodes A and B with bottom sur- face grounded. In Figure 2(a), a three-dimensional deforma- tion plot with bending strain along the x-direction ( exx)i s presented. The strain ( exx) represents the internal bending strain along the x-direction in the nano-dot. A 2D cross- sectional strain distribution plot is shown in Figure 2(b). This shows the internal strain ( exx) in the nano-dot and the deformation when the voltage is applied on both electrodesA and B. When a positive voltage is applied, a local bendingdeformation is produced in the nickel nano-dot that causes stresses and strains. The magnitude of the strain is on the order of 1000 lein the middle region of the nano-dot. Figure 3shows the mechanism of the bi-stable elliptical MRAM bit. Figure 3(a) shows the magnetization in the nano-dot with a bias ﬁeld ( H b¼492ðA=mÞ) before applying a voltage. The equilibrium magnetization was initially tiltedwith respect to the þx-direction by 5 /C14. Both electrodes A and B are initially energized, as shown in Figure 3(b). When a positive voltage is applied, a tensile strain is producedbelow the neutral axis of the nano-dot and a compressive strain above the neutral axis, i.e., a bending strain. A voltageis applied for a time period (time period /C244) until the mag- netization rotates close to 90 /C14. When the magnetization rotates to 90/C14, the voltage on electrode B is removed, and the voltage on electrode A remains on (during time ¼4–6). The removal of the voltage from electrode A causes the magnet- ization to rotate pass 90/C14in this process. Once the magnet- ization rotates pass 90/C14(at time /C246), the voltage on electrode A is switched off, and subsequently, the magnet-ization falls into the other stable energy well positioned at 170 /C14with respect to the þx-direction, as shown in Figure 3(d). When all voltages are removed from electrodes, the magnetization remains at 170/C14. By selecting a similar pro- cess of applying voltage to electrodes A and B, the magnet- ization can be switched back to 5/C14, i.e., the other stable state. Therefore, the magnetization can be switched deterministi-cally between these two states. Figure 4(a) shows the strain distribution ( e xx), and Figures 4(b)–4(d) show the magnetization components (m x, my,m z) for different layers in the nano-dot along the x- direction when the voltage is applied to both electrodes Aand B. Due to the bending effect in the nano-dot, internalstrains result from lateral deformation. Note that the strain distribution is symmetric in both electrode regions. As shown in Figure 4(a), the neutral axis is in the middle region of the nano-dot (at z ¼5 nm), where the stress/strain induced by bending vanishes. A tensile strain is induced below the neutral axis, and a compressive strain is induced above the neutral axis near the electrode region. This bending straindevelops from the localized out-of-plane bending effect near the electrodes when the voltage is applied on both electrodes A and B. These bending strains create a new strain-inducedeasy axis which causes the magnetization to rotate in thenickel nano-dot. Figures 4(b)–4(d) show that the magnetiza- tion components (m x,m y,m z) in each layer do not rotate coherently. Furthermore, due to the non-uniform straindistribution, the magnetization components also have non-uniform distribution. This is important and suggests that sin- gle spin models are inappropriate for evaluating the response of this design. Figure 5(a) shows the strain distribution ( e xx), and Figures 5(b)–5(d) show the magnetization components (m x, my,m z) for different layers in the nano-dot along the x-direction when the voltage is applied to electrode A only. Due to the one-sided bending effect in the nano-dot, internal FIG. 3. Response of the bi-stable elliptical memory bit. (Time in nanosecond) (color: voltage and arrow: magnetization). (a) Starting position. The magnetiza- tion stays one of the stable states at 5/C14with respect to the x-axis. (b) A positive voltage is applied on both electrodes A and B, the magnetization switches to 90/C14. (c) Switch off voltage on B and keep voltage on electrode A. The magnetization switches pass to 90/C14. (d) Switch off voltage on electrode A. The magnet- ization rotates to 170/C14with respect to the x-axis. By applying appropriate voltage to the electrodes, the magnetization can be switched back and forth between the two bi-stable states. FIG. 2. Simulation results (displacement scale exaggerated). (a) Voltage applied on both electrodes A and B. Two electrodes expand out-of-plane,and the bending effect is induced in the nano-dot. (b) Cross-section 2D plot along the x-direction.113903-5 Liang et al. J. Appl. Phys. 119, 113903 (2016) asymmetric strains result from lateral deformation. The strain distribution near the electrode A region is shown in Figure 5(a). As can be seen a one-side, a tensile strain is induced below the neutral axis, and a compressive strain is induced above the neutral axis near the electrode region.This is because the localized out-of-plane bending effect arises near region A when the voltage is applied on electrode A. Similar to the case of both electrodes being activated, themagnetization rotation in each layer is induced by symmetry breaking, when the voltage is applied only on electrode A, FIG. 4. Strain and magnetization component in different layers. Voltage a p p l i e do nb o t he l e c t r o d e sAa n dB .T w o electrodes expand out-of-plane, and the bending effect is induced in the nano- dot. (a) Strain for different layers in the nano-dot along the x-direction. Direct compressive strain is induced above the neutral axis of the nano-dot (z ¼6, 8, 10 nm), and direct tensile strain below the neutral axis of the nano-dot (z ¼0, 2, 4 nm). (b)–(d) Magnetization compo- nents (m x,my,mz) for different layers in the nano-dot along the x-direction. FIG. 5. Strain and magnetization component in different layers. Voltage applied on electrode A. Electrode A expands out-of-plane, and the bending effect is induced in the nano-dot. (a) Strain for different layers in the nano- dot along the x-direction. Asymmetric strain is induced at electrode Aregion. Direct compressive strain is induced above the neutral axis of the nano-dot (z ¼6, 8, 10 nm), and direct tensile strain below the neutral axis of the nano-dot (z ¼0, 2, 4 nm). (b)–(d) Magnetization components (m x,m y, mz) for different layers in the nano-dot along the x-direction.113903-6 Liang et al. J. Appl. Phys. 119, 113903 (2016) and affected by the shear lag effect, resulting in non-uniform reorientation in each layer, as shown in Figures 5(b)–5(d) . Figure 6(a)shows the temporal response of the magnet- ization when a voltage is applied. The magnetization was ini-tially in an equilibrium position, pointing to the þx-direction with 5 /C14tilt, which deﬁnes the “0” state in a representative memory device. The voltage is applied at time ¼0 on both electrodes A and B until time ¼4 ns. The magnetization switches from 5/C14to 90/C14as approaches time ¼6 ns. The mag- netization has a relative response time of approximately 2.5 ns which is inﬂuenced by choice of damping coefﬁcienta¼0.5. This value was chosen to expedite the computation time and does not alter the results with the exception of the temporal response. When using a smaller more realistic damping coefﬁcient a, the magnetization response is sub- stantially faster. When the voltage is removed from electrode B while the voltage on A remains, the magnetization contin- ues to rotate past 90 /C14. Once the magnetization passes 90/C14, the voltage on A is removed (at time ¼6 ns) and magnetiza- tion rotates to 170 at time ¼10 ns, which is deﬁned as the“1” state in a representative memory device. The magnetiza- tion can be rotated back and forth deterministically between5 and 170 using the appropriate voltage sequence. To deter-mine the energy to rotate the bit, the following process was used. Figure 6(b) shows the temporal mechanical strain ( e xx) changes in the nano-ellipse: Volume average strain response(blue line), middle point strain response (red line), and strainresponses at points 650 nm from the center (green and pur- ple lines, respectively). Three zones are shown in the straincurves. The ﬁrst zone (0–4 ns), when both electrodes A andB are on, shows negative average strain, negative strain forthe points at 650 nm, and positive strain at the middle point, therefore, symmetric bending. The second zone (4–6 ns),when the voltage is turned off from electrode B, shows nega- tive average strain, negative strain for the points at þ50 nm, positive strain at point /C050 nm, and positive at the middle point, therefore, unsymmetrical bending. The write energy for this bending switching mechanism is the energy required to generate voltage on the electrodes.This energy is equivalent to amount of charge delivered tothe electrodes on the PZT ﬁlm, i.e., capacitor charging. Thisenergy is called the “CV 2” energy, where C and V represent the capacitance of the piezoelectric ﬁlm and the applied volt-age, and is equivalent to QV/2. The total charge (Q) supplied to the electrodes is determined from the simulations. Using this approach for the results presented in Figure 3, the write energy is calculated to be 0.2 fJ. Here, it is important to pointout that the thickness of the PZT was not optimized in thisstudy and that the reduction in PZT ﬁlm thickness shouldreduce the write energy further. VI. CONCLUSION In this paper, an analytical model was used to determine the optimal nanodot dimensions, the electrode placement,and the voltage control mechanism to cause 170 /C14magnetiza- tion rotation of a magnetoelastic single domain. The design consisted of two stages where the ﬁrst stage determined the major and minor axis lengths to ensure thermal stability of asingle domain nanodot as well as electrode overlap, resultingin maximum localized strain with bending effect. The secondstage optimized the input voltage control scheme to produce170 /C14magnetization rotation. A physical description of the mechanism to produce the voltage induced magnetizationwas presented. The energy to reorient the single domain was200 (aJ) in this particular design. ACKNOWLEDGMENTS The authors would like to thank Andres Chavez for valuable discussions and editing. This work was supported by the NSF Nanosystems Engineering Research Center forTranslational Applications of Nanoscale MultiferroicSystems (TANMS) Cooperative Agreement Award (No.EEC-1160504). 1W. Eerenstein, M. Wiora, J. L. Prieto, J. F. Scott, and N. D. Mathur, “Giant sharp and persistent converse magnetoelectric effects in multifer- roic epitaxial heterostructures,” Nat. Mater. 6, 348–351 (2007). FIG. 6. Time response of magnetization (a) and strain (b) for the memory bit. (a) The magnetization starts at zero, and a positive voltage is applied to both electrodes A and B, in which the magnetization rotates from 5/C14to 90/C14. The voltage on electrode B is removed at time ¼4, and the voltage remains on electrode A. The magnetization keeps rotating pass 90/C14. 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1.3475501.pdf	Structural, static and dynamic magnetic properties of Co 2 MnGe thin films on a sapphire a -plane substrate M. Belmeguenai, F. Zighem, T. Chauveau, D. Faurie, Y. Roussigné, S. M. Chérif, P. Moch, K. Westerholt, and P. Monod Citation: Journal of Applied Physics 108, 063926 (2010); doi: 10.1063/1.3475501 View online: http://dx.doi.org/10.1063/1.3475501 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/108/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Growth, structural, and magnetic characterization of epitaxial Co2MnSi films deposited on MgO and Cr seed layers J. Appl. Phys. 113, 043921 (2013); 10.1063/1.4789801 Structural, electrical, and magnetic properties of Mn 2.52 − x Co x Ni 0.48 O 4 films J. Appl. Phys. 107, 053716 (2010); 10.1063/1.3309780 Ferromagnetic resonance properties and anisotropy of Ni-Mn-Ga thin films of different thicknesses deposited on Si substrate J. Appl. Phys. 105, 07A942 (2009); 10.1063/1.3075395 Influence of the L 2 1 ordering degree on the magnetic properties of Co 2 MnSi Heusler films J. Appl. Phys. 103, 103910 (2008); 10.1063/1.2931023 Growth, structural, and magnetic characterizations of nanocrystalline γ ′ - Fe Ni N ( 220 ) thin films Appl. Phys. Lett. 90, 032505 (2007); 10.1063/1.2430920 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.189.170.231 On: Sat, 20 Dec 2014 04:47:53Structural, static and dynamic magnetic properties of Co 2MnGe thin ﬁlms on a sapphire a-plane substrate M. Belmeguenai,1,a/H20850F. Zighem,2T. Chauveau,1D. Faurie,1Y. Roussigné,1S. M. Chérif,1 P. Moch,1K. Westerholt,3and P. Monod4 1LPMTM, Institut Galilée, UPR 9001 CNRS, Université Paris 13, 99 Avenue Jean-Baptiste Clément F-93430 Villetaneuse, France 2LLB (CEA CNRS UMR 12), Centre d’études de Saclay, 91191 Gif-Sur-Yvette, France 3Institut für Experimentalphysik/Festkörperphysik, Ruhr-Universität Bochum, 44780 Bochum, Germany 4LPEM, UPR A0005 CNRS, ESPCI, 10 Rue Vauquelin, F-75231 Paris Cedex 5, France /H20849Received 21 May 2010; accepted 7 July 2010; published online 27 September 2010 /H20850 Magnetic properties of Co 2MnGe thin ﬁlms of different thicknesses /H2084913, 34, 55, 83, 100, and 200 nm /H20850, grown by rf sputtering at 400 °C on single crystal sapphire substrates, were studied using vibrating sample magnetometry and conventional or microstrip line ferromagnetic resonance. Theirbehavior is described assuming a magnetic energy density showing twofold and fourfold in-planeanisotropies with some misalignment between their principal directions. For all the samples, theeasy axis of the fourfold anisotropy is parallel to the c-axis of the substrate while the direction of the twofold anisotropy easy axis varies from sample to sample and seems to be strongly inﬂuencedby the growth conditions. Its direction is most probably monitored by the slight unavoidable miscutangle of the Al 2O3substrate. The twofold in-plane anisotropy ﬁeld Huis almost temperature independent, in contrast with the fourfold ﬁeld H4which is a decreasing function of the temperature. Finally, we study the frequency dependence of the observed line-width of the resonant mode and weconclude to a typical Gilbert damping constant /H9251value of 0.0065 for the 55-nm-thick ﬁlm. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3475501 /H20852 I. INTRODUCTION Ferromagnetic Heusler half metals with full spin polar- ization at the Fermi level are considered as potential candi-dates for injecting a spin-polarized current from a ferromag-net into a semiconductor and for developing sensitivespintronic devices. 1Some Heusler alloys, like Co 2MnGe, are especially promising for these applications, due to their highCurie temperature /H20849905 K /H20850/H20849Ref. 2/H20850and to their good lattice matching with some technologically importantsemiconductors. 3Therefore, great attention was recently paid to this class of Heusler alloys.4–10 In a previous work,11we used conventional and micros- trip line /H20849MS /H20850ferromagnetic resonance /H20849FMR /H20850, as well as Brillouin light scattering to study magnetic properties of 34,55, and 83-nm-thick Co 2MnGe ﬁlms at room temperature. We showed that the in-plane anisotropy is described by thesuperposition of a twofold and of a fourfold term. The easyaxes of the fourfold anisotropy were found parallel to thec-axis of the Al 2O3substrate /H20849and, consequently, the hard axes lie at /H1100645° of c/H20850. The easy axes of the twofold aniso- tropy were found at /H1100645° of cfor the 34 and 55-nm-thick ﬁlms and slightly misaligned with this orientation in the caseof the 83-nm-thick sample. However, a detailed study of thein-plane anisotropy, involving temperature and thickness de-pendence, allowing for their physical interpretation was stillmissing. Therefore, it forms the aim of the present paper.Rather complete x-rays diffraction /H20849XRD /H20850measurements over a large thickness range of Co 2MnGe ﬁlms are reportedbelow in an attempt to ﬁnd correlations between in-plane anisotropies, thickness, and crystallographic textures. Thethickness-dependence and the temperature-dependence ofthese anisotropies are investigated using vibrating samplemagnetometry /H20849VSM /H20850and the above mentioned FMR tech- niques. In addition, we present intrinsic damping parametersdeduced from broadband FMR data obtained with the help ofa vector network analyzer /H20849VNA /H20850. 12–14 II. SAMPLE PROPERTIES AND PREPARATION Co2MnGe ﬁlms with 13, 34, 55, 83, 100, and 200 nm thickness were grown on sapphire a-plane substrates /H20849show- ing an in-plane c-axis /H20850by rf sputtering with a ﬁnal 4-nm- thick gold over layer. A more detailed description of thesample preparation procedure can be found elsewhere. 11,15 The static magnetic measurements were carried out at room temperature using a VSM. The dynamic magneticproperties were investigated with the help of 9.5 GHz con-ventional FMR and of MS-FMR. 11The conventional FMR set-up consists in a bipolar X-band Bruker ESR spectrometer equipped with a TE 102resonant cavity immersed is an Ox- ford cryostat, allowing for exploring the 4–300 K tempera-ture interval. The MS-FMR set-up is home-made designedand, up to now, only works at room temperature. The reso-nance ﬁelds /H20849conventional FMR /H20850and frequencies /H20849MS-FMR /H20850 are obtained from a ﬁt assuming a Lorentzian derivativeshape of the recorded data. The experimental results are ana-lyzed in the frame of the model presented in Ref. 11. XRD experiments were performed using four circles dif- fractometers in Bragg–Brentano geometry in order to deter- a/H20850Electronic mail: belmeguenai.mohamed@univ-paris13.fr.JOURNAL OF APPLIED PHYSICS 108, 063926 /H208492010 /H20850 0021-8979/2010/108 /H208496/H20850/063926/6/$30.00 © 2010 American Institute of Physics 108, 063926-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.189.170.231 On: Sat, 20 Dec 2014 04:47:53mine/H9258–2/H9258patterns and pole ﬁgures. The diffractometer de- voted to the /H9258–2/H9258patterns was equipped with a point detector /H20849providing a precision of 0.015° in 2 /H9258scale /H20850. The instrument used for recording pole ﬁgures was equipped withan Inel™ curved linear detector /H20849120° aperture with a preci- sion of 0.015° in 2 /H9258scale /H20850. The x-rays beams /H20849cobalt line focus source at /H9261=1.788 97 Å /H20850were emitted by a Bruker™ rotating anode. We deﬁne a direct macroscopic orthonormalreference /H208491, 2, 3 /H20850, where the 3-axis stands for the direction normal to the ﬁlm. /H9278and/H9274are the so-called rotation angles of samples used for pole ﬁgure measurements. /H9274is the dec- lination angle between the scattering vector and the 3-axis, /H9278 is the rotational angle around the 3-axis. The /H9258–2/H9258patterns /H20849not shown here /H20850indicate that, for all the Co 2MnGe thin ﬁlms, the /H20851110 /H20852axis can be taken along the 3-axis. The Co2MnGe deduced lattice constant /H20849a=5.755 Å /H20850is in good agreement with the previously published ones.6,16Due to the /H20851111 /H20852preferred orientation of the gold over layer along the 3-axis, only partial /H20853110 /H20854pole ﬁgures could be efﬁciently exploited. They behave as /H20853110 /H20854ﬁber textures containing well deﬁned zones showing signiﬁcantly higher intensities/H20851Figs. 1/H20849a/H20850and1/H20849b/H20850/H20852. These regions correspond to orientation variants which can be grouped into two families /H20849see Fig. 1/H20850. The ﬁrst one, where the threefold /H2085111¯1/H20852or the /H2085111¯1¯/H20852axis is oriented along the crhombohedral direction, consists of two kinds of distinct domains with the /H20851001 /H20852axis at /H1100654.5° from thec-axis. The second family, which is rotated around the 3-axis by 90° from the ﬁrst one, also contains two variants.This peculiar in-plane domain structure is presumably in-duced by the underlying vanadium seed layer. In the 200-nm-thick sample the anisotropy of the ﬁber is less markedbut the two families remain present. In the thinner sample/H2084913 nm thick /H20850the measured signal originating from the Co 2MnGe ﬁlm is very weak, thus preventing from a precise analysis of the distribution of the crystallographic orienta-tions. In the following, we then choose to preferentiallypresent results concerning the 55 and the 100-nm-thickspecimens which allow for quantitative analysis of the con-centration of the above mentioned variants. As illustrated inFig. 1/H20849b/H20850, which represents /H9278-scans at /H9274=60°, we do not observe major differences between the crystallographic tex-tures of the 55 and of the 100-nm-thick samples: the ﬁrstfamily shows a concentration twice larger than the secondone; at least for the ﬁrst family, which allows for quantitativeevaluations, the concentrations of the two variants do notappreciably differ from each other; ﬁnally, about 50% of thetotal scattered intensity arises from domains belonging tothese oriented parts of the scans. III. RESULTS AND DISCUSSION A. Static magnetic measurements In order to study the magnetic anisotropy at room tem- perature, the hysteresis loops were measured for all the stud-ied ﬁlms with an in-plane applied magnetic ﬁeld along vari-ous orientations as shown in Fig. 2/H20849 /H9272His the in-plane angle between the magnetic applied ﬁeld Hand the c-axis of the substrate /H20850. The variations in the coercive ﬁeld /H20849Hc/H20850and of the reduced remanent magnetization /H20849Mr/Ms/H20850were then investi-gated as function of /H9272H. The typical magnetic static behavior for all the studied samples is illustrated below through tworepresentative ﬁlms which present different anisotropies. Figure 2/H20849a/H20850shows the loops along four orientations for the 100-nm-thick sample. One observes differences in shapeof the normalized hysteresis loops depending upon the ﬁeldorientation. For Halong the c-axis /H20849 /H9272H=0° /H20850we observe a typical easy axis square-shaped loop with a nearly full nor- malized remanence /H20849Mr/Ms=0.9 /H20850, a coercive ﬁeld of about 20 Oe and a saturation ﬁeld of 100 Oe. As /H9272Hincreases away from the c-axis direction, the coercivity increases and the hysteresis loop tends to transform into a hard axis loop. When /H9272Hslightly overpasses 90° /H2084990°/H11021/H9272H/H11021100° /H20850the loop evolves into a more complicated shape: it becomes com- posed of three /H20849or two /H20850open smaller loops. Further increas- ing the in-plane rotation angle, it changes from such a split-open curve up to an almost rectangular shape. The results for /H9272H=45° and /H9272H=135° are different: they show a rounded loop with Mr/Msequal to 0.75 and 0.63 and with saturation ﬁelds of about 170 Oe and 200 Oe, respectively. This result FIG. 1. /H20849Color online /H20850/H20849a/H20850Partial /H20853110 /H20854x-rays pole ﬁgures /H20849around 60° /H20850of 13, 55, 100, and 200-nm-thick ﬁlms. /H20849b/H20850Display of the angular variations in the intensity derived from the above ﬁgures for the 55 and 100-nm-thicksamples /H20849the blue and pink vertical dashed lines, respectively, refer to the two expected positions of the diffraction peak relative to the two variantsbelonging to family 1 /H20850.063926-2 Belmeguenai et al. J. Appl. Phys. 108, 063926 /H208492010 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.189.170.231 On: Sat, 20 Dec 2014 04:47:53qualitatively agrees with a description of the in-plane aniso- tropy in terms of fourfold and twofold contributions withslightly misaligned easy axes. The variations in H candMr/Msversus /H9272Hare illustrated in Figs. 3/H20849a/H20850and3/H20849b/H20850for the 100-nm-thick ﬁlm. The pres- ence of a fourfold anisotropy contribution is supported by thebehavior of H c/H20851Fig.3/H20849a/H20850/H20852, since two minima appear within each period /H20849180°, as expected /H20850, as shown in Fig. 3/H20849a/H20850. The minimum minimorum is mainly related to the uniaxial aniso-tropy term. In the same way, as displayed in Fig. 3/H20849b/H20850, the behavior of M r/Msis dominated by the uniaxial anisotropy. It is worth to notice that the minimum minimorum positionslightly differs from 90° /H20849lying around 96° /H20850, thus arguing for a misalignment between the twofold and the fourfold aniso-tropy axes. Figure 2/H20849b/H20850shows a series of hysteresis loops, recorded with an in-plane applied ﬁeld, for the 55-nm-thick ﬁlm. Acareful examination suggests that the fourfold anisotropycontribution is the dominant one and that the related easyaxis lies along the c-axis. The M r/Msvariation versus /H9272H, reported in Fig. 3/H20849c/H20850, is consistent with an easy uniaxial axis oriented at 45° of this last direction. Both fourfold anduniaxial terms are smaller than for the 100-nm-thick sample.B. Dynamic magnetic properties As previously published,11the dynamic properties are tentatively interpreted assuming a magnetic energy densitywhich, in addition to Zeeman, demagnetizing and exchangeterms, is characterized by the following anisotropy contribu-tion: E anis=K/H11036sin2/H9258M−1 2/H208511 + cos 2 /H20849/H9272M−/H9272u/H20850/H20852Kusin2/H9258M −1 8/H208513 + cos 4 /H20849/H9272M−/H92724/H20850/H20852K4sin4/H9258M, /H208491/H20850 In the above expression, /H9258Mand/H9272M, respectively, represent the out-of-plane and the in-plane /H20849referring to the c-axis of the substrate /H20850angles deﬁning the direction of the magnetiza- tionMs;/H9272uand/H92724stand for the angles of the uniaxial axis and of the easy fourfold axis, respectively, with this c-axis. With these deﬁnitions KuandK4are necessarily positive. As done in Ref. 11, it is often convenient to introduce the effec- tive magnetization 4 /H9266Meff=4/H9266Ms−2K/H11036/Ms, the uniaxial in-plane anisotropy ﬁeld Hu=2Ku/Msand the fourfold in- plane anisotropy ﬁeld H=4K4/Ms. For an in-plane applied magnetic ﬁeld H, the studied model provides the following expression for the frequenciesof the experimentally observable magnetic modes: F n2=/H20873/H9253 2/H9266/H208742/H20875Hcos /H20849/H9272H−/H9272M/H20850+2K4 Mscos 4 /H20849/H9272M−/H92724/H20850 +2Ku Mscos 2 /H20849/H9272M−/H9272u/H20850+2Aex Ms/H20873n/H9266 d/H208742/H20876 /H11003/H20875Hcos /H20849/H9272H−/H9272M/H20850+4/H9266Meff +K4 2Ms/H208493 + cos 4 /H20849/H9272M−/H92724/H20850/H20850 +Ku Ms/H208491 + cos 2 /H20849/H9272M−/H9272u/H20850/H20850+2Aex Ms/H20873n/H9266 d/H208742/H20876. /H208492/H20850 In the above expression /H9253is the gyromagnetic factor: /H20849/H9253/2/H9266/H20850=g/H110031.397/H11003106Hz /Oe. The uniform mode corre- sponds to n=0. The other modes to be considered /H20849perpen- dicular standing modes /H20850are connected to integer values of n: their frequencies depend upon the exchange stiffness con-stant A exand upon the ﬁlm thickness d. For all the ﬁlms the magnetic parameters at room tem- perature were derived from MS-FMR measurements. The de-duced gfactor is equal to 2.17, as previously published. 11 The in-plane MS-FMR spectrum of the 100-nm-thick sample /H20851Fig.4/H20849a/H20850/H20852submitted to a ﬁeld of 520 Oe shows two distinct modes: a main one /H20849mode 2 /H20850, with a wide line-width /H20849about 0.6 GHz /H20850and a second weaker one /H20849mode 1 /H20850at lower frequency with a narrower line-width /H208490.2 GHz /H20850. Their ﬁeld- dependences are presented in Fig. 4/H20849b/H20850. In contrast with mode 2, which presents signiﬁcant in-plane anisotropy, themeasured resonance frequency of mode 1 does not vary ver-sus the in-plane angular orientation of the applied magneticﬁeld: such a different behavior prevents from attributingmode 1 to a perpendicular standing excitation. Consequently,mode 1 is presumably a uniform mode arising from the pres-ence of an additional magnetic phase in the ﬁlm, possessing-100 -50 0 50 100-1.0-0.50.00.51.0Normalized magnetization (M/Ms) Applied magnetic field (Oe)ϕΗ=0° ϕΗ=45° ϕΗ=90° ϕΗ=135°100 nm (a) -100 -50 0 50 100-1.0-0.50.00.51.0 (b)Normalized magnetization (M/Ms) Applied ma gnetic field (Oe)ϕH=0° ϕH=45° ϕH=90° ϕH=135°55 nm FIG. 2. /H20849Color online /H20850VSM magnetization loops of the /H20849a/H20850100-nm-thick and the /H20849b/H2085055-nm-thick samples. The magnetic ﬁeld is applied parallel to the ﬁlm surface, at various angles /H20849/H9272H/H20850with the c-axis of the sapphire substrate.063926-3 Belmeguenai et al. J. Appl. Phys. 108, 063926 /H208492010 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.189.170.231 On: Sat, 20 Dec 2014 04:47:53a lower effective demagnetizing ﬁeld. In the following, we focus on mode 2 which is assumed to be the uniform modearising from the main phase. As previously published, onlyone uniform mode is observed with the 55-nm-thick sample. Figures 5/H20849b/H20850and5/H20849d/H20850illustrate the experimental in-plane angular-dependencies of the resonance frequency of the uni-form mode for the 100 and 55-nm-thick samples, compared to the obtained ﬁts using Eq. /H208492/H20850. As expected from the VSM measurements, in the 100-nm sample the fourfold anduniaxial axes of anisotropy are misaligned: it results an ab-sence of symmetry of the representative graphs around /H9272H =90°. The best ﬁt is obtained for the following values of themagnetic parameters: 4 /H9266Meff=9800 Oe, Hu=55 Oe, H4 =110 Oe, /H92724=0°,/H9272u=12°. As previously published, in the case of the 55 nm sample the direction of the easy uniaxialaxis coincides with the hard fourfold axis: this parallelisminduces symmetry of the graphs around /H9272H=90°. The best ﬁt for this ﬁlm corresponds to: 4 /H9266Meff=9800 Oe, Hu=10 Oe, H4=54 Oe, /H92724=0°,/H9272u=45°. In both samples, the fourfold anisotropy easy direction is parallel to the c-axis of the sub- strate: this presumably results from an averaging effect of theabove described distribution of the crystallographic orienta-tions, in spite of the facts that such a conclusion requiresequal concentrations of the two main variants, a conditionwhich, strictly speaking, is not fully realized, and that theobserved value of /H92724does not derive from the probably over- simpliﬁed averaging model that we attempted to use, basedon individual domain contributions showing their principalaxis of anisotropy along their cubic direction.0 50 100 150 200 250 300 3501520253035VSM measurements (a)Coercive field (Oe) ϕΗ(degrees )100 nm 0 50 100 150 200 250 300 3500.20.40.60.8 (b)Reduced remanent magnetization (Mr/Ms) ϕΗ(degrees)VSM measurements100 nm 0 50 100 150 200 250 300 3500.750.800.850.900.95Reduced remanent magnetization (Mr/Ms) ϕΗ(degrees)VSM measurements55 nm (c) FIG. 3. /H20849a/H20850Coercive ﬁeld and /H20849b/H20850reduced remanent magnetization of the 100-nm-thick sample as a function of the in-plane ﬁeld orientation /H20849/H9272H/H20850./H20849c/H20850 Reduced remanent magnetization of the 55-nm-thick ﬁlm.6.5 7.0 7.5 8.0 8.5 9.0 9.5 10. 0-400-300-200-1000100200300Amplitude (arb. units ) Frequency (GHz)(a)H=520 Oe ϕΗ=0° Mode 2 Mode 1 0 300 600 900 1200 15002468101214 (b)Frequency (GHz) Applied ma gnetic field (Oe)Mode 2 Mode 1 Fit mode 2 Fit mode 1 ϕΗ=0° FIG. 4. /H20849Color online /H20850/H20849a/H20850MS-FMR spectrum under a magnetic ﬁeld applied /H20849H=520 Oe /H20850parallel the c-axis and /H20849b/H20850ﬁeld-dependence of the resonance frequency of the uniform excited modes, in the 100-nm-thick thin ﬁlm. Theﬁts are obtained using Eq. /H208492/H20850with the parameters indicated in Table I.063926-4 Belmeguenai et al. J. Appl. Phys. 108, 063926 /H208492010 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.189.170.231 On: Sat, 20 Dec 2014 04:47:53As usual, attempts to interpret the in-plane hysteresis loops using the coherent rotation model do not provide aquantitative evaluation of the anisotropy terms involved inthe expression of magnetic energy density. However, the ex-perimentally measured M r/Msangular variation, which, with this model, is given by cos /H20849/H9272M−/H9272H/H20850in zero-applied ﬁeld and is easily calculated knowing /H9272u,/H92724andHu/H4, is in agree- ment with the values of these coefﬁcients ﬁtted from reso-nance data, as shown in Figs. 5/H20849a/H20850and5/H20849c/H20850. The magnetic parameters deduced from our resonance measurements are given in Table Ifor the complete set of the studied ﬁlms. In contrast with the direction of the fourfoldaxis which does not vary, the orientation of the uniaxial axisis sample dependent: for some of them /H2084934 and 55 nm /H20850the easy uniaxial direction lies at 45° from the c-axis of the substrate /H20849thus coinciding with the hard fourfold direction /H20850; for other ones /H2084913, 83, 100 nm /H20850it shows a variable misalign- ment; ﬁnally, the uniaxial anisotropy ﬁeld vanishes for thethickest sample /H20849200 nm /H20850. We tentatively attribute at least a fraction of the uniaxial contribution as originating from aslight misorientation of the surface of the substrate. The am- plitudes of both in-plane anisotropies are sample dependentand cannot be simply related to the ﬁlm thickness. It shouldbe mentioned that some authors 17have reported on strain- dependent uniaxial and fourfold anisotropies in Co 2MnGa. This suggests a forthcoming experimental x-rays study of thestrains present in our ﬁlms. In addition, it is useful to get information about the damping terms involved in the dynamics of magnetic excita-tions in the above samples. Notice that in order to integratethese ﬁlms in application devices like, for instance, magneticrandom access memory, it is important to make sure thattheir damping constant is small enough. The damping of the55-nm-thick ﬁlm was studied by VNA-FMR: 12–14it is ana- lyzed in terms of a Gilbert coefﬁcient /H9251in the Landau– Lifschitz–Gilbert equation of motion. The frequency line-width /H9004fof the resonant signal around f robserved using this technique is related to the ﬁeld line-width /H9004Hmeasured with conventional FMR excited with a radio-frequency equal to fr through the equation:18 /H9004H=/H20879/H9004f/H11509H/H20849f/H20850 /H11509f/H20879 f=fr. /H208493/H20850 /H9004His given by: /H9004H=/H9004H0+4/H9266fr /H20841/H9253/H20841/H9251 /H208494/H20850 /H20849where /H9004H0stands for a small contribution arising from in- homogeneous broadening /H20850. The measured linear dependence of/H9004His shown versus frin Fig. 6. We then obtain the damping coefﬁcient: /H9251=0.0065. This value lies in the range observed in the Co 2MnSi thin ﬁlms.19–21 Finally, the temperature dependence was studied for the 55-nm-thick sample using conventional FMR. The ﬁts of themagnetic parameters were performed assuming that gprac- tically does not vary versus the temperature T, as generally expected. We then take: g=2.17. The results for the uniaxial and for the fourfold in-plane anisotropy ﬁelds are reported inFig.7.H uis temperature independent while H4is a signiﬁ- cantly decreasing function of T. This behavior of H4is pre- sumably related to the magnetocrystalline origin of this an-isotropy term.0 50 100 150 200 250 300 3506.87.27.68.00.00.30.60.9 100 nm (b)Frequency (GHz) ϕH(degrees)MS-FMR measurements Fit100 nm (a)Mr/Ms Fit VSM measurements H=520 Oe 0 50 100 150 200 250 300 3502.73.03.33.60.60.81.0 (d)55 nm H=130 OeFrequency (GHz) ϕH(degrees)MS-FMR Fit(c)55 nmMr/Ms VSM measurements Fit FIG. 5. Reduced remanent magnetization of the /H20849a/H20850100-nm-thick and of the /H20849c/H2085055-nm-thick ﬁlms. The simulations are obtained from the energy mini- mization using the parameters reported in Table I./H20849b/H20850and /H20849d/H20850show the compared in-plane angular-dependences of the resonance frequency of theuniform modes. The ﬁt is obtained using Eq. /H208492/H20850with the parameters indi- cated in Table I.TABLE I. Magnetic parameters obtained from the best ﬁts to our experi- mental results. /H9272uand/H92724are the angles of in-plane uniaxial and of fourfold anisotropy easy axes, respectively. Thickness /H20849nm /H208504/H9266Meff /H20849Oe /H20850Hu=2Ku/Ms /H20849Oe /H20850H4=4K4/Ms /H20849Oe /H20850/H9272u /H20849deg /H20850/H92724 /H20849deg /H20850 13 8000 45 40 12 0 34 9000 5 20 45 055 9800 10 54 45 083 9200 15 22 /H1100250 100 9800 60 110 12 0200 9900 /H1101502 4 0063926-5 Belmeguenai et al. J. Appl. Phys. 108, 063926 /H208492010 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.189.170.231 On: Sat, 20 Dec 2014 04:47:53IV. SUMMARY The static and dynamic magnetic properties of Co 2MnGe ﬁlms of various thicknesses sputtered on a-plane sapphire substrates have been studied. The present work focused onthe dependence of the parameters describing the magneticanisotropy upon the crystallographic texture and upon thethickness of the ﬁlms. The crystallographic characteristicswere obtained through XRD which reveals the presence of amajority of two distinct /H20849110 /H20850domains. Magnetometric mea- surements were performed by VSM and magnetization dy-namics was analyzed using conventional and MS resonances/H20849FMR and MS-FMR /H20850. The main results concern the in-planeanisotropy which contributes to the magnetic energy density through two terms: a uniaxial one and a fourfold one. Theeasy axis related to the fourfold term is always parallel to thec-axis of the substrate while the easy twofold axis shows a variable misalignment with the c-axis. The fourfold aniso- tropy is a decreasing function of the temperature: it is pre-sumably of magnetocrystalline nature and its orientation isrelated to the above noticed domains. The observed mis-alignment of the twofold axis is tentatively interpreted asinduced by random slight miscuts affecting the orientation ofthe surface of the substrate. The twofold anisotropy does notsigniﬁcantly depend on the temperature. There is no evi-dence of a well-deﬁned dependence of the anisotropy versusthe thickness of the ﬁlms. Finally, we show that the dampingof the magnetization dynamics can be interpreted as arisingfrom a Gilbert term in the equation of motion that we evalu-ate. 1S. Tsunegi, Y. Sakuraba, M. Oogane, K. Takanashi, and Y. Ando, Appl. Phys. Lett. 93, 112506 /H208492008 /H20850. 2S. Picozzi, A. Continenza, and A. J. Freeman, Phys. Rev. B 66, 094421 /H208492002 /H20850. 3S. Picozzi, A. Continenza, and A. J. Freeman, J. Phys. Chem. Solids 64, 1697 /H208492003 /H20850. 4T. Ambrose, J. J. Krebs, and G. A. Prinz, J. Appl. Phys. 89,7 5 2 2 /H208492001 /H20850. 5T. Ishikawa, T. Marukame, K. Matsuda, T. Uemura, M. Arita, and M. Yamamoto, J. Appl. Phys. 99, 08J110 /H208492006 /H20850. 6F. Y. Yang, C. H. Shang, C. L. Chien, T. Ambrose, J. J. Krebs, G. A. Prinz, V. I. Nikitenko, V. S. Gornakov, A. J. Shapiro, and R. D. Shull, Phys. Rev. B65, 174410 /H208492002 /H20850. 7H. Wang, A. Sato, K. Saito, S. Mitani, K. Takanashi, and K. Yakushiji, Appl. Phys. Lett. 90, 142510 /H208492007 /H20850. 8Y. Sakuraba, M. Hattori, M. Oogane, Y. Ando, H. Kato, A. Sakuma, T. Miyazaki, and H. Kubota, Appl. Phys. Lett. 88, 192508 /H208492006 /H20850. 9T. Marukame, T. Ishikawa, K. Matsuda, T. Uemura, and M. Yamamoto, Appl. Phys. Lett. 88, 262503 /H208492006 /H20850. 10D. Ebke, J. Schmalhorst, N.-N. Liu, A. Thomas, G. Reiss, and A. Hütten, Appl. Phys. Lett. 89, 162506 /H208492006 /H20850. 11M. Belmeguenai, F. Zighem, Y. Roussigné, S.-M. Chérif, P. Moch, K. Westerholt, G. Woltersdorf, and G. Bayreuther, Phys. Rev. B 79, 024419 /H208492009 /H20850. 12M. Belmeguenai, T. Martin, G. Woltersdorf, M. Maier, and G. Bayreuther, Phys. Rev. B 76, 104414 /H208492007 /H20850. 13T. Martin, M. Belmeguenai, M. Maier, K. Perzlmaier, and G. Bayreuther, J. Appl. Phys. 101, 09C101 /H208492007 /H20850. 14M. Belmeguenai, T. Martin, G. Woltersdorf, G. Bayreuther, V. Baltz, A. K. Suszka, and B. J. Hickey, J. Phys.: Condens. Matter 20, 345206 /H208492008 /H20850. 15U. Geiersbach, K. Westerholt, and H. Back, J. Magn. Magn. Mater. 240, 546 /H208492002 /H20850. 16T. Ambrose, J. J. Krebs, and G. A. Prinz, J. Appl. Phys. 87,5 4 6 3 /H208492000 /H20850. 17M. J. Pechan, C. Yua, D. Carrb, and C. J. Palmstrøm, J. Magn. Magn. Mater. 286, 340 /H208492005 /H20850. 18S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, and J. P. Nibarger, J. Appl. Phys. 99, 093909 /H208492006 /H20850. 19R. Yilgin, M. Oogane, Y. Ando, and T. Miyazaki, J. Magn. Magn. Mater. 310, 2322 /H208492007 /H20850. 20R. Yilgin, Y. Sakuraba, M. Oogane, S. Mizumaki, Y. Ando, and T. Miyazaki, Jpn. J. Appl. Phys., Part 2 46, L205 /H208492007 /H20850. 21S. Trudel, O. Gaier, J. Hamrle, and B. Hillebrands, J. Phys. D: Appl. Phys. 43, 193001 /H208492010 /H20850.456789 1 0253035404550Field linewidth ΔH( O e ) Frequenc y(GHz)VNA-FMR measurements Fit FIG. 6. Line-width /H9004Has a function of the resonance frequency for 55-nm- thick ﬁlm. /H9004His derived from the experimental VNA-FMR frequency- swept line-width. 0 50 100 150 200 250 300102030405060708090100Anisotropy fields (Oe) Temperature (K)Hu H4 FIG. 7. /H20849Color online /H20850Temperature-dependence of the fourfold anisotropy ﬁeld /H20849H4/H20850and the unixial anisotropy ﬁeld /H20849Hu/H20850of the 55-nm-thick ﬁlm, measured by FMR at 9.5 GHz.063926-6 Belmeguenai et al. J. Appl. Phys. 108, 063926 /H208492010 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.189.170.231 On: Sat, 20 Dec 2014 04:47:53
1.3651792.pdf	Vibro-acoustics of organ pipes—Revisiting the Miller experiment (L) F . Gautier,a)G. Nief, J. Gilbert, and J. P . Dalmont Laboratoire d’Acoustique de l’Universite ´ du Maine, UMR CNRS 6613 Avenue O. Messiaen, 72085 Le Mans Cedex 9, France (Received 23 November 2010; revised 6 July 2011; accepted 6 July 2011) A century ago, Science published a spectacular experimental study on the physics of organ pipes. Dayton C. Miller observed experimentally that the sound produced by an organ pipe can depend onthe vibration of its walls, in addition to its internal geometry and the interaction between the air jet and the labium. The Miller experiment has been repeated and an interpretation is now proposed in terms of vibroacoustic coupling mechanisms between walls and internal ﬂuid, which can lead to“pathological” behavior. VC2012 Acoustical Society of America . [DOI: 10.1121/1.3651792] PACS number(s): 43.75.Np, 43.75.Yy [JW] Pages: 737–738 I. INTRODUCTION How does the wall material of a wind instrument affect its sound? This question has engendered a long-lasting debate among scientists, musicians, and instrument makers.One way the material might have an effect is related to vibrations of the instrument walls. Acousticians explain that the behavior of a wind instrument is governed by the acousti-cal response of the pipe—its input impedance—which is mainly determined by its internal geometry—the bore. This gives the wall vibrations a negligible role. On the contrary,some makers or musicians think they play a signiﬁcant part. The literature on this subject has thus sometimes provided various and outwardly contradictory results. In fact, the wall vibrations are easily felt or measured on most wind instruments. However, their inﬂuence on the pro- duced sound is more difﬁcult to bring to light, because theﬂuid-structure couplings involved are weak. A century ago in Science, Dayton C. Miller, 1one of the founding members of the Acoustical Society of America,published an experimental study 2about materials and vibra- tions of organ pipes. He compared ﬂue pipes having rectan- gular cross section, of identical internal geometry, but withdifferent thicknesses and construction materials. In one of these experiments using a double walled organ pipe, the space between the two walls could be ﬁlled with water whilethe pipe was sounded, presumably damping the wall vibra- tions. Miller then observed, without explaining it, that the ﬁlling led to unusual behavior of the pipe, clearly audible.Some heights of the water jacket produced pitch changes or inharmonic and unstable tones. The aim of this paper is to present experimental results obtained from a copy of thisexperiment and to interpret them according to recent results on vibroacoustics of musical wave-guides. II. EXPERIMENTAL RESULTS The experiment [Fig. 1(a)], often referenced in musical acoustics, was reproduced [Fig. 1(b)] in the Laboratoired’Acoustique de l’Universite ´ du Maine, UMR CNRS, France. An organ zinc pipe, whose cross section is rectangular (5.8 cm /C27.1 cm) and whose thickness is 0.5 mm was surrounded by a pipe of larger cross section to form a double- walled pipe; the space between the walls could be ﬁlled with water. Figure 1(c) presents a time-frequency analysis of the sound recorded at 15 cm outside the labium while the double wall is continuously ﬁlled with water. In the experiment, the relationship between time and water height is hðtÞ¼ _ht, where the ﬁlling speed _his equal to 5 mm s/C01. The fundamental frequency rises about a semi-tone throughout the whole experiment, due to thermal effects.During the ﬁlling some “accidents” occur that are more note- worthy. Pitch changes and unstable tones are indeed clearly audible. In Fig. 1(c) the arrows on the second harmonic show respectively a strong pitch change (1), an unstable tone (2), and a silence (3). The silence corresponds to a break up of the self-sustained oscillation. This phenomenon was notobserved by Miller in his original paper. III. INTERPRETATION These spectacular experimental facts are due to an inter- nal vibroacoustic coupling mechanism. The walls have been set in vibration by the internal acoustic ﬁeld, but in these cases acoustic radiation from the vibrating walls to the exter-nal ﬁeld does not have a signiﬁcant effect. Instead, the oscil- lation is disturbed by strong sound waves produced inside the instrument by the vibrating walls. This occurs when oneof the acoustical resonance frequencies of the air column and one of the mechanical resonance frequencies of the wall coincide. In the experiment, the mechanical resonance fre-quencies vary as the water rises, because it affects the effec- tive mass and stiffness of the walls. The variations of the mechanical resonance frequencies has been conﬁrmed bymodal testing made with an impact hammer and an acceler- ometer located at the top of pipe. In a parallel experiment described in Ref. 3, a rigid slide is connected to a test tube. In such a way, the acoustical resonance can be changed, without modifying the mechanical resonance frequencies of the test tube, in order to satisfy or not the coincidence a)Author to whom correspondence should be addressed. Electronic mail: gautier@univ-lemans.fr J. Acoust. Soc. Am. 131(1), Pt. 2, January 2012 VC2012 Acoustical Society of America 737 0001-4966/2012/131(1)/737/2/$30.00condition. When it is satisﬁed, some tone color changes are also measured. The input impedance of the pipe can be calculated using a model that includes the coupling between the air column and the vibrating walls.3This model shows that a deformable wall can sometimes shift acoustical resonance frequencies enough to disturb t he harmonic relationships that are important for self-sustaining oscillations of the ﬂuepipe. IV. CONCLUSION The debate about the inﬂuence of wall materials will continue, because they are important in some contexts and less so in others. However, a re-examination of the vibroa- coustics of Miller’s experiment shows convincingly how thecoupling between the wall vibrations and the air column can be strong enough to cause drastic effects when the acoustical and mechanical resonances match. The spectacular effect ofwall vibrations on the sound produced by the organ pipe is explained by a coincidence phenomenon between acousticand mechanical resonances. ACKNOWLEDGMENTS The authors are grateful to M. Walther (Centre National de Formation des Apprentis Facteurs d’Orgues, Eschau, France), to E. Boyer, G. Estienne, R. Le Goaziou, C. Pineau(Ecole Nationale Supe ´rieure d’Inge ´nieurs du Mans, France), and to P. Hoekje for their contribution. 1P. Hoekje, Dayton C. Miller, Echoes 13(1), 1–7 (2003). 2D. Miller, “The inﬂuence of the material of wind-instruments on the tone quality,” Science 29(735), 161–171 (1909). 3G. Nief, F. Gautier, J. P. Dalmont, and J. Gilbert, “Inﬂuence of wall vibra- tions on the behavior of a simpliﬁed wind instrument,” J. Acoust. Soc. Am.124, 1320–1331 (2008). 4D. Miller, The Science of Musical Sounds (MacMillan, New York, 1926), p. 180. FIG. 1. (Color online) (a) Photograph of Miller’s historical experiment (from Ref. 4). (b) Diagram of experiment set-up using an organ pipe, as similar as pos- sible to the one used by Miller. (c) Time-frequency analysis of the sound. In this spectrogram, time has been converted to water height (vertical axis) using the constant ﬁlling rate. Arrows indicate a strong pitch change (1), an unstable tone (2), and silence (3). 738 J. Acoust. Soc. Am., Vol. 131, No. 1, Pt. 2, January 2012 Gautier et al.: Letters to the EditorCopyright of Journal of the Acoustical Society of America is the property of American Institute of Physics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
1.4961371.pdf	Exceptionally high magnetization of stoichiometric Y 3Fe5O12 epitaxial films grown on Gd3Ga5O12 James C. Gallagher , Angela S. Yang , Jack T. Brangham , Bryan D. Esser , Shane P. White , Michael R. Page , Keng-Yuan Meng , Sisheng Yu , Rohan Adur , William Ruane , Sarah R. Dunsiger , David W. McComb , Fengyuan Yang , and P. Chris Hammel Citation: Appl. Phys. Lett. 109, 072401 (2016); doi: 10.1063/1.4961371 View online: http://dx.doi.org/10.1063/1.4961371 View Table of Contents: http://aip.scitation.org/toc/apl/109/7 Published by the American Institute of Physics Exceptionally high magnetization of stoichiometric Y 3Fe5O12epitaxial films grown on Gd 3Ga5O12 James C. Gallagher,1Angela S. Yang,1Jack T. Brangham,1Bryan D. Esser,2 Shane P . White,1Michael R. Page,1Keng-Yuan Meng,1Sisheng Yu,1Rohan Adur,1 William Ruane,1Sarah R. Dunsiger,1David W. McComb,2Fengyuan Yang,1 and P . Chris Hammel1 1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA 2Center for Electron Microscopy and Analysis, Department of Materials Science and Engineering, The Ohio State University, Columbus, Ohio 43210, USA (Received 31 May 2016; accepted 7 August 2016; published online 15 August 2016) The saturation magnetization of Y 3Fe5O12(YIG) epitaxial ﬁlms 4 to 250 nm in thickness has been determined by complementary measurements including the angular and frequency dependencies ofthe ferromagnetic resonance ﬁelds as well as magnetometry measurements. The YIG ﬁlms exhibit state-of-the-art crystalline quality, proper stoichiometry, and pure Fe 3þvalence state. The values of YIG magnetization obtained from all the techniques signiﬁcantly exceed previously reportedvalues for single crystal YIG and the theoretical maximum. This enhancement of magnetization, not attributable to off-stoichiometry or other defects in YIG, opens opportunities for tuning magnetic properties in epitaxial ﬁlms of magnetic insulators. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4961371 ] Y 3Fe5O12(YIG) is one of the most thoroughly studied magnetic materials and has been widely used in microwave applications in the past several decades due to its exceptionally low magnetic damping.1In recent years, YIG has played a cen- tral role in the emerging ﬁelds of ferromagnetic resonance (FMR) spin pumping and therm ally driven spin calori- tronics.2–9Previous studies and applications almost exclusively used bulk YIG crystals or lm-thick YIG ﬁlms synthesized by liquid phase epitaxy. More recently, high quality thin YIG ﬁlms of a few to hundreds of nm thickness grown by pulsed laser deposition10,11and sputtering3,12,13have attracted much attention and revealed exciting phenomena, particularly in spin pumping.2,12–14The static and dynamic magnetic characteris- tics of the YIG thin ﬁlms depend on crystalline ordering, stoi- chiometry, and defect level. These characteristics in turn determine the behavior of spin transport in YIG-based hetero-structures. Despite the extensive use of YIG thin ﬁlms for these studies, a thorough characteriz ation of the magnetization is generally lacking. Here, we report a systematic study of high quality YIG ﬁlms grown by off-axis sputtering using comple- mentary characterization techn iques, which exhibit a surpris- ingly high magnetization. YIG epitaxial ﬁlms, 4 to 250 nm thick, were deposited on Gd 3Ga5O12(GGG) (111) substrates using an ultrahigh vacuum off-axis sputter deposition system.12,13A sputtering gas of Ar þ1.05% O 2with a total pressure of 11.5 mTorr was used. EPI-polished single crystal GGG substrates with a surface roughness of <5A˚were purchased from MTI Corporation. The GGG substrates were heated to 750/C14C for YIG deposition and rotated at 10/C14/s. The growth rate was 16 nm/h at a radio-frequency power of 60 W. We extensively characterized the quality and purity of the YIG ﬁlms. These studies revealed highly ordered ﬁlms essentially free of defects and impurities. The structuralquality of the YIG ﬁlms was examined by X-ray diffraction (XRD) using a Bruker D8 triple-axis X-ray diffractometer.Figure 1shows the 2 h-xscans for ﬁve YIG ﬁlms 16, 24, 40, 80, and 164 nm thick, all of which exhibit distinct Laue oscillations, indicating that the ﬁlms are highly crystalline,ordered, and uniform. Although the YIG(444) peak is over- shadowed by the GGG(444) peak, the YIG(444) peak posi- tion can be accurately determined from the Laue oscillationsatellite peaks. The out-of-plane spacing between adjacent YIG (111) planes is 7.175, 7.169, 7.161, 7.154, and 7.153 A ˚, corresponding to YIG cubic lattice constants of 12.427,12.417, 12.403, 12.391, and 12.390 A ˚for the 16, 24, 40, 80, FIG. 1. XRD 2 h-xscans of YIG ﬁlms of thickness tYIG¼16, 24, 40, 80, and 164 nm grown on the GGG(111) substrate near the YIG(444) peak (indi- cated by the arrows), from which the out-of-plane spacing of the YIG(111) planes is determined. The spectra are shifted vertically for clarity. Inset: 2h-xscan of a 160 nm YIG ﬁlm epitaxially grown on a YAG(001) substrate. 0003-6951/2016/109(7)/072401/5/$30.00 Published by AIP Publishing. 109, 072401-1APPLIED PHYSICS LETTERS 109, 072401 (2016) and 164 nm ﬁlms, respectively. Given the dislocation-free, full epitaxy shown below by STEM images, the in-plane lat-tice constant of the YIG ﬁlms should equal that of GGG(12.383 A ˚), resulting in minimal epitaxial strain with very small tetragonal distortions between 0.36% (16 nm) and 0.057% (164 nm). The crystalline ordering of the 164 nm YIG ﬁlm was fur- ther characterized by high-angle annular dark ﬁeld scanning transmission electron microscopy (HAADF-STEM) using an FEI probe-corrected Titan 380–300S/TEM. Figure 2(a)shows a large-area STEM image of the (111)-oriented YIG/GGGviewed along the h110idirection, which demonstrates the high uniformity of the YIG ﬁlm without any detectabledefects. It directly measures the thickness of the ﬁlm (164 nm). We also performed energy-dispersive X-ray (EDX) spectroscopy over the area marked by the yellow dashed box,which gives an atomic (at. %) composition of Y: 14.9 61.2 at. %, Fe: 25.0 63.4 at. %, and O: 60.1 63.9 at. % using experimentally determined Cliff-Lorimer k-factors from astandard of equal thickness. This conﬁrms proper stoichiome-try (3:5:12 ¼15%:25%:60%) within the instrument sensitiv- ity. A high-resolution STEM image in Fig. 2(b) reveals clear atomic ordering of Y and Fe in the garnet lattice. In HAADF-STEM or “ Z-contrast” ( Z: atomic number) imaging, scatter- ing is Rutherford-like in nature leading to an intensity that isapproximately proportional to Z 2: the most intense columns are pure Y ( Z¼39), the least intense are pure Fe ( Z¼26), and the intermediate-intensity columns contain alternating Y/Fe atoms. Thus, we identify the alternating pure Y andpure Fe columns along the blue dashed lines, while the greenbox marks a triplet of (Y/Fe)-Fe-(Y/Fe), which matches theoverlaying h110iprojection of the YIG lattice. FIG. 2. HAADF-STEM images of a 164 nm YIG ﬁlm grown on GGG (111) viewed along the h110idirection. (a) A low magniﬁcation STEM imagereveals that the ﬁlm is highly uniform without any detectable defects. The yellow dashed box is the region of the EDX measurement. (b) An atomic res- olution STEM image of the YIG ﬁlm, where the two perpendicular dashed lines indicate the alternating Y (bright)and Fe (dim) atomic columns. The green box encloses a (Y/Fe)-Fe-(Fe/Y) triplet chain. (c) A STEM image of the YIG/GGG epitaxial interface shows excellent continuity and no defects. The yellow box highlights a chain of - Ga-Gd-Fe-Y- atoms across the inter-face. (d) An EELS scan of the Fe L 2,3 edge in the YIG ﬁlm, where the dashed line indicates the Fe3þmaximum on the Fe L3peak at 709.5 eV and no detectable Fe2þat 707.8 eV. (e) EDX line scans for Y, Fe, Gd, and Ga across the interface as indicated by the reddashed line in (c).072401-2 Gallagher et al. Appl. Phys. Lett. 109, 072401 (2016) The STEM image of the YIG/GGG interface shown in Fig.2(c)demonstrates a smooth transition from GGG to YIG without any visible transition lay er or detectable defects such as dislocations. This high quality epitaxy arises from the factthat YIG and GGG are nearly perfectly matched with a lattice mismatch of only 0.057%, which is also why GGG is the ideal substrate for YIG growth. It clearly shows the atomic ordering of Gd/Ga in the GGG and Y/Fe in the YIG. The yellow box in Fig.2(c)highlights a chain of -Ga-Gd-Fe-Y- atoms across the interface. The atomic ordering o f YIG near the interface is the same as anywhere else deep in the YIG ﬁlm. In stoichiometric YIG, all Fe atoms should be Fe 3þwhere oxygen deﬁciency can lead to the presence of Fe2þions. To test for oxygen deﬁciency, we used electron energy loss spec- troscopy (EELS) to measure the valence state of Fe in the YIG ﬁlm. Figure 2(d) shows an EELS scan of the Fe L2,3edge in t h eY I Gﬁ l m ,w h e r eo n l yt h eF e3þL3(L2) peak is present with the maximum at 709.5 eV (722.6 eV) and no Fe2þat 707.8 eV (720.4 eV) is detected.15,16Quantitative analysis of the L2,3 edge gives 99.0 63.9% Fe3þ, indicating a stoichiometric oxi- dation state. To probe whether there is interdiffusion at theYIG/GGG interface, we performed EDX line scans across the interface as indicated by the red dashed line in Fig. 2(c). Figure 2(e)shows the atomic percentages of Y, Fe, Gd, and Ga as a function of distance from the interface, which provides evidence of an interfacial transition region (from 0 to fullintensity) of 1.4, 4.9, 2.8, and 4.8 nm, respectively. There is no detectable Gd and Ga in the YIG ﬁlm beyond a few nm from the interface. The widths of the interfacial transition region may be due to delocalization of the X-ray emission signal or interdiffusion. As shown below, since the 4 nm YIG ﬁlmexhibits high magnetization (2052 G) similar to that of the thicker ﬁlms, the interdiffusion layer should be much thinner than 4 nm; thus, the widths of the EDX transition region are mostly due to delocalization of the X-ray emission. Ferromagnetic resonance is a precise spectroscopic tech- nique for quantitative measurement of magnetization and magnetic anisotropy. The angular dependencies of the FMRabsorption for all of the YIG samples were measured in a microwave cavity using an X-band Bruker electron paramag- netic resonance (EPR) spectrometer. Figure 3(a) shows the derivative FMR absorption spectra for the 16 nm YIG ﬁlm at a resonance frequency f¼9.61 GHz with the orientation of the applied magnetic ﬁeld varying from in-plane ( h H¼90/C14) to out-of-plane ( hH¼0/C14) [see Fig. 3(b) for the experimental setup]. The in-plane linewidth is DH¼4.3 G as shown in Fig. 3(c), which is very narrow for a 16 nm YIG ﬁlm on GGG and indicates high magnetic uniformity. By plotting the angulardependence ( h H) of the FMR resonance ﬁeld ( Hres), we can determine the effective saturation magnetization, 4 pMeff ¼4pMsþH2?, where Msis the saturation magnetization and H2?is the out-of-plane uniaxial anisotropy. The free energy for a cubic crystal structure in the presence of anapplied magnetic ﬁeld can be calculated using, 13,17 E¼/C0H/C1Mþ1 2M4pMeffcos2h/C01 2H4?cos4h/C26 /C01 8H4jj3þcos4/ ðÞ sin4h/C0H2jjsin2hsin2//C0p 4/C18/C19/C27 ;(1)where H4?is the out-of-plane cubic anisotropy, H4jjis the in-plane cubic anisotropy, H2jjis the in-plane uniaxial anisot- ropy, and hand/are the angles describing the orientation of the equilibrium magnetization ( M) with respect to the ﬁlm normal and in-plane easy axes, respectively. The equilibrium orientation ( h;/) at each hHcan be obtained by minimizing Ein Eq. (1)13,18 2pf c/C18/C192 ¼1 M2sin2hd2E dh2d2E du2/C0d2E dhdu !22 43 5; (2) where cis the gyromagnetic ratio, from which the effective saturation magnetization can be extracted. Figures 3(d) and FIG. 3. (a) Derivative FMR absorption spectra of a 16 nm YIG ﬁlm taken at various angles ranging from in plane ( hH¼90/C14) to out of plane ( hH¼0/C14), as schematically shown in (b). (c) The FMR spectrum for the 16 nm YIG ﬁlm in an in-plane ﬁeld exhibiting a narrow linewidth of 4.3 G. The angular ( hH) dependence of the FMR resonance ﬁe ld for (d) a 16 nm and (e) a 164 nm YIG ﬁlm, from which the effective saturation magnetization (4 pMeff) is extracted. (f) The effective saturation magnetization (4 pMeff) for YIG ﬁlms 4–250 nm thick obtained from the angular and frequency dependencies of Hres,t h es a t u r a - tion magnetization (4 pMs) of the 164 nm and 250 nm YIG ﬁlms measured from in-plane hysteresis loops by VSM, and the saturation ﬁeld ( Hsat) of the 164 nm and 250 nm ﬁlms determined from the out-of-plane hysteresis loops by VSM. As a comparison, the room -temperature value of 4 pMsfor YIG bulk crystals is shown as a dashed line. All results s hown here are for room temperature.072401-3 Gallagher et al. Appl. Phys. Lett. 109, 072401 (2016) 3(e) show the angular dependencies of Hresfor the 16 and 164 nm YIG ﬁlms at room temperature, which give4pM eff¼2172613 and 2141 632 G, respectively. The sat- uration magnetization ( Ms) can be calculated by subtracting H2?from 4 pMeff. Previously, we studied the magnetocrys- talline anisotropy as a function of the tetragonal distortion ofYIG ﬁlms grown on Y 3Al5O12(YAG).13From this depen- dence and the tetragonal distortion of the YIG ﬁlms reportedhere—between 0.36% and 0.057%—the induced uniaxialanisotropy term would be rather small: 166 G for the 16 nm YIG ﬁlms and 38 G for the 164 nm ﬁlm. Thus, 4 pM effnearly equals 4 pMs, especially for the thicker ﬁlms. The values of 4pMeffobtained from the angular dependencies of Hresfor YIG ﬁlms from 4 to 250 nm thick are shown in Fig. 3(f), ranging between 2052 G (4 nm) and 2261 G (250 nm) atroom temperature. All of these values are signiﬁcantly higherthan the previously reported saturation magnetization of 1797 G for YIG single crystals, 19motivating further investi- gation to conﬁrm these results. A second method to determine the YIG saturation mag- netization is through the frequency dependence of Hres.W e measured FMR absorption of the YIG ﬁlms at frequencies ranging from 4 to 20 GHz using a microwave stripline. For each measurement, the magnetic ﬁeld was swept with themicrowave frequency ﬁxed. The FMR signal was detectedby applying a small modulation to the magnetic ﬁeld andmeasuring the differential reﬂected microwave power with aStanford Research 850 Lock-In ampliﬁer after passing thereﬂected microwave signal through a DC-blocked, zero- biased Schottky diode detector. 4 pM effcan be determined from the frequency dependence of Hresby ﬁtting to the Kittel formula,20,21x¼cﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HresðHresþ4pMeffÞp . Figures 4(a) and 4(b) show two representative fvs.Hresplots for the 16 and 164 nm YIG ﬁlms, from which we obtain 4 pMeff¼227368 and 2156 612 G, respectively. The values of 4 pMefffor YIG ﬁlms of 8–250 nm thickness, obtained from thefrequency dependence of Hres, are also shown in Fig. 3(f). All of these (2156 to 2347 G) are again well above thereported value of 1797 G for YIG single crystals, 19even after subtraction of the out-of-plane anisotropy ﬁeld. From the fre- quency dependence of the FMR linewidth, DH, we can deter- mine the Gilbert damping constant, a, of the YIG ﬁlms using,9,21DH¼DHinhþ4pafﬃﬃ 3p c, where DHinhis the inhomoge- neous broadening. Figures 4(c) and4(d) show the DHvs.f plots for the 16 and 164 nm YIG ﬁlms, yieldinga¼6.6/C210 /C04and 9.4 /C210/C04, respectively, which are repre- sentative for all of our YIG ﬁlms. The inhomogeneousbroadening of the YIG ﬁlms ranges from 1.6 to 4.4 G. These low values of aandDH inhare additional veriﬁcation of the quality of these YIG ﬁlms and well controlled oxidationstates since oxygen deﬁciency increases damping. 9,21 To further conﬁrm the surprisingly large magnetization of our YIG ﬁlms obtained from the angular and frequency depen- dencies of Hres, we measured the saturation magnetization (4pMs) of the 164 and 250 nm YIG ﬁlms using a LakeShore vibrating sample magnetometer (VSM) at room temperature.Given the large paramagnetic background of the GGG sub-strate, these two thick ﬁlms gi ve the highest accuracy in the magnetization measurements. Figure 5(a)shows the room tem- perature in-plane hysteresis loop for the 164 nm YIG ﬁlm after subtraction of the paramagnetic background, which gives4pM s¼2020650 G. Similarly, the 250 nm YIG ﬁlm was f o u n dt oh a v e4 pMs¼2085650 G. Both are considerably higher than the 1797 G reported for single crystal YIG at roomtemperature, conﬁrming a surprisingly large saturation magne- tization of our YIG ﬁlms. In addition, the YIG ﬁlms exhibit very small coercivity ( H c) with sharp magnetic reversal, such as in our earlier report of YIG ﬁlms with Hc¼0.35 Oe and a nearly ideal square hysteresis loop, further indicating the highmagnetic uniformity and low defect density of the YIG ﬁlms. 22 Figure 5(b)shows an out-of-plane magnetic hysteresis loop of the 164 nm YIG ﬁlm, where the saturation ﬁeld, Hsat¼2070 G, equals the effective saturation m agnetization, corroborating the values obtained from th e FMR measurements. We also measured the temperature dependence of satu- ration magnetization for the 164 and 250 nm ﬁlms in the VSM, as shown in Figs. 5(c)and5(d), which exhibit a Curie temperature of 530 and 520 K, respectively, slightly belowthe value of 559 K reported for single crystal YIG. 19In addi- tion, the angular dependence of the FMR was measured atlow temperatures down to 20 K and the results are shown in Figs. 5(c) and5(d). Despite the small differences between the saturation magnetizations obtained from the angulardependence of H res, in-plane VSM, and out-of-plane VSM measurements, all of the data show low temperature satura-tion magnetization around 3000 G. This is well above thevalue of 2470 G reported for bulk YIG at 4.2 K, 19and more surprisingly, clearly higher than the maximum theoretical value of 2459 G at 0 K for Y 3Fe5O12. Similarly, high magnetization has been previously reported for YIG ﬁlms grown by pulsed-laser deposition.11,23 For example, Kelly et al. obtained 4 pMs¼2100 G for 20 and 7 nm YIG ﬁlms grown on GGG,23which was attributed to an off-stoichiometry in their YIG ﬁlms. However, ourSTEM, EDX, and EELS results demonstrate that our YIGFIG. 4. Frequency vs. resonance ﬁeld plots of the (a) 16 nm and (b) 164 nm YIG ﬁlms give the effective magnetization by ﬁtting to the Kittel formula. Frequency dependencies of FMR linewidths of (c) a 16 nm and (d) 164 nmYIG ﬁlm, from which the damping constants a¼(6.660.4)/C210 /C04and (9.460.5)/C210/C04, respectively, are extracted.072401-4 Gallagher et al. Appl. Phys. Lett. 109, 072401 (2016) ﬁlms on GGG are stoichiometric without any detectable Gd or Ga in YIG and all Fe atoms are in the 3 þstate (no Fe2þ) within the resolution of the techniques. Thus, off- stoichiometry is not the cause of the high magnetization mea- sured by multiple techniques in our YIG ﬁlms over a broadrange of thicknesses, but the underlying mechanism for this interesting phenomenon remains a tantalizing puzzle deserv- ing further study. To elucidate whether the GGG substrateaffects the YIG magnetization, we grew a 160 nm YIG ﬁlm on the Y 3Al5O12(YAG) (001) substrate. The measured mag- netization of the YIG/YAG sample at various temperaturesusing VSM is shown in Fig. 5(c), which is clearly lower than that of the YIG ﬁlms on GGG. A possible explanation for thisdecrease is the lower quality of YIG on YAG due to the larger lattice mismatch (3%). However, the XRD scan in the inset toFig.1for this YIG/YAG sample demonstrates that the 160 nm YIG ﬁlm is still of high crystalline quality with clear Laueoscillations and fully relaxed with a lattice constant of12.386 A ˚. Thus, the surprisingly high magnetization observed in YIG/GGG could be due to some unexpected effect of theGGG substrate on the YIG ﬁlms. Future characterization ofthese YIG ﬁlms with high magnetization, such as element-speciﬁc magnetic moment measurements, and theoreticalinsights will be needed to reveal the underlying mechanismfor this surprising observation. Nevertheless, the ability totune the magnetization of a technologically important mag-netic material such as YIG via epitaxy can potentially openopportunities for microwave and spintronic applications. This work was primarily supported by National Science Foundation under Grant No. DMR-1507274 (sample growth andcharacterization, ISHE measurements and analysis). This workw a ss u p p o r t e di np a r tb yU . S .D e p a r t m e n to fE n e r g y( D O E ) ,Ofﬁce of Science, Basic Energy Sciences, under Award No. DE-FG02–03ER46054 (FMR measurements and modeling) and theCenter for Emergent Materials, an NSF-funded MRSEC, underGrant No. DMR-1420451 (STEM characterization). 1V. Cherepanov, I. Kolokolov, and V. Lvov, Phys. Rep. 229, 81 (1993). 2C. H. Du, H. L. Wang, F. Y. Yang, and P. C. Hammel, Phys. Rev. B 90, 140407(R) (2014). 3H. L. Wang, C. H. Du, P. C. Hammel, and F. Y. Yang, Appl. Phys. Lett. 104, 202405 (2014). 4C. H. Du, H. L. Wang, F. Y. Yang, and P. C. Hammel, Phys. Rev. Appl. 1, 044004 (2014). 5T. Kikkawa, K. Uchida, S. Daimon, Z. Y. Qiu, Y. Shiomi, and E. Saitoh,Phys. Rev. B 92, 064413 (2015). 6C. H. Du, H. L. Wang, Y. Pu, T. L. Meyer, P. M. Woodward, F. Y. Yang, and P. C. Hammel, Phys. Rev. Lett. 111, 247202 (2013). 7H. Y. Jin, S. R. Boona, Z. H. Yang, R. C. Myers, and J. P. Heremans, Phys. Rev. B 92, 054436 (2015). 8D. J. Sanders and D. Walton, Phys. Rev. B 15, 1489 (1977). 9H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014). 10C. Hauser, T. Richter, N. Homonnay, C. Eisenschmidt, M. Qaid, H. Deniz, D. Hesse, M. Sawicki, S. G. Ebbinghaus, and G. Schmidt, Sci. Rep. 6, 20827 (2016). 11S. A. Manuilov, S. I. Khartsev, and A. M. Grishin, J. Appl. Phys. 106, 123917 (2009). 12H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang,Phys. Rev. B 88, 100406(R) (2013). 13H. L. Wang, C. H. Du, P. C. Hammel, and F. Y. Yang, Phys. Rev. B 89, 134404 (2014). 14O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer,S. D. Bader, and A. Hoffmann, Phys. Rev. B 82, 214403 (2010). 15P. A. van Aken, B. Liebscher, and V. J. Styrsa, Phys. Chem. Miner. 25, 323 (1998). 16L. A. J. Garvie, A. J. Craven, and R. Brydson, Am. Miner. 79, 411 (1994). 17X. Liu, W. L. Lim, L. V. Titova, M. Dobrowolska, J. K. Furdyna, M. Kutrowski, and T. Wojtowicz, J. Appl. Phys. 98, 063904 (2005). 18H. Suhl, Phys. Rev. 97, 555 (1955). 19P. Hansen, P. Roschman, and W. Tolksdor, J. Appl. Phys. 45, 2728 (1974). 20C. Kittel, Phys. Rev. 73, 155 (1948). 21S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, and J. P. Nibarger, J. Appl. Phys. 99, 093909 (2006). 22C. H. Du, H. L. Wang, P. C. Hammel, and F. Y. Yang, J. Appl. Phys. 117, 172603 (2015). 23O. d’Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef, C. Hahn, A. H. Molpeceres, C. Carretero, E. Jacquet, C. Deranlot, P. Bortolotti, R. Lebourgeois, J. C. Mage, G. de Loubens, O. Klein, V. Cros, and A. Fert,Appl. Phys. Lett. 103, 082408 (2013).FIG. 5. (a) Room temperature in-plane hysteresis loop of a 164 nm YIG ﬁlm after subtraction of the GGG substrate background, which gives a saturation magnetization of 2020 G. Inset: raw M-H loop (b) Room temperature out- of-plane hysteresis loop of the 164 nm YIG ﬁlm after subtraction of GGG background. The arrows indicate the saturation ﬁeld Hsat¼2100 Oe which corresponds to the value of 4 pMeff. Temperature dependencies of the effec- tive saturation magnetization obtained from the angular dependence of the FMR resonance ﬁeld, the saturation magnetization measured by VSMin-plane hysteresis loops, and the saturation ﬁeld obtained from VSM out-of-plane hysteresis loops for (c) a 164 nm and (d) 250 nm YIG ﬁlm. The saturation magnetization of a 160 nm YIG ﬁlm grown on YAG measured by VSM in-plane hysteresis loops is also shown in (c).072401-5 Gallagher et al. Appl. Phys. Lett. 109, 072401 (2016)
1.4978435.pdf	Orientation and characterization of immobilized antibodies for improved immunoassays (Review) Nicholas G. Welch Judith A. Scoble and Benjamin W. Muir Paul J. Pigram Citation: Biointerphases 12, 02D301 (2017); doi: 10.1116/1.4978435 View online: http://dx.doi.org/10.1116/1.4978435 View Table of Contents: http://avs.scitation.org/toc/bip/12/2 Published by the American Vacuum SocietyOrientation and characterization of immobilized antibodies for improved immunoassays (Review) Nicholas G. Welch Centre for Materials and Surface Science and Department of Chemistry and Physics, School of Molecular Sciences, La Trobe University, Melbourne, VIC 3086, Australia and CSIRO Manufacturing, Clayton, VIC 3168, Australia Judith A. Scoble and Benjamin W. Muir CSIRO Manufacturing, Clayton, VIC 3168, Australia Paul J. Pigrama) Centre for Materials and Surface Science and Department of Chemistry and Physics, School of Molecular Sciences, La Trobe University, Melbourne, VIC 3086, Australia (Received 19 January 2017; accepted 28 February 2017; published 16 March 2017) Orientation of surface immobilized capture proteins, such as antibodies, plays a critical role in the performance of immunoassays. The sensitivity of immunodiagnostic procedures is dependent onpresentation of the antibody, with optimum performance requiring the antigen binding sites be directed toward the solution phase. This review describes the most recent methods for oriented antibody immobilization and the characterization techniques employed for investigation of theantibody state. The introduction describes the importance of oriented antibodies for maximizing biosensor capabilities. Methods for improving antibody binding are discussed, including surface modiﬁcation and design (with sections on surface treatments, three-dimensional substrates, self-assembled monolayers, and molecular imprinting), covalent attachment (including targeting amine, carboxyl, thiol and carbohydrates, as well as “click” chemistries), and (bio)afﬁnity techniques (with sections on material binding peptides, biotin-streptavidin interaction, DNA directed immobi-lization, Protein A and G, Fc binding peptides, aptamers, and metal afﬁnity). Characterization tech- niques for investigating antibody orientation are discussed, including x-ray photoelectron spectroscopy, spectroscopic ellipsometry, dual polarization interferometry, neutron reﬂectometry,atomic force microscopy, and time-of-ﬂight secondary-ion mass spectrometry. Future perspectives and recommendations are offered in conclusion. VC2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license(http://creativecommons.org/licenses/by/4.0/ ).[http://dx.doi.org/10.1116/1.4978435 ] I. INTRODUCTION Immunodiagnostics, protein biochips, and biosensors employed for antigen detection and quantiﬁcation from bio-logical samples often employ recognition proteins such as antibodies. 1–4Assay sensitivity is dependent on immobiliza- tion of the capture antibodies onto a solid support with a suf-ﬁcient surface density, a conformation that is representative of their native, solution-phase state, and an orientation that maximizes their antigen capture potential. Antibodies are biopolymers of approximately 150 kDa molecular mass and with dimensions of approximately 14/C210/C24 nm (Refs. 3and5) comprised of amino acids whose sequence and composition, like other proteins, deﬁne the three-dimensional structure. 6An antibody is comprised of two fragment antigen binding (Fab) regions and a fragmentcrystallizable region (Fc). The Fab regions, joined by the hinge region, is known as F(ab 0)2fragment. The Fab regions are dis- similar in their composition, isoelectric point, and physicalstructure to the Fc region of the antibody (Fc), allowing deter- mination of antibody orienta tion on the surface. Ideally, the immobilized antibody is oriente d such that the Fc is substratefacing, sometimes referred to as “end-on”; however, randomly immobilized antibodies may assume various surface orienta- tions, including those loosely referred to as “head-on,” “side-on,” and “lying-on” (see Fig. 1). 7 Generally, hydrophobic amino acids are internalized in a correctly folded protein structure, leaving hydrophilic resi-dues at the antibody surface, with chemically reactive func-tionalities, including amine, carboxyl, and hydroxyl groups.Disulﬁdes, such as those that contribute to the hinge region,can be reduced to make thiol species that can also be conju- gated. Further, sites for targeted immobilization (including natural or non-natural amino acids) can be introducedrecombinantly to antibodies. 8–10 Physical adsorption of antibodies onto traditional immu- noassay solid supports, such as polystyrene, occurs viahydrophobic and electrostatic interactions. 11While this method offers the simplest attachment pathway, it is uncon-trollable, and antibodies can be immobilized in a randomly oriented manner, denatured, or displaced in later steps by washing. 12–14The substrate design has increased the capabil- ities of immunoassays by improving antibody binding capac-ity and reducing denaturation at the surface. 15Covalent attachment of the antibody, via its functional groups, tochemically engineered substrates has resulted in further a)Electronic mail: p.pigram@latrobe.edu.au 02D301-1 Biointer phases 12(2), June 2017 1934-8630/2017/12(2)/02D301/13 VCAuthor(s) 2017. 02D301-1 improvements in antibody density, though often these meth- ods are not site-directed and unfavorable random orientation can occur.16In an ideal scenario, antibodies should be immobilized in their native form, without the need for intro-duced functional groups, in a homogeneous arrangement such that their antigen binding sites are free from steric hin- drance and are oriented so as to maximize complementarybinding. A method that truly provides the ideal scenario has yet to be realized; however, recent developments, as dis- cussed in this review, offer improved control over antibodyimmobilization and orientation at the interface. In parallel with the development and evolution of immo- bilization methods, a signiﬁcant need has emerged for sur-face characterization techniques that can accurately identify antibody orientation at a substrate. Current techniques that rely on indirect analysis, or inference from complex models,make deﬁnitive conclusions regarding orientation difﬁcult. 17 Advances in data processing and multivariate analysis haveprovided an improved level of understanding of complexsurfaces, and direct surface analysis techniques, such as time-of-ﬂight secondary ion mass spectrometry (ToF-SIMS), provide molecular information that has the potential to deter-mine antibody orientation with conﬁdence. This review is structured into two parts. First, a review of current antibody immobilization strategies, including surfacemodiﬁcation, antibody targeting, and coupling, will be pre- sented. Second, advances in characterization techniques for investigating these systems will be explored, including indi-rect and direct analyses. The advantages and shortfalls of strategies and techniques will be addressed, and the review will conclude with future perspectives and recommendations.II. IMPROVING ANTIBODY BINDING AT THE INTERFACE A. Surface modification and design strategies Physical adsorption is the simplest method for the immo- bilization of antibodies to immunoassay solid supports,such as microtiter plates. However, this method does notallow control of the antibody orientation and is typically associated with poor binding and denaturation. 18Microtiter plate manufacturers utilize polymers such as polystyrene,polypropylene, polyethylene, and cyclic oleﬁn copolymerblends, employing surface modiﬁcation methods to increasethe hydrophilicity of substrates. The increase in hydrophilic- ity can increase antibody binding (density) and decrease the amount of denatured protein. Polymer substrates are the tra-ditional immobilization platform for immunoassay as theyprovide a cheap, stable, reproducible substrate that is easy tomanufacture with precision. 1. Plasma treatment and plasma polymers Plasma treatment is a surface modiﬁcation method that uses radio-frequency glow-discharge to generate a plasma ofa gas or monomer vapor. Microtiter plates have been treated with oxygen, nitrogen, and other gas plasmas to create chem- ical functionalities at the surface, thereby reducing thehydrophobicity of the plastic. 19Similarly, nonreactive gases such as argon can be used to “activate” the surface by intro-ducing radicals, which react with atmospheric species upon exposure to air. 20Plasma treatment offers a method to func- tionalize existing substrates’ increasing hydrophilicity21 and reducing denaturation of bound proteins or to provide astarting point for further chemical treatment and covalentgrafting of proteins. 22In a recent example, P ^aslaru et al.23 plasma treated poly(vinylidene ﬂuoride) (PVDF) with CO 2, N2and N 2/H2(25/75) gases to attach carboxyl or amine functionality for subsequent covalent immobilization ofproteins. N-ethyl- N-(3-dimethylaminopropyl) carbodiimide (EDC)/ N-hydroxysuccinimide (NHS) chemistry was used to attach IgG or Protein A (with subsequent IgG binding) to the PVDF treated surfaces. Possible preferential end-on orienta- tion of IgG was achieved via the PVDF surfaces treated withN 2/H2and grafted with the Protein A. Plasmas can also be used to produce polymer thin ﬁlms that retain some of the chemistry associated with the mono- mer. The radicalized monomer fragments bind to the substratesurface, and to one another, creating a ubiquitous surfacecoating referred to as a plasma polymer. This methodology ofcreating polymers allows adherent and continuous coatings tobe formed on a broad range of substrates, including microtiter plates. 24Plasma polymers have been produced from a diverse range of monomers, including allylamine,25,26cyclopropyl- amine,27bromine,28polyethylene glycol (PEG), diethylene glycol dimethyl ether (diglyme),29–31a n dm a n yo t h e r s ,32,33 providing a broad spectrum of chemical functionalities for subsequent protein grafting steps, including the option for pat- terning.34–36Overall, a signiﬁcant range of polymers have been used to improve biomolecule immobilization properties FIG. 1. Antibody orientation, dimensions, and important chemical species for targeting.02D301-2 Welch et al. : Orientation and characterization of immobilized antibodies 02D301-2 Biointer phases , Vol. 12, No. 2, June 2017of substrates for use in microarray and protein assay applica- tions.33,37Hasan and Pandey38provide an excellent review of polymers and plasma techniques for producing materials forprotein immobilization. 2. Three-dimensional substrates Three-dimensional substrates are attractive as they offer increased surface area for antibody binding and can mini-mize steric hindrance that may prevent antigen capture. Porous three-dimensional substrates have been prepared from a variety of materials, including various polymers, sili-con, glass slides, metals, and gels. Wang and Feng 15provide an excellent review on three-dimensional substrates with a focus on the orientation of proteins. Similarly, gels and sol–gels manufactured from agarose or dextran have found improved antibody binding due to their high surface area.1,39In a recent study, Orlov et al.40 employed three-dimensional immunochromatographic nitro- cellulose membranes impregnated with magnetic nanopar-ticles for use in a strip sensor immunoassay, providing a solid phase with a large surface area for antibody immobili- zation. The sensor had a limit of detection of approximately740 fM and had a strong correlation with a standard enzyme-linked immunosorbent assay (ELISA) for the detection of prostate speciﬁc antigen from serum, though with improved dynamic range (3.5 orders). One example from Feng et al. 41 utilized repeat units of Protein A genetically fused to the nickel chelating His-tag. Adjacently immobilized (via a nickel matrix substrate) the fused proteins order as columnsprotruding from the substrate and could immobilize ﬁve anti-bodies via their Fc regions. This three-dimensional protein construct had a 64-fold increase in antigen detection sensi- tivity relative to standard IgG immobilization. 3. Self-assembled monolayers The formation of a self-assembled monolayer (SAM) pro- vides another means of modifying surface chemistry to pro-mote antibody adsorption or to produce functional groupsfor subsequent covalent attachment. 42SAMs are typically formed from molecules that contain active functional head- groups at either end of a hydrocarbon chain and a linear car-bon chain which promote self-assembly when they attach toa surface. The anchoring head-group has an afﬁnity for the surface, while the other provides a solution-facing chemistry for protein adsorption or attachment. The central hydrocar-bon chains provide stabilization of the SAM by interchain hydrophobic interactions. 38 The gold–thiol interaction has been exploited most commonly by utilizing alkanethiols to provide a linkerthat can bind gold substrates via the thiol group, and offer customizable chemistry for adsorption or coupling anti- bodies to SAMs. 43,44Lebec et al.45used alkanethiol SAMs formed from 11-mercaptoundecanoic acid (11-MUA) and1-undecanethiol on gold substrates to produce COOH and CH 3surface chemistries, respectively. Antibodies were adsorbed to both substrate types with an increase shown forthe CH 3surface. However, this antibody was found to have no antigen recognition indicating denaturation or poor ori- entation, with the latter supported by ToF-SIMS ﬁndings. Chen et al.44demonstrated preferred orientation of mouse IgG1 (and to a lesser extent IgG2a) by exploiting the anti- body dipole and the use of charged SAM surfaces. IgG1 adsorbed to a NH 2terminated SAM produced from 11- amino-1-undecanethiol had a higher antigen/antibody ratio than a COOH terminated SAM produced from 16- mercaptohexadecanoic acid. Vashist et al.46provide a good review of antibody immobilization using silane SAMs on various substrates for improved surface densities. 4. Molecularly imprinted polymers Molecular imprinting is a polymerization technique that uses a molecular template to produce target-speciﬁc binding regions.47Once formed, the target-speciﬁc binding sites in the polymer substrate may selectively immobilize the target, such as an antibody, from a complex matrix. Bereli et al.48 imprinted a poly(hydroxyethyl methacrylate) cryogel with the Fc portion of anti-human-IgG to create an antibody ori- enting substrate. The Fc portion was then ﬂushed from the cryogel, which was subsequently activated with carbodii-mide for whole antibody binding. Anti-human-IgG was then used as a capture antibody to bind IgG from human plasma and the imprinted cryogel was demonstrated to be at least three times better than using a nonimprinted cryogel. This was despite similar amounts of the capture anti-human- IgG being immobilized, via the nonspeciﬁc carbodiimide method, indicating preferential antibody orientation. In addition to providing orientating substrates, molecularly imprinted polymers utilize gel structures with aqueous envi- ronments that are thought to reduce the probability of protein denaturation. 49 B. Covalent binding targets While substrate design is important, the ability to reliably attach the antibody to a solid support underpins the success of an immunoassay. The covalent coupling of capture anti- bodies ensures robust immobilization and can improve den- sity and orientation outcomes at the substrate. However, for oriented immobilization, site-directed attachment to the anti- body is required. The covalent attachment can proceed via various chemistries dependent on the substrate functionality, target group on the antibody, and the physical restraints of the system, i.e., pH, temperature, and degree of conjuga- tion.33This section will discuss functional groups on anti- bodies for covalent attachment, including common targets such as amine, carboxyl, thiol, and carbohydrate moieties (see Fig. 2for overview). The covalent attachment methods of proteins are covered in good detail in reviews by Rao et al.50and Liu and Yu.16 1. Amine and carboxyl groups Amine and carboxyl groups are ubiquitous throughout an antibody’s structure and are common at the antibody’s02D301-3 Welch et al. : Orientation and characterization of immobilized antibodies 02D301-3 Biointer phases , Vol. 12, No. 2, June 2017surface due to their polar nature. Amino acids such as lysine with reactive primary amine side chains, and aspartate and glutamate with carboxyl side chains, can be targeted forcovalent attachment. Due to their prevalence throughout theantibody surface, site-directed covalent antibody immobili-zation targeting these groups is difﬁcult. Amine and carboxylcoupling (between the substrate and protein) is commonly attained with carbodiimide chemistry that utilizes EDC com- bined with succinimidyl esters (such as NHS). 33This method is known as EDC/NHS coupling and results in robust amidebond formation. EDC/NHS chemistry has been employed asa covalent attachment methodology for immobilization of proteins and also as a method for the preparation of sub- strates. Carrigan et al. 51used EDC/NHS in two ways, (1) for the cross-linking of polyethylenimine and carboxymethylcellulose, and (2) for activation of this substrate for proteinbinding targeting amine and carboxyl functionality. Sun et al. 52took a novel approach to EDC/NHS coupling by introducing NHS reactive groups to random sites on an anti-body then using an electric ﬁeld to preferentially orient theantibody, via its intrinsic dipole, before reaction to the freeamines on cysteine immobilized to a gold electrode. 2. Thiol groups Disulﬁde-bridged cysteines, such as those present in the hinge region of antibodies, can also be targeted by reducingagents such as tris(2-carboxyethyl)phosphine (TCEP) or 2-mercaptoethylamine (2-MEA) to form reactive thiols, 8,53 which may subsequently react with maleimide or iodoacetyl activated surfaces. However, as the cysteines are internal to the antibody tertiary structure, covalent attachment via thismethod can disrupt the conformation of the antibody 54whilesteric hindrance may limit antigen binding. Exploitation of the gold–thiol bond makes this immobilization strategy useful for gold substrates and nanoparticles in techniques such assurface plasmon resonance (SPR). 55UV-excitation has also been used to initiate photoreduction of disulﬁde bridges in hinge regions of antibodies according to an approach known as the photonic immobilization technique.56Employing UV pulses at 258 nm and 10 kHz, free thiols can be produced thatare then able to bind gold substrates for quartz crystal micro- balance (QCM) measurements. 57Antibody fragments, such as Fab0, can also be produced with reactive thiols and used in immunosensors;5,42however, this review aims to focus pri- marily on whole antibody immobilization and will not coverantibody fragments speciﬁcally. Alternatively, primary amine groups can be converted to thiol functionality using, for example, 2-iminothiolane (Traut’s reagent) for subsequent immobilization using the same maleimide or iodoacetyl chemistry. 58For example, lipid PEG, functionalized with maleimide, was incorporated into lipid nanocapsules for coupling thiolated antibodies or Fab0.59 3. “Click” chemistry More recently, “click chemistry” exploiting 1,3-dipolar cycloaddition between an azide and an alkyne has demon- strated utility for the conjugation of single-domain camelidantibodies, known as VHH, to a dextran substrate. 9This technique produced an oriented system via the site-speciﬁc insertion of azidohomoalanine into the VHHs. 4. Carbohydrate groups Antibodies are glycosylated at the Fc region and can pro- vide a target for site-directed immobilization.60,61Periodate FIG. 2. Oriented antibody immobilization strategies. (a) EDC/NHS coupling of antibody amine and carboxyl groups to surface carboxyl and amine groups. ( b) Reduction of antibody disulﬁdes, with TCEP or 2-MEA, to reactive thiols for binding gold substrates. (c) Periodate oxidation of carbohydrates in the Fc region of antibodies followed by coupling to hydrazide surface chemistry.02D301-4 Welch et al. : Orientation and characterization of immobilized antibodies 02D301-4 Biointer phases , Vol. 12, No. 2, June 2017oxidation can be used to oxidize diols in carbohydrates into aldehyde groups that can react with amines and hydra- zides.62Diols can also be targeted by boronic acids to form boronate ester intermediates reversibly.63Song et al.64dem- onstrated end-on immobilization of antibodies using 3- aminophenylboronic acid to ﬁrst couple a NHS-derivatizedsurface and second bind the Fc carbohydrate. Due to the reversible nature of boronate esters, Adak et al. 61employed a single molecule with two functional groups to ﬁrst form aboronate ester with the carbohydrate, and second to cova- lently attach the antibody via photoinitiated cross-linking. UV exposure causes the (triﬂuoromethyl)phenyldiazirine functional group to form reactive carbenes that irreversibly bind the antibody. Alternatively, Huang et al . 65oxidized the carbohydrate of anti-alpha-fetoprotein to an aldehyde and covalently linked the protein to a 3-aminopropyltri ethoxysilane (APTES) modiﬁed silicon substrate. Theamount of antibody immobilized increased by 32% and the antigen binding amount by 16% relative to physical adsorp- tion. Yuan et al. 66used periodate oxidation of carbohydrates on the anti-CD34 antibody to promote oriented immobiliza- tion. First, stainless steel substrates were coated with ethyl- ene vinyl acetate, then treated with O 2plasma, and silanated with APTES to create amine groups (labeled SCA-SS). Amines were then coupled with the oxidized carbohydrates and successful binding was assessed via cell uptake by the anti-CD34 antibody. Prieto-Sim /C19onet al.67used thiolated hydrazide SAM linkers or electrografting of diazonium saltsto immobilize periodate-oxidized carbohydrates of anti- bodies via hydrazide chemistry onto functionalized gold substrates. C. Affinity immobilization techniques While covalent methods provide robust antibody immobi- lization and can achieve site-directed immobilization,68 they may be unsuited due to the high prevalence of a particu- lar functional group in the antibody (amine and carboxyl), or may cause conformation changes upon attachment(thiol), without the use protein engineering. Afﬁnity immobi- lization techniques provide alternative and potentially favorable strategies to promote site-directed antibodyimmobilization. 15 1. Material binding peptides Peptide sequences that will preferentially immobilize to substrates, metal ions, and other biomolecules have been developed for enhanced protein orientation.69These peptides can be incorporated as tags, into proteins of interest, via chemical conjugation or genetic fusion. Phage screening techniques have been developed to produce peptides withspeciﬁcity to a broad range of materials and proteins. 70A large number of material binding peptides exist, including those speciﬁc to polystyrene [PS-tag,71Lig1 (Refs. 72and 73)] and hydrophilic polystyrene (Phi-PS) [PS19-1 and PS19-6 (Ref. 74)], silicon (Si-tag75,76), glass slides or silica resin [R9 (Ref. 77)], poly(methylmethacrylate) (PMMA)(c02,78PM-OMP25,79PMMA-tag80), polycarbonate [PC- OMP6 (Ref. 79)], poly- L-lactide [c22 (Ref. 81)], gold [GBP (Ref. 82)], and the well known nickel and copper speciﬁc His-tag.83,84 2. Biotin–streptavidin interaction Oriented immobilization can be realized by exploiting the biotin–avidin/streptavidin interaction.85Antibodies can be easily conjugated to biotin using biotin-NHS chemistry that targets amines; however, this results in randomly biotiny-lated antibodies. Paek’s group 86compared randomly biotiny- lated IgG using biotin-NHS, with IgG biotinylated at the hinge disulﬁdes via competitive maleimide chemistry for immobilization to streptavidin treated microwells and glassslides. The authors found a two-fold improvement in antigen detection for the hinge disulﬁde biotinylated IgG, relative to the random system. The group also demonstrated a twofoldimprovement for the same system by employing a gold sub- strate with a thiol SAM and biotin–streptavidin linker. 87 3. DNA directed immobilization DNA-directed immobilization (DDI) of proteins is another afﬁnity method that can produce oriented systems. This method requires the binding of short nucleotide sequen-ces to the substrate and to the protein of interest, allowing direct binding between the two. However, to truly achieve an oriented system, care must be taken to ensure that thecovalent attachment of the DNA to the antibody occurs in a site-direct manner. Wacker et al . 88investigated antibody immobilization and ﬂuorescent immunoassay performanceof antibodies bound via DDI, physical adsorption, and by streptavidin–biotin interactions. IgG immobilized with DDI was found to require 100-fold less antibody for the sameﬂuorescence detection of the analyte; however, orientationof the antibody was not independently determined, and site- directed biotinylation of the IgG was not stated. More recently, Seymour et al. 89compared anti-ebola virus glyco- protein (EBOV GP) antibody immobilized directly to NHS containing copolymer or via DDI. The DDI immobilized antibody was found to be an order of magnitude more sensi-tive to the EBOV GP antigen. Glavan et al. 90synthesized single-stranded DNA onto paper substrates and investigated antihuman C-reactive protein (hCRP) immobilized via DDIfor binding hCRP from serum in a sandwich ELISA. However, DNA conjugation to the antibody utilized non- site-directed NHS chemistry. Boozer et al. 91prepared mixed SAMs containing ssDNA thiols and oligo(ethylene glycol) thiols on gold SPR chips. The complementary DNA strand was cross-linked usingNHS chemistry to the antibodies and immobilized by the SAM. This system generated a 50-fold improvement on their previous work 92using biotinylated immobilization. However, by binding DNA to the antibody nonspeciﬁcallywith NHS chemistry, it is more likely that the improvement arises due to the antibody being separated from the surface, than through orientation.02D301-5 Welch et al. : Orientation and characterization of immobilized antibodies 02D301-5 Biointer phases , Vol. 12, No. 2, June 20174. Protein A and Protein G Protein A and Protein G are small proteins, derived from bacteria, which can speciﬁcally bind the Fc portion of anti-bodies allowing oriented systems to be obtained. 93–96The IgG binding domain of Protein A, known as the Z-domain or ZZ-domain, is also used as a smaller synthetic option for Fcbinding. 97This technique offers a method for truly obtaining oriented antibodies as binding can only occur via the Fc por- tion. Due to its effectiveness, Protein A, or its derivatives,has been exploited with many surface immobilization strate- gies including biotin-streptavidin, 98SAMs,99,100EDC/NHS chemistry,100,101glutaraldehyde,102tyrosinase chemistry,102 non-natural amino acid insertion,10gold binding peptide82or polystyrene afﬁnity ligand fusion,73and additional protein linkers.103The ZZ-domain has also been coupled to carbohy- drate binding modules for selective immobilization to cellu-lose coated slides 104and to paper105substrates, and also the metal afﬁnity His-tag.106,107It is important to note that sev- eral issues may arise with such systems: (1) the Protein Acapture of the Fc is reversible, (2) Protein A has been reported to bind Fab regions and albumin (although to a much lesser extent), and (3) it is required that the Fc bindingsite of the Protein A is correctly oriented at the substrate to permit antibody binding. 15,16More recently, Yang et al.108 engineered a photoactivatable Z-domain variant that incor- porated the UV-active amino acid benzoylphenylalanine and a biotin molecule. The Z-domain preferentially bound the Fc portion of IgG that was irreversibly attached using UV-activation of the benzoylphenylalanine. When immobilized to a streptavidin coated substrate, this system had a ﬁvefold lower antigen detection limit than randomly NHS-biotinylated IgG. 5. Fc-binding peptides and aptamers Short peptides with speciﬁcity to the Fc domain of anti- bodies have been used to promote oriented immobilization.Jung et al . 109used an Fc-binding peptide to immobilize human, rabbit, goat, and mouse antibodies, with strong selectivity for human IgG1 and IgG2. Anti-C-reactive pro-tein (anti-CRP) antibody immobilized with the Fc-binding peptide was compared with randomly immobilized (EDC/ NHS coupling) antibody immobilized to SPR chip substratesand found that a 1.6-fold increase in the CRP/anti-CRP ratio when employing the Fc-binding peptide. More recently, Tsai et al. 110demonstrated that molecular dynamics can be used to design a short peptide (RRGW) with high speciﬁcity to the mouse IgG2a antihuman prostate speciﬁc antigen (PSA) antibody. PSA binding was monitored via SPR demonstrat-ing good antibody orientation. Yoo and Choi 111used a phage biopanning to screen for peptides speciﬁc to the Fc portion of rabbit anti-goat IgG. The peptide (KHRFNKD) immobilized with biotin to an avidin-QCM surface showed improved IgG binding relativeto physical adsorption. Dostalova et al. 112used the Fc bind- ing peptide HWRGWVC to immobilize antiprostate speciﬁc membrane antigen antibodies to gold coated doxorubicinnanocarriers with solution-phase orientation. The inclusion of the peptide gave a 1.4-fold improvement in signal during the immunoassay. Lee et al.113developed photoactivatable Fc-speciﬁc antibody binding proteins (FcBPs) expressed in Escherichia coli that undergo photo-crosslinking (via photo- methionine) with antibodies upon UV irradiation. FcBPswere immobilized on maleimide-coated slides and the epi-dermal growth factor receptor (EGFR)-hmAb antibody cross-linked with UV exposure. Dose-dependent antigen (EGFR) binding was observed at above 110 fmol. Aptamers are single chain DNA, or RNA, oligonucleoti- des that fold to form complex three dimensional structures and can speciﬁcally immobilize proteins of interest. 114 Miyakawa et al.115developed an RNA aptamer that selec- tively binds the Fc portion of human IgG1 through IgG4, but not other nonhuman IgGs. SPR was used to assess the bind-ing site of the aptamer on IgG, and it was found that the site was similarly positioned to that of the Protein A binding site, making it suitable for promoting antibody orientation. Maet al. 116produced a DNA aptamer capable of binding the Fc domain of multiple mouse subclasses. This area has potential to develop aptamers capable of universal Fc binding sub-strates, hence promoting antibody orientation and immuno- diagnostic sensitivity. 6. Nucleotide binding site Antibodies, even across different isotypes, contain largely conserved sequences between the heavy and light chains of the Fab region known as a nucleotide binding site (NBS).Targeting the NBS using a small molecule, indole-3-butyricacid, and UV irradiation, Alves et al. 117were able to immo- bilize antibodies selectively, offering a 7.9-fold increase in antigen sensitivity, compared with physical adsorption. Thegroup also demonstrated the utility of this technique with Fab fragments. 118,119 7. Metal affinity Perhaps the simplest option is to take advantage of the endogenous metal binding properties of antibodies. In addi- tion to recombinant peptide tags for metal coordination,native tag-free IgG have been puriﬁed using metal afﬁnity. The interaction of histidine and cysteine with metals, particu- larly copper and zinc, has been exploited for protein fraction-ation and IgG puriﬁcation using immobilized metal-afﬁnity chromatography. 120,121Hale122then went on to demonstrate that Co(II) loaded resin could be used to irreversibly bindIgG in an oriented manner via the Fc region. Todorova- Balvay et al. 123used computational modeling and immobi- lized metal-ion afﬁnity chromatography to investigate thetransition metals copper (II), nickel (II), zinc (II) and cobalt (II), to determine a native metal-binding target in the Fc por- tion of whole human IgG1. The histidine cluster His433–X–His 435 was found to be surface accessible to afﬁnitybinding using these metals without the need for recombinant tags. Muir et al. 124prepared metal coordinating polymer sub- strates and screened a large range of transition metals to02D301-6 Welch et al. : Orientation and characterization of immobilized antibodies 02D301-6 Biointer phases , Vol. 12, No. 2, June 2017assess their antibody binding capabilities. This work covered a library of 1600 different metal immobilized surface chemis-tries and identiﬁed chromium perchlorate with ethylenedi-amine as the “lead” combination. Immobilization ofantitumor necrosis factor alpha (anti-TNF a) to Luminex beads with chromium perchlorate and ethylenediamine wascompared with carbodiimide coupling chemistry. Thechromium-mediated immobilization has an approximatelyninefold improvement in TNF aantigen sensitivity relative to the carbodiimide method. Pingarr /C19on’s group used a metallic- complex chelating polymer (Mix&Go TM) to achieve oriented immobilization of native anti-adiponectin antibodyon carboxyphenyl multiwalled carbon nanotubes 125and graphene oxide-carboxymethyl cellulose hybrid.126Recently, Welch et al .127employed the chromium complex, [Cr(OH) 6]3/C0, buffered with ethylenediamine as a wet chemi- cal modiﬁcation to ten commercial microtiter plates, postpro-duction, to improve antibody immobilization and ELISAperformance. For an anti-EGFR, x-ray photoelectron spec- troscopy (XPS) analysis indicated that the chromium modi- ﬁed microtiter plate bound twice the amount of antibodyrelative to the unmodiﬁed plate, and the ELISA signal morethan tripled indicating improved antibody orientation. Thechromium modiﬁcation was demonstrated for use with ﬁveother antigen capture ELISAs. Welch et al. 128also demon- strated optimization of the chromium complex by varying themetal salt and buffering base compounds and ratios. The opti-mized complex (1:1 chromium perchlorate hexahydrate toethylenediamine) was used to improve the antigen detectionlimits of a bovine tumor necrosis factor alpha (TNF a) ELISA by an order of magnitude relative to untreated plates. In a recent study, Welch et al. 29employed a traditionally low fouling diethylene glycol dimethyl ether plasma polymer(DGpp) as a substrate for binding [Cr(OH) 6]3/C0with subse- quent antibody immobilization. When equivalent amounts ofantibody were immobilized on the DGpp and the chromiumfunctionalized DGpp substrates, a tenfold improvement in ELISA signal intensity was observed for the chromium func- tionalized system indicating an oriented system. ToF-SIMSanalysis identiﬁed that chromium may be binding the anti-body through lysine, methionine, arginine, and histidineresidues. III. IMMOBILIZED ANTIBODY CHARACTERIZATION OVERVIEW An immunoassay is the most widely employed method to assess the state of an immobilized antibody, in that it repre-sents the practical application of successful immobilization. ELISAs require that immobilized antibodies maintain the correct orientation, unperturbed conformation, and adequatedensity at the substrate surface to maximize signal produc-tion and thus antigen quantiﬁcation. However, as this is anindirect analysis technique, it only allows the antibody state and orientation to be determined by inference. As a result, complementary and independent methods for assessing theantibody have been developed. While there are a number oftechniques for quantifying the density of adsorbed pro-teins, 129the number of techniques that can probe antibody orientation is much smaller. Table Ipresents an overview of current characterization techniques, and schematic represen-tations are shown in Fig. 3. A. X-Ray photoelectron spectroscopy XPS is a spectroscopic technique that uses incident x-ray photons to probe the elemental and chemical composition of the top 10 nm of the sample surface. This technique can be TABLE I. Overview of surface analysis techniques employed for investigating antibody. Technique Input Output Information Comments References A) XPS Monochromatic x-raysPhotoelectrons Elemental and chemical Quantiﬁcation of anti- body surface density23,29,127,128, and131 B) SE Elliptically polar- ized lightChange in light phase or intensityThickness, refractive index, surface roughnessModel based analysis, inferred state of antibody132–137 C) Dual polarization interferometry (DPI)Laser light Evanescent wave changeMass, ﬁlm thickness, refractive index, densityInferred state of antibody based on mass and ﬁlm thickness64,138, and 139 D) SPR Monochromatic multiangle laser lightChange in reﬂected and absorbed lightRefractive index, ﬁlm thicknessInferred state based on antibody and antigen adsorption characteristics55and140–142 E) NR Neutron beam Change in reﬂec- tion of neutron beamRefractive index, ﬁlm thickness, surface roughnessModel based analysis, inferred state of antibody143–146 F) AFM Feedback driven cantilevered tipz-height in 2D and tip/surface forceSurface roughness, phase information, imagingCorrectly oriented anti- bodies have 14 nm height56,142, and 147–149 G) QCM Resonance fre- quency of microbalanceChange in fre- quency and amplitudeMass of adsorption, bioafﬁnityInferred state of antibody based on adsorption and mass150and151 H) ToF-SIMS Ionized metal clusters, “primary- ions”Ionized sample fragments, “secondary-ions”Semiquantitative elemen- tal, chemical, and molecularAmino acid composition of F(ab0)2and Fc varies and can be distinguished45,130, and 152–15702D301-7 Welch et al. : Orientation and characterization of immobilized antibodies 02D301-7 Biointer phases , Vol. 12, No. 2, June 2017used to quantify the amount of nitrogen present, which in turn can be correlated to the amount of immobilized anti-body. 29,127,128,130Antibody orientation can then be inferred based on the immunoassay signal. Radadia et al.131com- pared XPS and ELISA results to infer stability of antibodyimmobilized to glass and diamond ﬁlms. P ^aslaru et al. 23pre- pared plasma treated PVDF membranes for adsorption orgrafting of Protein A (followed by IgG) or IgG alone andused XPS to investigate sulfur and nitrogen content in thesamples. The nitrogen concentration was used to quantifyoverall protein content, and sulfur concentration (present due to disulﬁde bonds in the IgG) was used to monitor IgG binding. B. Spectroscopic ellipsometry Spectroscopic ellipsometry (SE) is a surface sensitive optical technique that monitors the polarization change in light reﬂected from the sample surface (typically a metal or ceramic).132,133Polarization changes occur due to varia- tions in the dielectric or refr active index properties of the sample. Balevicius et al.134used total internal reﬂection ellipsometry to demonstrate t hat antibodies reduced with 2-MEA, immobilized to a gold substrate, bind 2.5 times the amount of antigen as compared with their intact whole anti- body counterparts immobilize d randomly and covalently to SAMs. Bae et al .135compared IgG immobilized to thio- lated Protein G (to represent an oriented antibody) with chemically bound IgG to an 11-MUA SAM, to represent randomly oriented binding. SE was used to estimate the IgG ﬁlm thickness and together with atomic force micros- copy (AFM) and SPR inference could be made regarding different orientations of the antibodies. Wang and Jin136 ﬁrst utilized Protein A adsorbed to silicon to immobilize anti-IgG and compared with anti-IgG adsorbed to the sili- con. In a kinetic manner, SE was used to investigate the ﬁlm thickness and found an increase in the anti-IgG bound by Protein A, inferring a prefe rentially oriented system. Wang then went on to investigate three different silane modiﬁcations to silicon for IgG binding and identiﬁed with SE that APTES/methyltriethoxys ilane functionalized sili- con covalently bound IgG using glutaraldehyde gave the largest increase in antibody and antigen binding as com- pared to either APTES and glutaraldehyde, or APTES alone.137 FIG. 3. Illustrative schematic of antibody orientation characterization techniques: (a) X-ray photoelectron spectroscopy, (b) spectroscopic eMips ometry, (c) dual polarization interferometry, (d) surface plasmon resonance, (e) neutron reﬂectometry, (f) atomic force microscopy, (g) quartz crystal micro balance, and (h) time-of-ﬂight secondary-ion mass spectrometry.02D301-8 Welch et al. : Orientation and characterization of immobilized antibodies 02D301-8 Biointer phases , Vol. 12, No. 2, June 2017C. Dual polarization interferometry DPI is an optical waveguide technique that utilizes changes in the evanescent wave of the sample beam of laserlight relative to a reference beam. The combined beams cre-ate an interference pattern that provides information regard-ing mass, ﬁlm thickness, refractive index, and density. 138 Song et al.139used DPI to investigate anti-PSA antibody (anti-PSA) immobilized covalently via lysines to thiolatedDPI chips, or captured via the Fc using immobilized ProteinG. The thickness of the two systems was monitored as theantibody bound the PSA antigen and a detection limit of 10 pg/ml was achieved. Song went on to investigate different methods for anti-PSA immobilization with DPI, comparingboronate chelation, TCEP reduction with maleimide cova-lent linkage, Protein G, and random immobilization methods to PEG:Thiol or amine modiﬁed DPI chips. 64They found that while the mass of anti-PSA loaded onto the DPI chipwas lower for TCEP than boronate chelation, the antigensensitivity was 20 times higher inferring preferential orienta-tion with the TCEP method using amine modiﬁed chips. D. Surface plasmon resonance SPR is an optical technique used to monitor protein adsorption and surface interactions. Monochromatic laserlight is used to stimulate a surface plasmon wave in the sens-ing surface and the reﬂected light angle is measured. Upon absorption on the sensing surface, the refractive index of the material changes causing the reﬂected light angle and thesurface plasmon resonance angle to change accordingly. Arecent review by Mauriz et al. 55covers SPR-based assays in great detail. Vashist et al.140utilized SPR as an assay method to inves- tigate antibody immobilization strategies; random, covalent (EDC/NHS), oriented with adsorbed Protein A followed by antibody binding, covalent-oriented Protein A followed byantibody binding, and last covalent-CM5-dextran binding.SPR determined that the mass of antibody immobilizedwas greatest for the CM5 system. However, the covalent- oriented system bound the greatest amount of antigen and indicated preferential orientation. Zhang et al. 141used SPR to investigate antigen binding capabilities of gold–grapheneoxide (Au/GO) composites compared with gold coated withProtein A. By tracking the resonant wavelength change as a function of antigen concentration, the authors demonstrated the oriented Protein A on Au/GO system had improved anti-gen sensitivity compared with Protein A on Au alone. E. Neutron reflectometry Neutron reﬂectometry (NR) is a diffraction technique used to investigate ﬁlm thickness. Neutrons are reﬂected off the sample surface and assessed as a function of change inangle or wavelength. Lu’s team have used NR to investigateseveral antibody binding and orientation effects. 143–145NR was used in conjunction with AFM to determine a ﬂat-on orientation of anti- b-hCG antibody (speciﬁc to the bunit of human chorionic gonadotrophin) to silicon-oxide, asobserved by the ﬁlm thickness.143Then using anti-PSA immobilized to silicon-oxide substrate, the combination of NR and DPI was used to demonstrate ﬂat-on antibody orien-tation. 145NR and SE were also used complementarily to assess the mass of antibody immobilized at different concen- trations.144Schneck et al.146used NR to conﬁrm the orienta- tion of anti-(polyethylene glycol) (anti-PEG) antibodies immobilized to PEG polymer brushes of varying their graft- ing density. They noted that increased grafting densitycaused the distance between the two Fab regions to decrease and overall orients the antibodies such that the Fc region faced away from the PEG brush substrate into the bulk solution. F. Atomic force microscopy AFM is a topographic analysis technique that employs the scanning of a nanoscale tip across a sample surface. The tip is bound to an oscillating cantilever and by accuratelymonitoring the cantilever’s change in resonance, a nanome- ter scale resolution image of the surface can be achieved. When surface bound, the asymmetrical dimensions of anti-bodies (14 /C210/C24n m 3) allow changes to the surface topog- raphy, i.e., ﬁlm thickness and surface roughness, to be measured and can be representative of different antibody ori-entations. Chen et al. 142used AFM and SPR to conﬁrm end- on antibody orientation to the ProLinkerTMSAM by plotting the height proﬁle of the immobilized antibody and monitor-ing antigen uptake. Coppari et al. 147investigated a monoclo- nal antibody adsorbed on mica and, by combining height traces and images, were able to determine different antibody orientations with AFM. By incorporating molecular dynamic simulations with AFM images, Vilhena et al.148demon- strated ﬂat-on, head-on, side-on, and end-on orientations of IgG adsorbed to graphene. Funari et al.56characterized phys- isorbed and irradiation-coupled antibodies on extremelysmooth (root-mean-squared roughness 0.15 60.01 nm) gold- coated silicon wafers. The irradiation causes photo-reduction of disulﬁde bridges that yield free-thiols for binding gold.Using AFM, the authors found that the irradiated antibodies had a smaller contact area with the surface and a larger height distribution indicating a side-on orientation relative tothe physisorbed system being ﬂat-on. Marciello et al . 149 used AFM to investigate the orientation of antibodies at the surface of lipase-coated magnetic nanoparticles. The authors assessed the surface of the nanoparticles and after immobili- zation. Two immobilized antibody systems were investi-gated; the optimized sample, S1, with a recovered immune activity (from immunoassay) of 80%, and S2, with a lower recovered immune activity (about 3%). The peak-to-valleyheight determined with AFM of S1 was found to be 9 nm as compared with only 5 nm for S2, indicating preferential ori- entation of the antibody on S1. G. Quartz crystal microbalance QCM is a mass sensitive technique that monitors the change in frequency and damping of a resonant quartz02D301-9 Welch et al. : Orientation and characterization of immobilized antibodies 02D301-9 Biointer phases , Vol. 12, No. 2, June 2017piezoelectric crystal. As the material is adsorbed to the crys- tal, the resonance frequency decreases, allowing the amount of material to be determined very precisely. Thus, if the mass of the antibody and its complementary antigen areknown, then antibody orientation can be inferred from anti-gen binding. Recently, Deng et al . 150have developed a QCM chip modiﬁcation that is used to orient biotin-labeled antibodies for use in an immunoassay. QCM was used to quantify antibody binding and antigen coupling. Comparedwith the control surface, the biotinylated graphene oxide-avidin surface modiﬁcation was found to bind slightly loweramounts of antibody, although it demonstrated improved antigen capture, suggestive of antibody orientation. Dissipation monitoring during QCM can provide insight into the density and permeability of immobilized (bio)mole-cules at the interface. 51,158Via this method, protein orienta- tion may be investigated; for instance, an antibody immobilized close to the surface in a ﬂat-on orientation will have a low viscoelastic dissipation, while in contrast, an anti-body in head-on or end-on orientation will have a higher vis-coelastic dissipation. Another QCM mass-based immunoassay was proposed by Akter et al. 151who employed Protein A as the antibody orienting component. Using QCM, the authors were able tomonitor antibody binding, antigen selectivity, and demon-strate the beneﬁts of their precipitation mass ampliﬁcationsystem as applicable in immunoassay. H. Time-of-flight secondary-ion mass spectrometry ToF-SIMS employs a focused beam of ionized metal atoms (or clusters of atoms), large molecules such as fuller-enes (C 60), or gas clusters, to bombard the sample and remove fragmented material from the surface. A small per- centage of the fragments are ionized, known as secondary-ions, which are collected as a representative sample of theelemental and molecular species from the top few mono-layers of the surface (on the order of 10 A ˚). 159The 14 nm long axis of antibodies, combined with this limited sampling depth, provides differentiation between orientations ofimmobilized antibody due to different amino acid composi-tions in the portions being analyzed. ToF-SIMS analysis of known peptides has been used to determine most common mass fragments arising from partic- ular amino acids. 152,160,161These amino acid speciﬁc lists, typically of up to 40 mass fragments, allow differentiationbetween protein and substrate. 153,154Foster et al.130analyzed bovine IgG adsorbed to gold and sodium styrenesulfonate coated gold surfaces. As the prevalence of serine is higher in the F(ab0)2region of the IgG, and aspartate and valine preva- lence higher in the Fc region, then a ratio of these aminoacids mass fragment intensities could be used to assess therelative orientation of IgG immobilized to the substrates.Wang et al. 155investigated anti-hCG IgG, F(ab0)2, and Fc immobilized on gold and also orientation promoting SAMs (COOH and NH 2). Principal component analysis (PCA) identiﬁed amino acid fragments that were more prevalent inthe F(ab0)2and Fc portions, and these were correlated against the antibody amino acid composition. A ratio of the impor- tant amino acids could then be used as predictors for anti-body orientation. More recently, Welch et al. has employed larger peak lists incorporating over 700 mass fragments for characterizing adsorbed whole antibody and antibody frag-ments. 156The increased peak list signiﬁcantly improved the ability to identify and classify the samples using multivariate analysis techniques. One potential drawback is that ToF- SIMS operates in an ultrahigh vacuum environment (UHV)that is likely to denature proteins. 157One possible strategy for avoiding UHV induced denaturation is by the ﬁxation of proteins at surfaces with trehalose. Trehalose is a disaccha-ride that has been used to ﬁx the state of proteins at interfa- ces to preserve their conformation and minimize changes to their orientation. 153,162However, coating samples prior to ToF-SIMS analysis, or other surface analysis techniques such as XPS, may cause difﬁculties in subsequent data analysis. I. Multivariate analysis It is now a common practice to employ multivariate anal- ysis techniques such as PCA to reduce the dimensionality ofcomplex data sets, such as those derived from ToF-SIMS.PCA is used to identify the variables that contribute to the largest amount of variance in the dataset. In the case of ToF- SIMS analysis, the intensity or number of counts obtainedfor each mass fragment comprise the input variables for PCA. PCA has been used to investigate antibody orientation on various substrates to good effect. 163,164Liu et al .165 immobilized Fab and Fc fragments to both gold and polymer-coated slide substrates and used PCA to investigate each of the four systems. Principal component 1 (PC1) sepa-rated the samples based on substrate, and principal compo- nent 2 (PC2) separated samples based on antibody fragment. The loadings plot for PC2, showing the contributions fromeach of the amino acid variables, correlated strongly with natural amino acid composition differences in the antibody fragments. Park et al . 164interrogated randomly and site- directed IgG and F(ab0)2with ToF-SIMS and PCA. Using known amino acid related mass fragments, PC1 showed that site-directed IgG and F(ab0)2were more commonly in the end-on orientation than the randomly immobilized proteins.Kosobrodova et al. 166investigate antibodies immobilized to untreated and plasma treated polycarbonate with ToF-SIMS and PCA and found that the F(ab0)2component of the anti- body was preferentially exposed on the plasma treated surface. Artiﬁcial neural networks (ANN) are another class of multivariate analysis techniques and classify and group sam- ples based on their similarities and differences across the input variables. Sanni et al.167employed ANN to differenti- ate between 13 different protein ﬁlms, including two typesof antibodies, using nominal mass values of all mass frag- ments available in the spectra. ANN was able to identify the key mass fragments associated with each of the proteins to02D301-10 Welch et al. : Orientation and characterization of immobilized antibodies 02D301-10 Biointer phases , Vol. 12, No. 2, June 2017distinguish between them. More recently, Welch et al .156 employed ANNs to discriminate between an antibody and its proteolysis fragments adsorbed to silicon substrates, based solely on their ToF-SIMS spectra. The ANN analysismethod holds promise for investigating antibody orientation at interfaces due to its ability to incorporate a broad range of mass fragments and investigate complex relationships between variables. IV. CONCLUSIONS AND PERSPECTIVES In summary, existing and new methods for oriented anti- body immobilization have been developed over the past few years with good progress made on improving the established methods. Nevertheless, some shortcomings and challenges still exist. Oriented systems based on novel surface chemis-try, protein engineering, or both, require complex and time- consuming production steps, which may also be expensive. Ideally, antibodies truly representative of their native solution-phase state, i.e., without protein engineered tags, would be immobilized to a simple, cheap, and easy to pro- duce substrate, homogeneously arranged and site-directed as to maximize their antigen capture. However, it is likely thatsmaller antibody fragments and aptamers may be favored over whole antibodies in the future as they can be prepared recombinantly and are easily modiﬁed genetically or chemi- cally with the ability to maximize capture events due to increased packing and binding site density. 168Additionally, camelid antibodies with one single domain for antigen bind- ing (known as VHH) have attracted attention due to theirhigh solubility and stability, and may provide an opportunity for incorporation into sensors. 8Also, peptides and aptamers provide a highly customizable method for producing cova- lent attachment of antibodies, or fragments thereof, either by active targeting of the protein, or the substrate, or both. In the future, it may be seen that metallic thin ﬁlms are used to produce homogeneous substrates that can be targeted simplyby endogenous antibody epitopes or material binding pepti- des coupled to Fc speciﬁc aptamers or peptides. In the case of the latter, such a system could be near universally applica- ble to native state antibodies. It is clear that surface analysis techniques and multivari- ate analysis tools will play a prevalent role in identifying, investigating, and predicting antibody orientation at sub- strates of interest. Characterization of antibody orientation in situor in the native “wet” environment permits a more accu- rate presentation of the antibody state without the potential for conﬁrmation changes. XPS and ToF-SIMS require UHVconditions and are ex situ . AFM is typically performed in a dry state (however progress has been made regarding wet analysis) and is performed ex situ. QCM, SPR, NR, SE, and DPI can be performed in solution and with the correct appa- ratus in situ also. Additionally, complementary techniques such as sum frequency generation promise to provide in situ characterization of protein orientation via changes in theinfrared absorption patterns; however, complex analysis is still required. 169,170Nevertheless, ToF-SIMS has the potential to characterize and differentiate antibody properties including orientationand denaturation state, yielding molecularly speciﬁc infor-mation from the uppermost surface. The information-rich data greatly stands to beneﬁt from interrogation with multi- variate analysis techniques PCA and ANN. Not only can thiscombination offer new insight into the state of immobilizedproteins, but it is also directly applicable to new material dis-covery and will greatly assist in the development and optimi- zation of immunoassay performance. 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5.0059800.pdf	AIP Advances ARTICLE scitation.org/journal/adv Analytical study of the sth-order perturbative corrections to the solution to a one-dimensional harmonic oscillator perturbed by a spatially power-law potential Vper(x)=λxα Cite as: AIP Advances 11, 085310 (2021); doi: 10.1063/5.0059800 Submitted: 12 June 2021 •Accepted: 24 July 2021 • Published Online: 6 August 2021 Tran Duong Anh-Tai,1,a) Duc T. Hoang,2Thu D. H. Truong,2 Chinh Dung Nguyen,3,4Le Ngoc Uyen,5 Do Hung Dung,6Nguyen Duy Vy,7,8,b) and Vinh N. T. Pham2,c) AFFILIATIONS 1Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan 2Department of Physics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam 3Institute of Fundamental and Applied Sciences, Duy Tan University, 6 Tran Nhat Duat St., District 1, Ho Chi Minh City 700000, Vietnam 4Faculty of Natural Sciences, Duy Tan University, 03 Quang Trung St., Hai Chau District, Danang 550000, Vietnam 5Department of Engineering Science, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan 6Department of Natural Science, Dong Nai University, Dong Nai, Vietnam 7Laboratory of Applied Physics, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam 8Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam a)Electronic mail: tai.tran@oist.jp b)Electronic mail: nguyenduyvy@tdtu.edu.vn c)Author to whom correspondence should be addressed: vinhpnt@hcmue.edu.vn ABSTRACT In this work, we present a rigorous mathematical scheme for the derivation of the sth-order perturbative corrections to the solution to a one-dimensional harmonic oscillator perturbed by the potential Vper(x)=λxα, whereαis a positive integer, using the non-degenerate time-independent perturbation theory. To do so, we derive a generalized formula for the integral I=+∞ ∫ −∞xαexp(−x2)Hn(x)Hm(x)dx, where Hn(x)denotes the Hermite polynomial of degree n, using the generating function of orthogonal polynomials. Finally, the ana- lytical results with α=3 andα=4 are discussed in detail and compared with the numerical calculations obtained by the Lagrange-mesh method. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0059800 I. INTRODUCTION Approximation methods play a crucial role in quantum mechanics since the number of problems that are exactly solv- able is small in comparison to those that must be solved approxi- mately. To our knowledge, the hydrogen atom, harmonic oscillators, and quantum particles in some specific potential wells have exactsolutions,1–4and two cold atoms interacting through a point-like force in a three-dimensional harmonic oscillator potential5can also be solved analytically. Recently, Jafarov et al. reported an exact solution to the position-dependent effective mass harmonic oscil- lator model.6Because of this, approximation methods have been developed early since the dawn of quantum mechanics. One of the essential approximation methods is the perturbation theory AIP Advances 11, 085310 (2021); doi: 10.1063/5.0059800 11, 085310-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv (PT) established by Schrödinger in 1926.7Later, it was immedi- ately used to interpret the LoSurdo–Stark effect of the hydrogen atom by Epstein.8Although the PT is not well convergent at higher- order corrections,1–3,9,10many more efficient approximation meth- ods have been developed to treat quantum-mechanically complex problems,11it is still a paramount and elementary approximation method in quantum physics. For instance, the PT contributes signif- icantly to quantum optics and quantum field theory, as discussed in Ref. 12. For this reason, the PT is usually presented clearly and dis- cussed in detail in quantum mechanics textbooks for undergraduate students.1–3,9 The one-dimensional harmonic oscillator is not only a rich pedagogical example for approximation theories in quantum mechanics13–24but also an excellent candidate for numerical25,26and analytical or algebraic methods,6,27–30owing to its simple calcula- tion and exact solution. Moreover, the one-dimensional harmonic oscillator potential has also played a key role in studies of ultra- cold atomic quantum gases for the last two decades.5,31–51Moshinsky and Smirnov52provided a deep review of the role of quantum har- monic oscillators in modern physics. In standard quantum mechan- ics textbooks,1–3a one-dimensional anharmonic oscillator is usu- ally presented as an application of PT because the solutions can be constructed based on the exact solution to a one-dimensional harmonic oscillator. The two perturbative potentials that are usu- ally considered are Vper(x)=λxand Vper(x)=λx4. It is common to present the calculation of the first- and second-order correc- tions of the energy; however, the corrections to the wave function are usually not provided in detail.1–3Moreover, physicists have also studied the generalized case Vper(x)=λx2β, whereβis a positive integer. The case β=2 has been studied using the WKB method,53 the intermediate Hamiltonian,54the Padé approximation,55–57the Heisenberg matrix mechanics,58and the variational perturbation.59 In addition, the cases β=3 andβ=4 were studied in Refs. 60 and 61, and the considered potential of the harmonic oscillator is V(x)=x2 instead of V(x)=0.5x2. The authors intended to establish efficient methods to solve the problem mathematically, regardless of their physical meaning. Recently, the problem has been extended to sex- tic (x6) and decatic ( x10) potentials using polynomial solutions62,63 and a polynomial perturbative potential.64Interestingly, we real- ized that in the above-mentioned works, the authors only consid- ered even values of the power of x, while the odd cases have not been studied. It is also interesting to note that the wave function was not considered in the above-mentioned articles. Essentially, the applications of the one-dimensional anisotropic oscillator can be found in chemistry, in which the perturbative potential is used to study the vibration in molecules.65–69In addition, the pertur- bative potential λx4has recently been used to model the Brown- ian motion of particles in optical tweezers.70Consequently, it is necessary to compute the approximated wave function and the energy of a one-dimensional harmonic oscillator perturbed by the potential Vper(x)=λxαfor arbitrary eigenstates with arbitrary values ofα. The goal of this study is to present a systematic and complete treatment of the sth-order perturbative corrections to the solution to a one-dimensional harmonic oscillator perturbed by the poten- tialVper(x)=λxα. To achieve this goal, we derived a formula for I=∫+∞ −∞xαexp(−x2)Hn(x)Hm(x)dx, with Hn(x)being the Her- mite polynomial of degree n. Our scheme is based on the so-calledgenerating functions of orthogonal polynomials.71Because the potential depends solely on the spatial coordinate and the states are non-degenerate, the non-degenerate time-independent PT is used to derive the corrections to the wave function and energy. Note that our results can be used for arbitrary eigenstates of a one-dimensional anharmonic oscillator with an arbitrary power coefficient α. This is significantly different from previous works, as discussed above. The remainder of this paper is organized as follows: Sec. II briefly outlines the time-independent PT for non-degenerate states. Section III presents the main results and discussion. Finally, conclu- sions are presented in Sec. IV. For simplicity, we use atomic units in which h=m=ω=1 throughout this study. In addition, the notation Xα n,sdenotes the sth-order perturbation correction to the physical quantity Xin the state with the quantum number nand the power coefficientα. II. NON-DEGENERATE TIME-INDEPENDENT PERTURBATION THEORY AND THE 2s+1RULE This section presents the time-independent PT for non- degenerate states and a recurrence relation to obtain higher-order corrections to the wave function and energy. We followed the procedure of Fernandez.9The Schrödinger equation describing a one-dimensional quantum system is as follows: ˆHψn=Enψn, (1) where ˆHis the Hamiltonian operator and Enis the eigenvalue corre- sponding to the eigenfunction ψn. The Hamiltonian can be split into two parts, ˆH=ˆH0+λˆH′, (2) with ˆH0being the Hamiltonian operator, whose eigenvalues and eigenfunctions are analytically solvable and satisfy the equation ˆH0ψn,0=En,0ψn,0, (3) and ˆH′being sufficiently small and considered as a small perturba- tion with parameter λ. The Taylor formula is used to expand the energy and the eigenfunction of the Hamiltonian ˆHas a function of perturbation parameter λ, En=∞ ∑ s=0En,sλs,ψn=∞ ∑ s=0ψn,sλs, (4) where sis the order of the perturbative correction. Substituting Eq. (4) into Eq. (1), we obtain a recurrence equation expressing the relation between the corrections to the eigenfunction ψn,sand the energy En,sas follows: [ˆH0−En,0]ψn,s=s ∑ j=1En,jψn,s−j−ˆH′ψn,s−1. (5) The perturbation correction to the wave function, ψn,s, is then expanded as a linear combination of the eigenfunctions of the non-perturbative Hamiltonian, ˆH0, as follows: ψn,s=∑ mcmn,sψm,0, (6) AIP Advances 11, 085310 (2021); doi: 10.1063/5.0059800 11, 085310-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv where cmn,s=⟨ψm,0∣ψn,s⟩is the expanding coefficient. Substituting Eq. (6) into Eq. (5) and then integrating over the whole space after multiplying both sides by ψ∗ m,0yield [Em,0−En,0]cmn,s=s ∑ j=1En,jcmn,s−j−∑ kˆH′ mkckn,s−1, (7) where ˆH′ mk=⟨ψm,0∣ˆH′∣ψk,0⟩. For non-degenerate states, Em,0≠En,0 if and only if m≠n, we can derive the general expression for the correction to the energy by letting m=nin Eq. (7); hence, we obtain En,s=⟨ψn,0∣ˆH′∣ψn,s−1⟩−s−1 ∑ j=1En,jcnn,s−j. (8) The normalization condition ⟨ψn∣ψn⟩=1 results in a constraint on the correction to the wave function, s ∑ j=0⟨ψn,j∣ψn,s−j⟩=δs0. (9) Finally, in the case of m≠n, we can derive the expanding coefficients for the correction to the wave function as follows: cmn,s=1 En,0−Em,0⎛ ⎝∑ kˆH′ mkckn,s−1−s ∑ j=1En,jcmn,s−j⎞ ⎠(10) and cnn,1=0,cnn,s=−1 2s−1 ∑ j=1∑ mcmn,jcmn,s−j,s>1 (11) for the case of m=n. Substituting s=1 into Eq. (8) derives the for- mula for the first-order correction to the energy, which is the average of the perturbation potential with respect to the eigenfunction ψn,0, En,1=⟨ψn,0∣ˆH′∣ψn,0⟩, (12) and the expanding coefficient for the first-order correction to the wave function is then derived as follows: cmn,1=⟨ψm,0∣ˆH′∣ψn,0⟩ En,0−Em,0. (13) Obtaining the second-order correction to the energy is also straight- forward. It is obtained as follows: En,2=⟨ψn,0∣ˆH′∣ψn,1⟩=∑ m≠n∣⟨ψm,0∣ˆH′∣ψn,0⟩∣2 En,0−Em,0. (14) These results can be found in standard quantum mechanics text- books.1–3To derive higher-order corrections to the energy, it is obvi- ous that one can use the recurrence given by Eq. (8). However, there is another way to quickly compute the corrections to the energy. It is called the 2 s+1 rule, in which, once we know the sorder of the correction to the wave function, we are allowed to compute the corrections to the energy up to the 2 s+1 order. For the detailed derivation of the rule, one should refer to the textbook.9Below, we list the formula for the third-, fourth-, and fifth-order corrections to the energy used for calculations in Sec. III, En,3=⟨ψn,1∣ˆH′−En,1∣ψn,1⟩, (15)En,4=⟨ψn,2∣ˆH′−En,1∣ψn,1⟩−En,2(⟨ψn,2∣ψn,0⟩+⟨ψn,1∣ψn,1⟩), (16) En,5=⟨ψn,2∣ˆH′−En,1∣ψn,2⟩−En,2(⟨ψn,1∣ψn,2⟩+⟨ψn,2∣ψn,1⟩). (17) III. RESULTS AND DISCUSSION A. Derivation of the sth-order perturbative corrections to the solution to a one-dimensional anharmonic oscillator The Schrödinger equation describing a one-dimensional har- monic oscillator induced by a perturbation potential λxαis (−1 2d2 dx2+1 2x2+λxα)ψα n(x)=Eα nψα n(x), (18) whereλis the strength of the external field, which gives rise to the perturbation, and αis a positive integer. In the absence of the perturbation, Eq. (18) is the well-known equation describing a one-dimensional harmonic oscillator with a wave function ψ0 n,0(x)=Anexp(−x2 2)Hn(x), (19) where An=1√ 2nn!√πis the normalization constant, nis the quantum number, and Hn(x)is the Hermite polynomial of degree n, and the energy is given by E0 n,0=n+1 2. (20) Since Eq. (18) cannot be analytically solvable, the PT is then cho- sen to approximate the solutions. As discussed above, the first-order correction of the wave function is given by the following equation: ψα n,1(x)=∑ m≠ncmn,1ψ0 m,0(x), (21) where cmn,1=⟨ψ0 m,0∣xα∣ψ0 n,0⟩ En,0−Em,0(22) is the expanding coefficient of the first-order correction to the wave function. Making use of Eq. (A12) (see the Appendix), we obtain the following equation: ⟨ψ0 m,0∣xα∣ψ0 n,0⟩=k≤α/2 ∑ k=0α−2k ∑ ℓ=0(α 2k)(α−2k ℓ)n!m!2n−ℓAnAm (m−α+2k+ℓ)! ×Γ(k+1 2)δm,n+α−2(k+ℓ), (23) whereδm,ndenotes the Kronecker delta, satisfying δm,n=⎧⎪⎪⎨⎪⎪⎩1, m=n, 0, m≠n.(24) By substituting Eq. (23) into Eq. (21), the first-order correction to the wave function for arbitrary states is obtained by AIP Advances 11, 085310 (2021); doi: 10.1063/5.0059800 11, 085310-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv ψα n,1(x)=∑ m≠nk≤α/2 ∑ k=0α−2k ∑ ℓ=0(α 2k)(α−2k ℓ)n!m!2n−ℓAnAm (m−α+2k+ℓ)! ×Γ(k+1 2) (n−m)ψ0 m,0(x)δm,n+α−2(k+ℓ). (25) The first-order correction to the energy is then computed by the following equation: Eα n,1=⟨ψ0 n,0∣xα∣ψ0 n,0⟩. (26) Combining the known wave function of a one-dimensional har- monic oscillator and Eq. (A12), the first-order correction to the energy is obtained by Eα n,1=k≤α/2 ∑ k=0α−2k ∑ ℓ=0(α 2k)(α−2k ℓ)n!2−ℓ (n−α+2k+ℓ)! ×Γ(k+1 2)√πδn,n+α−2(k+ℓ). (27) It is interesting to note that En,1is non-zero only if k+ℓ=α 2, (28) owing to the mathematical property of the Kronecker delta. Because kand ℓare integers, the above equation has the solutions for even αsolely. This means that in the case of odd α, the first-order cor- rection to energy always equals to zero. Therefore, the anharmonic oscillator does not feel the presence of the external field in this case ifonly the first-order approximation is considered. Therefore, it is nec- essary to compute higher-order corrections to the energy. Regard- ing the second-order correction of energy, it can be computed as follows: Eα n,2=⟨ψ0 n,0∣xα∣ψα n,1⟩. (29) By substituting Eqs. (19) and (25) into Eq. (29), we obtain the following equation: Eα n,2=∑ m≠nk≤α/2 ∑ k=0α−2k ∑ ℓ=0(α 2k)(α−2k ℓ) ×n!m!2n−ℓAnAm (m−α+2k+ℓ)!Γ(k+1 2) (n−m)δm,n+α−2(k+ℓ) ×+∞ ∫ −∞ψ0 m,0(x)ψ0 n,0(x)dx. (30) Once again, the integral can be treated by making use of Eq. (A12), +∞ ∫ −∞ψ0 m,0(x)ψ0 n,0(x)dx=k′≤α/2 ∑ k′=0α−2k′ ∑ ℓ′=0(α 2k′)(α−2k′ ℓ′) ×n!m!2n−ℓ′AnAm (m−α+2k′+ℓ′)!Γ(k′+1 2) ×δm,n+α−2(k′+ℓ′). (31) Substituting back into Eq. (30), the general second-order correction to the energy is obtained as follows: Eα n,2=∑ m≠nk≤α/2 ∑ k=0α−2k ∑ ℓ=0(α 2k)2 (α−2k ℓ)2(n!m!)222(n−ℓ)A2 nA2 m (m−α+2k+ℓ)!2(n−m)Γ(k+1 2)2δm,n+α−2(k+ℓ). (32) Using Eq. (15), the third-order correction to the energy for arbitrary states could be derived as follows: Eα n,3=⟨ψα n,1∣xα−Eα n,1∣ψα n,1⟩=∑ m1≠n∑ m2≠nk1≤α/2 ∑ k1=0α−2k1 ∑ ℓ1=0k2≤α/2 ∑ k2=0α−2k2 ∑ ℓ2=0(α 2k1)(α−2k1 ℓ1)(α 2k2)(α−2k2 ℓ2) ×n!m1!Γ(k1+1 2)Γ(k2+1 2) (m1−α+2k1+ℓ1)!(n−m1)π√πδm1,n+α−2(k1+ℓ1)⎡⎢⎢⎢⎢⎣k′≤α/2 ∑ k′=0α−2k′ ∑ ℓ′=0(α 2k′)(α−2k′ ℓ′)m2!2n−m2−ℓ1−ℓ2−ℓ′Γ(k′+1 2) (m2−α+2k2+ℓ2)!(m2−α+2k′+ℓ′)!(n−m2) ×δm2,n+α−2(k2+ℓ2)δm2,m1+α−2(k′+ℓ′)−k≤α/2 ∑ k=0α−2k ∑ ℓ=0(α 2k)(α−2k ℓ)n!2n−m1−ℓ1−ℓ2−ℓΓ(k+1 2) (m1−α+2k2+ℓ2)!(n−α+2k+ℓ)!(n−m1) ×δm1,n+α−2(k2+ℓ2)δn,n+α−2(k+ℓ)⎤⎥⎥⎥⎥⎦. (33) The wave function of a one-dimensional anharmonic oscil- lator with first-order correction is given by the following equation: ψα n(x)=ψα n,0(x)+λψα n,1(x), (34)and the corresponding energy with third-order correction is as follows: Eα n=Eα n,0+λEα n,1+λ2Eα n,2+λ3Eα n,3. (35) Obviously, the above results can be used to approximate the wave function and the energy for arbitrary states and arbitrary power α. AIP Advances 11, 085310 (2021); doi: 10.1063/5.0059800 11, 085310-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv Since the calculation for the higher-order corrections is more tedious for the general case α, we constrained our calculation for the general powerαat this point. In the following, we discuss two particular circumstances in which α=4 andα=3 in detail and then extend the calculation to the second-order correction to the wave function and the fourth- and fifth-order corrections to the energy for these cases. B. Typical cases: α=4and α=3 In this section, we discuss two particular circumstances where α=4 andα=3 as two typical examples of PT. In addition, to val- idate the significance of the higher-order corrections to the energy obtained in the textbook,9we calculated the second-order correc- tion to the wave function for these two cases and then computed the energy up to the fifth-order approximation. The numerical results obtained by the Lagrange-mesh method25,26were then used as the benchmark to validate the applicable range of the analytically approximated results. First, let us discuss the case where α=4 explicitly. According to Eq. (25), the first-order correction to the wave function is given bythe following equation: ψ4 n,1(x)=1 4[1 4√ (n−3)4ψ0 n−4,0(x)−1 4√ (n+1)4ψ0 n+4,0(x) +(2n−1)√ (n−1)2ψ0 n−2,0(x)−(2n+3) ×√ (n+1)2ψ0 n+2,0(x)], (36) where(a)n=a(a+1)⋅⋅⋅(a+n−1)is the Pochhammer symbol. The first-, second-, and third-order corrections to the energy are obtained by using Eqs. (27), (32), and (33), respectively, with α=4, E4 n,1=3 4(2n2+2n+1), (37) E4 n,2=−1 8(34n3+51n2+59n+21), (38) E4 n,3=(375 16n4+375 8n3+177 2n2+1041 16n+333 16). (39) The second-order correction can be derived by the 2 s+1 rule, which was presented in Sec. II. The calculation shows that ψ4 n,2(x)=1 512√ (n−7)8ψ0 n−8,0(x)+1 192(6n−11)√ (n−5)6ψ0 n−6,0(x)+1 16(2n−7)(n−1)√ (n−3)4ψ0 n−4,0(x) +1 2√ (n−1)2(−1 16n3−129 32n2+107 32n−33 16)ψ0 n−2,0(x)+1 2√ (n+1)2(−1 16n3+123 32n2+359 32n+75 8)ψ0 n+2,0(x) +1 16(2n+9)(n+2)√ (n+1)4ψ0 n+4,0(x)+1 192(6n+17)√ (n+1)6ψ0 n+6,0(x)+1 512√ (n+1)8ψ0 n+8,0(x) −1 2(65 128n4+65 64n3+487 128n2+211 64n+39 32)ψ0 n,0(x). (40) Consequently, the fourth- and fifth-order corrections to the energy are obtained, respectively, as follows: E4 n,4=−10 689 64n5−53 445 128n4−71 305 64n3−80 235 64n2 −111 697 128n−30 885 128, (41) E4 n,5=87 549 64n6+262 647 64n5+3 662 295 256n4+2 786 805 128n3 +3 090 693 128n2+3 569 679 256n+916 731 256. (42) Next, we compare the analytically approximated formulas with numerical calculations to validate the exactness and determine the applicable range of the approximated formulas. For this purpose, the relative deviation is given by the following equation: σ=∣Enum−Eana Enum∣, (43)where Enum and Eanaare the numerical and analytical results, respectively. As shown in Fig. 1, in the small λregime (0≤λ≤0.05), the approximated formulas match well to the numerical calculation. In this regime, the tenth-order corrected formula has the lowest relative deviation, approximately zero. However, the higher-order formulas diverge rapidly as the perturbative parameter increases. Nevertheless, in the large λregime, the first-order corrected formula has, in general, a smaller relative deviation compared to others. Finally, we explicitly present the corrections to the energy and wave function in the case of α=3. The first- and second-order cor- rections to the wave function of this case are given, respectively, as follows: ψ3 n,1(x)=√ 2 12√ (n−2)3ψ0 n−3,0(x)−√ 2 12√ (n+1)3ψ0 n+3,0(x) +3√ 2 4n√nψ0 n−1,0(x)−3√ 2 4(n+1)√ n+1ψ0 n+1,0(x) (44) AIP Advances 11, 085310 (2021); doi: 10.1063/5.0059800 11, 085310-5 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 1. The first row shows the plots of the energy of the ground state and three excited states as a function of perturbative parameter λwith different orders of correction in the case ofα=4. Meanwhile, the second row depicts the relative deviation between the analytical formulas and the numerical calculation obtained by the Lagrange-mesh method (black solid line). Note that the tenth-order corrected energy (pink star line) is taken from Ref. 9. and ψ3 n,2(x)=1 144√ (n+1)6ψ0 n+6,0(x)+1 32√ (n+1)4(4n+7)ψ0 n+4,0(x) +1 16√ (n+1)2(7n2+33n+27)ψ0 n+2,0(x) +1 144√ (n−5)6ψ0 n−6,0(x) +1 32√ (n−3)4(4n−3)ψ0 n−4,0(x) +1 16√ (n−1)2(7n2−19n+1)ψ0 n−2,0(x) −1 2(41 18n3+41 12n2+32 9n+29 24)ψ0 n,0(x). (45)Straightforwardly, we obtain E3 n,1=0, (46) E3 n,2=−15 4n2−15 4n−11 8, (47) E3 n,3=0, (48) E3 n,4=−705 16n3−2115 32n2−1635 32n−465 32, (49) E3 n,5=0. (50) Similarly, the analytically approximated formulas were compared to the numerical calculation. Because the odd-order corrections to FIG. 2. Same as Fig. 1 but for α=3. AIP Advances 11, 085310 (2021); doi: 10.1063/5.0059800 11, 085310-6 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv the energy are zero, the second- and fourth-order corrections are considered. The results are depicted in Fig. 2. The comparison shows that the second- and fourth-order corrected energies are relatively identical to the numerical results. The deviations of the ground and first-excited state are less than 1%, while those of higher-excited states are larger, ∼5%, which is still acceptable. IV. CONCLUSION In summary, we have provided a mathematical procedure to derive the wave function and energy for any arbitrary states of a one- dimensional anharmonic oscillator with the general perturbation potential Vper(x)=λxαusing the time-independent non-degenerate PT. Subsequently, the explicit results of two particular cases in which α=3 andα=4 are discussed in detail. In addition, the second-order correction to the wave function and up to fifth-order correction to the energy for these cases were computed. The analytical results were then compared with the numerical solutions obtained using the Lagrange-mesh method. The comparison indicates that in the regime in which the perturbation parameter is small, the analytical results agree well with those obtained numerically and then dramat- ically diverge as the perturbation parameter increases. The higher- order corrections to the energy diverge rapidly as the perturbative parameter increases. In addition, the fifth-order corrected energy in the case of α=4 is the same as that printed in Ref. 9; however, the procedure is more comfortable to approach. The results in the present article are anticipated to be a useful reference. ACKNOWLEDGMENTS This work was supported by the Ministry of Education and Training of Vietnam (Grant No. B2021-SPS-02-VL). Mr. Tran Duong Anh-Tai acknowledges support provided by the Okinawa Institute of Science and Technology Graduate University (OIST). The authors are thankful to Mr. Mathias Mikkelsen for introducing the Lagrange-mesh method to them. APPENDIX: THE INTEGRAL I=+∞ ∫ −∞xαexp(−x2)Hn(x)Hm(x)dx In this work, the Hermite polynomials of degrees mandnare expanded by generating function,71respectively, g(t,x)=+∞ ∑ n=0Hn(x)tn n!=exp(−t2+2tx), (A1) f(z,x)=+∞ ∑ m=0Hm(x)zn m!=exp(−z2+2zx). (A2) Then, multiplying both sides of the above two equations by xαexp(−x2)and taking the integral from −∞ to+∞, we obtain +∞ ∑ n=0+∞ ∑ m=0⎡⎢⎢⎢⎢⎣+∞ ∫ −∞xαexp(−x2)Hm(x)Hn(x)dx⎤⎥⎥⎥⎥⎦tnzm n!m! =exp(2tz)+∞ ∫ −∞xαexp[−(x−(t+z))2]dx. (A3)To simplify the calculation, let us introduce a new variable y=x −(t+z), and then the right-hand side is rewritten as A=exp(2tz)+∞ ∫ −∞[y+(t+z)]αexp(−y2)dy. (A4) The binomial in (A4) is expanded by the binomial formula [y+(t+z)]α=α ∑ i=0(α i)(t+z)α−iyi, (A5) and hence, A=exp(2tz)α ∑ i=0(α i)(t+z)α−i+∞ ∫ −∞yiexp(−y2)dy. (A6) The integral in Eq. (A6) can be straightforwardly deduced as +∞ ∫ −∞xkexp(−x2)dx=⎧⎪⎪⎪⎨⎪⎪⎪⎩0 if kis odd, Γ(1+k 2) ifkis even,(A7) therefore solely the integrals with even coefficients i=2kare considered A=exp(2tz)k≤α/2 ∑ k=0(α 2k)Γ(k+1 2)(t+z)α−2k. (A8) It can be seen in (A3) that the value of the integral is the coefficient oftnzm; thus, we need to find that expanding coefficient. To do so, we expand exp(2tz)=∞ ∑ j=0(2tz)j j!(A9) by the Taylor formula and (t+z)α−2k=α−2k ∑ ℓ=0(α−2k ℓ)tℓzα−2k−ℓ(A10) by the binomial formula. Substituting into (A4), we obtain A=+∞ ∑ j=0⎡⎢⎢⎢⎢⎣k≤α/2 ∑ k=0α−2k ∑ ℓ=0(α 2k)(α−2k ℓ)Γ(k+1 2)2j j!⎤⎥⎥⎥⎥⎦tj+ℓzα−2k−ℓ+j. (A11) Combining (A11) and (A3), the desired result is obtained, +∞ ∫ −∞xαexp(−x2)Hm(x)Hn(x)dx =k≤α/2 ∑ k=0α−2k ∑ ℓ=0(α 2k)(α−2k ℓ)n!m!2n−ℓ (m−α+2k+ℓ)! ×Γ(k+1 2)δm,n+α−2(k+ℓ). 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1.5023118.pdf	Pseudomorphic spinel ferrite films with perpendicular anisotropy and low damping R. C. Budhani , Satoru Emori , Zbigniew Galazka , Benjamin A. Gray , Maxwell Schmitt , Jacob J. Wisser , Hyung- Min Jeon , Hadley Smith , Piyush Shah , Michael R. Page , Michael E. McConney , Yuri Suzuki , and Brandon M. Howe Citation: Appl. Phys. Lett. 113, 082404 (2018); doi: 10.1063/1.5023118 View online: https://doi.org/10.1063/1.5023118 View Table of Contents: http://aip.scitation.org/toc/apl/113/8 Published by the American Institute of PhysicsPseudomorphic spinel ferrite films with perpendicular anisotropy and low damping R. C. Budhani,1,2,a)Satoru Emori,3,4Zbigniew Galazka,5Benjamin A. Gray,1 Maxwell Schmitt,1Jacob J. Wisser,3,6Hyung-Min Jeon,7Hadley Smith,1Piyush Shah,1 Michael R. Page,1Michael E. McConney,1YuriSuzuki,3,6and Brandon M. Howe1 1Materials and Manufacturing Directorate, Air Force Research Laboratory, Dayton, Ohio 45433, USA 2Department of Physics, Morgan State University, Baltimore, Maryland 21251, USA 3Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA 4Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA 5Leibniz Institute for Crystal Growth—Forschungsverbund Berlin eV, 12489 Berlin, Germany 6Department of Applied Physics, Stanford University, Stanford, California 94305, USA 7Department of Electrical Engineering, Wright State University, Dayton, Ohio 45435, USA (Received 21 January 2018; accepted 6 August 2018; published online 22 August 2018) We report on epitaxial thin ﬁlms of spinel ferrite Ni 0.65Zn0.35Fe1.2Al0.8O4with strain-induced per- pendicular magnetic anisotropy (PMA) and low magnetic damping. Static magnetometry and broadband ferromagnetic resonance experiments show a distinct change in the preferred direction of magnetization from in-plane to out-of-plane when the coherent strain in ﬁlms changes from/C242% compressive on (001) MgAl 2O4to/C240.5% tensile on (001) MgGa 2O4substrates. Signiﬁcant deviations from the spin-only value (2.0) of the g-factor suggest spin-orbit effects and further sup- port our conclusion of strain-driven magnetic anisotropy in these ﬁlms. The low Gilbert dampingparameter of a¼5/C210 /C03in these ferrite ﬁlms, combined with their PMA, makes them promising for spintronic and frequency-agile microwave device applications. Published by AIP Publishing. https://doi.org/10.1063/1.5023118 Thin magnetic ﬁlms possessing perpendicular magnetic anisotropy (PMA) are essential material platforms for highdensity data recording technology 1–3and spin-torque-driven magnetic memory and logic devices.4–7While conventional spin-torque devices have been based on metallic ferromag-nets, more recently, control of magnetization by spin-orbittorque (SOT) due to the spin-Hall effect has been realized inmagnetic insulators. 8–14For instance, magnetic insulator thin ﬁlms with PMA, such as barium hexaferrite (BaFe 12O19)15 and thulium iron garnet (Tm 3Fe5O12),16have been used to demonstrate tuning of coercivity and magnetic switching bySOT. 12–14One potential merit of magnetic insulators is that their magnetic damping may be lower than that of metallicferromagnets. Since the threshold current density for spin-torque switching scales with the Gilbert damping parameter,low-damping magnetic insulators may be excellent platformsfor SOT driven memory and nano-oscillator devices.However, while thin ﬁlms of yttrium iron garnet (Y 3Fe5O12) without PMA exhibit a low Gilbert damping of a<10/C03,17–20 damping in thin ﬁlms of PMA insulators is rather high, e.g., a>0.01 with ferromagnetic resonance (FMR) linewidths on the order of /C2410 mT at 10 GHz in Tm 3Fe5O12.21,22It is there- fore an outstanding challenge to develop an insulating PMAmaterial system with low damping, which would be suitablefor power-efﬁcient spintronic memory and logic devices. Here, we achieve the desirable combination of PMA and low damping in pseudomorphic thin ﬁlms of insulating spinelNiZnAl-ferrite (NZAFO) via epitaxial strain. Speciﬁcally, weshow that tensile strain in NZAFO on MgGa 2O4(MGO)substrates causes the magnetic easy axis to point out of plane , in contrast with the previous report of easy-in-plane anisotropy in compressively strained NZAFO on MgAl 2O4(MAO) sub- strates.23We also show that the tensile-strained NZAFO/MGO ﬁlms exhibit a Gilbert damping parameter of a<5/C210/C03, which is among the lowest reported for thin-ﬁlm magneticinsulators with PMA. Our ﬁndings identify epitaxial spinel fer-rites as an attractive insulating platform for spintronic applica-tions that require both PMA and low damping. Compared to iron garnets and hexaferrites, spinel ferrites have a relatively simple crystal structure and are amenable to arange of chemical substitutions to tune their epitaxial strain andmagnetism. 24–29In this study, we focus on spinel NZAFO ﬁlms with nominal composition Ni 0.65Zn0.35Fe1.2Al0.8O4, derived from the parent compound NiFe 2O4.30Here, Zn2þsubstitution of Ni2þleads to softer magnetism, and Al3þsubstitution of Fe3þdecreases structural defects by improving the lattice match with the single-crystal spinel substrate (MAO or MGO).This particular composition of NZAFO enables high-qualitypseudomorphic ﬁlm growth, while keeping the Curie tempera-ture well above room temperature. We note that the Fe cationsin NZAFO are mostly in the 3 þvalence state, 23according to X-ray magnetic circular dichroism measurements on thin ﬁlmsof this compound deposited under conditions similar to this study. Thin epitaxial ﬁlms of NZAFO were deposited in a load- locked vacuum chamber by ablating a 2.5-cm-diameter sin-tered target of Ni 0.65Zn0.35Al0.8Fe1.2O4with a KrF excimer laser operating at 248 nm. The typical deposition parametersfor the ferrite ﬁlms are the laser pulse repetition rate of 4 Hz, laser ﬂuence of 4 J/cm 2, target to substrate distance of /C245 cm, substrate temperature of 600–700/C14C, oxygen pressure duringa)Author to whom correspondence should be addressed: ramesh.budhani@ morgan.edu 0003-6951/2018/113(8)/082404/5/$30.00 Published by AIP Publishing. 113, 082404-1APPLIED PHYSICS LETTERS 113, 082404 (2018) growth of /C24250–300 mTorr, and post-deposition cooling rate of 10/C14C/min to ambient temperature. These deposition parameters give rise to a growth rate of /C240.02 nm/pulse. The ﬁlms were deposited on 5 /C25m m2and/or 10 /C210 mm2(001) oriented single crystal substrates of MAO and MGO. While the MAO substrates were acquired from a commercial source, the MGO substrates were prepared from bulk crystalsobtained by the Czochralski method. 31Here, we focus on results from 19-nm and 114-nm thick ﬁlms deposited on MGO substrates. These ﬁlms are compared with 27-nm and 84-nm thick NZAFO ﬁlms on MAO substrates. The crystalline quality of the ﬁlms on these two types of substrates was compared by high resolution four circle x-ray diffractometry. In Fig. 1(a), we compare the x-2hscans of four ﬁlms, two each on one kind of substrate, in the angularrange covering the (004) reﬂection of the substrate and the ferrite. The diffraction proﬁles show prominent Laue oscilla- tions, from which the thickness values of the ﬁlms werededuced. The presence of Laue oscillations in the ﬁlms indi- cates an atomically ﬂat surface and interface. Taking note of the relative positions of the substrate and ﬁlm diffraction peaks and the lattice parameter of NZAFO inthe bulk form (a ¼0.8242 nm), we found that the ﬁlms are com- pressively strained on MAO (lattice parameter, a ¼0.8083 nm), whereas they are tensile strained on MGO (a ¼0.828 nm). Speciﬁcally, the c-axis lattice parameters of the 27-nm and84-nm thick ﬁlms on MAO are 0.8395 nm and 0.8373 nm,respectively, whereas those of the 19-nm and 114-nm thickﬁlms on MGO are 0.8169 nm and 0.8185 nm, respectively. The c-axis lattice parameter of each thicker ﬁlm being closer to the lattice parameter of bulk NZAFO suggests some structuralrelaxation with the increasing ﬁlm thickness. However, in thereciprocal space maps of the thicker ﬁlms shown in Fig. 1(b), the in-plane lattice parameter of the ﬁlm coincides with that of the substrate, suggesting fully strained pseudomorphic growth. While this observation appears incompatible with the differenceseen in the c-axis lattice parameters of the thick and thin ﬁlms,we believe the elasticity of the lattice accommodates this difference. The deformation of the lattice due to coherent strain affects the magnetic anisotropy in these ﬁlms signiﬁcantly as seen from the magnetization loops ( M(H)) presented in Fig.2. The saturation magnetization of the ﬁlms is in close agreement with the value reported for the bulk NZAFO. 30 This suggests minimal changes in the site occupancies ofmetal ions in the ﬁlms. The epitaxial ﬁlms of NZAFO on MAO display easy-plane anisotropy, with a very large out-of- plane saturation ﬁeld of >1 T. The magnitude of effective in- plane uniaxial anisotropy energy density, jK u,effj, calculated from the difference in the area ( M.H) of Figs. 2(a)and2(b) under the ﬁrst quadrant of M-Hresponse is 1.7 /C2105J/m3for the 27-nm thick ﬁlm on MAO and 2.3 /C2105J/m3for the 84- nm thick ﬁlm on MAO. These results compare well with thevalues of the anisotropy energy derived from ferromagneticresonance measurements, 23and its origin is primarily the strain related magneto-elastic interaction. FIG. 1. (a) X-ray x/C02Hdiffraction proﬁles of four NZAFO ﬁlms in the angular range covering the (004) reﬂection of the ferrite and the substrate. (b)Reciprocal space map of the ( /C221/C2215) asymmetric diffraction for the 84-nm ﬁlm on MAO and the 114-nm ﬁlm on MGO (marked b 1and b 2, respectively).FIG. 2. Static magnetization ( M) measured as a function of ﬁeld at 300 K in a SQUID magnetometer. Panels (a) and (b) are for 27-nm and 84-nm thick ﬁlms, respectively, on MAO. The in-plane ﬁeld in both the cases was directedalong the (110) direction. Panels (c) and (d) show the M(H) response of the 19-nm and 114-nm thick ﬁlms on MGO.082404-2 Budhani et al. Appl. Phys. Lett. 113, 082404 (2018)A dramatically different M(H) response is seen in ﬁlms deposited on MGO where the ferrite is subjected to a tensile stress due to the /C240.5% larger lattice parameter of the sub- strate. Here, the magnetization in the out-of-plane ﬁeld geom-etry [see Figs. 2(c)and2(d)] saturates at a much lower ﬁeld for both of the ﬁlms compared to the ﬁlms on MAO [Figs. 2(a) and2(b)], whereas the magnetization in the in-plane geometry shows hard-axis behavior. The M(H) response seen here is indicative of out-of-plane magnetic anisotropy. The value of K u,effcalculated from the difference in area enclosed by the M-H curve in the ﬁrst quadrant is /C251.4/C2104J/m3for the 19-nm and 114-nm thick ﬁlms and comparable to the val-ues reported for epitaxial Tm 3Fe5O12with PMA.21,22 To gain further insight into the magnetic anisotropy as well as spin-orbit coupling and magnetization dynamics inthese PMA ﬁlms deposited on MGO, we performed ferromag- netic resonance (FMR) measurements with a broadband coplanar-waveguide-based spectrometer with both in-planeand out-of-plane dc ﬁeld directions. Some typical examples of the FMR spectra are shown in Fig. 3.E a c hs p e c t r u mw a sﬁ t with the derivative of a sum of the symmetric and antisym-metric Lorentzians to extract the resonance ﬁeld, H fmr, and the half-width-at-half-maximum linewidth, DH. Under applied ﬁelds higher than the saturation ﬁeld, most spectra could be ﬁtwith a single mode of a Lorentzian derivative, implying that the observed signal arises from the uniform FMR mode. However, multiple modes were observed at all excita- tion frequencies in the out-of-plane FMR spectra of the 114- nm thick ﬁlm [Fig. 3(b)]. These modes are indicative of per- pendicular standing spin waves (PSSW) with indicesn¼odd, consistent with the case where the spins are pinned at both ﬁlm surfaces under uniform microwave excitation. Further studies of PSSW and their evolution with tempera-ture will be reported in the future. We quantify the Land /C19eg-factor and effective anisotropy ﬁelds by ﬁtting the frequency dependence of H fmrwith theappropriate Kittel equation.32For the out-of-plane case, the Kittel equation is f¼goplB hl0Hfmr/C0Meff ðÞ ; (1) where gopis the out-of-plane g-factor and l0Meff¼l0(Ms /C0Hpma) is the total uniaxial out-of-plane anisotropy ﬁeld, which includes the demagnetizing ﬁeld ( l0Ms/C250.13 T) and the perpendicular anisotropy ﬁeld, l0Hpma. The results of the ﬁts with Eq. (1)are shown in Fig. 4. We ﬁrst discuss our ﬁndings for Meff, while deferring the discussion of the g-factor to compare with the in-plane results. The fact that l0Meffisnegative signiﬁes that l0Hpma is large enough to overcome the shape anisotropy, thus allowing the magnetic easy axis to be out of plane. Theeffective PMA energy density jK u,effj¼l0jMeffjMs/2 is /C251.2/C2104J/m3for the 19-nm thick ﬁlm and /C251.4/C2104J/m3 for the 114-nm thick ﬁlm. These values are in excellent agreement with the anisotropy energy deduced from thestatic M(H) measurements of Fig. 2. This PMA ﬁeld stems from the epitaxial strain state of the ferrite ﬁlm, where the tetragonal distortion of the ferrite lattice subject to an in- plane biaxial strain of e biand an out-of-plane uniaxial strain ezzleads to a magneto-elastic response. The magnetoe- lastic coefﬁcient B1, which quantiﬁes the coupling, is given asB1¼l0HMEMs/(2(ezz/C0ebi)), where the magneto-elastic anisotropy ﬁeld HMEis assumed to be equivalent to the PMA ﬁeld HPMA. Following the linear elastic response assumption, we arrive at ebi¼0.005 and ezz¼/C00.007. These values yield B1¼1.5/C2106J/m3, which is in good agreement with previ- ously reported NZAFO on MAO.23 For the in-plane FMR, the Kittel equation is f¼giplB hl0HfmrþH4;ipcos 4U ðÞ1 2 /C2HfmrþMeffþ1 4H4;ip3þcos 4U ðÞ/C18/C19 1 2 ; (2) FIG. 3. Exemplary FMR spectra at f¼15 GHz measured under out-of-plane ﬁeld (a) and (b) and in-plane ﬁeld (c) and (d) for the 19-nm thick (a) and (c) and 114-nm thick (b) and (d) NZAFO ﬁlms on MGO. Note the presence ofdistinct multiple modes, attributed to perpendicular standing spin waves, in the out-of-plane FMR spectrum in the 114-nm thick ﬁlm (b). FIG. 4. Frequency dependence of resonance ﬁeld, Hfmr, with out-of-plane ﬁeld (a) and (b) and in-plane ﬁeld (c) and (d) for the 19-nm thick (a) and (c) and 114-nm thick (b) and (d) NZAFO ﬁlms on MGO. For the 114-nm thickﬁlm in an out-of-plane ﬁeld (b), the data for the uniform FMR ( n¼0) mode are shown.082404-3 Budhani et al. Appl. Phys. Lett. 113, 082404 (2018)where gipis the in-plane g-factor, H4,ipis the in-plane cubic anisotropy ﬁeld, and Uis the in-plane magnetization angle with respect to the [100] crystallographic axis. The in-planeFMR measurements were conducted with the ﬁeld appliedalong [100], i.e., U¼0. To constrain the number of free parameters in the ﬁt using Eq. (2),w eﬁ x l 0Meffto the value found from the out-of-plane analysis [Eq. (1), Figs. 4(a)and 4(b)]. We note that l0jHpmaj>200 mT whereas l0jH4,ipj <10 mT, conﬁrming that the perpendicular anisotropy from the tetragonal distortion is much stronger than the in-planeanisotropy from the in-plane four-fold crystallographic struc-ture of the ﬁlm. From both out-of-plane and in-plane mea-surements, as shown in Fig. 4, we ﬁnd that the g-factor of >2.1 is systematically greater than the spin-only value of 2.0. This supports the existence of spin-orbit coupling thatgives rise to the strain-induced anisotropy. However, theg-factors found here in tensile-strained NZAFO ﬁlms on MGO are smaller than g/C252.3 found previously in compres- sively strained ﬁlms on MAO. 23This discrepancy in gpossi- bly arises from the different strain state, which may change the occupation of orbitals and hence the magnitude of orbitalangular momentum of magnetic cations. We now quantify the Gilbert damping parameter, a, from the frequency dependence of DH, using the relation DH¼DH 0þh gl0lBaf; (3) whereDH0is the zero-frequency linewidth and g¼goporgip depending on the ﬁeld direction. Figure 5summarizes our results. We ﬁnd a/C254/C210/C03for both the 19- and 114-nm thick NZAFO ﬁlms in out-of-plane and in-plane ﬁelds. Thedamping in NZAFO thin ﬁlms with PMA is signiﬁcantlylower than what has been reported previously in insulatingPMA thin ﬁlms. 21Our ﬁndings also show that it is possible to simultaneously achieve PMA and low damping in pseu- domorphic spinel ferrite thin ﬁlms, thus introducing anadditional family of magnetic insulators for spin-torque- driven applications. While epitaxial magnetic ﬁlms in general can exhibit sub- stantial non-Gilbert damping (e.g., due to two-magnon scatter-ing) mediated by defects, 33–35damping in NZAFO ﬁlms is dominated by the Gilbert mechanism [captured by Eq. (3)]. First, the linear scaling of linewidth with the wide range of fre-quencies suggests that two-magnon scattering does not play arole. The Gilbert damping parameters for the in-plane and out-of-plane conﬁgurations are comparable, which also suggests anegligible role played by two-magnon scattering. However, wenote that the damping parameters for the 19-nm thick NZAFOon MGO are higher than those for a ﬁlm of comparable thick-ness on MAO ( a/C253/C210 /C03).23The exact reason for this dis- crepancy is unknown but may be due to the differentmicrostructure of the thin ﬁlms grown on MGO compared to MAO. In summary, it is shown that a tensile strain in NZAFO epitaxial ﬁlms grown on the (001) surface of MGO spinelresults in perpendicular magnetic anisotropy with an energydensity of /C251.4/C210 4J/m3, derived from static magnetometry and frequency dependent FMR measurements. This is in con-trast with compressively strained NZAFO on MAO that exhib-its large easy-plane anisotropy. The NZAFO ﬁlms with PMAdisplay among the lowest Gilbert damping ( a<5/C210 /C03) reported so far in insulating PMA thin ﬁlms. This demonstra- tion of low-damping spinel ferrite ﬁlms with sizable PMA is promising for future spintronic applications such as spin-orbit-torque nano-oscillators and high density magnetic memorydevices. This material is based upon work supported by the Air Force Ofﬁce of Scientiﬁc Research under Award No.FA9550-15RXCOR198. R.C.B. thanks the NationalResearch Council, Washington DC, for the award of a senior fellowship. Work at Stanford was supported by the Vannevar Bush Faculty Fellowship Program sponsored by the BasicResearch Ofﬁce of the Assistant Secretary of Defense forResearch and Engineering and funded by the Ofﬁce of NavalResearch through Grant No. N00014-15-1-0045. 1S. Iwasaki and K. Ouchi, IEEE Trans. Magn. MAG 14, 849 (1978). 2A. Moser, K. Takano, D. T. Margulier, M. Albrecht, Y. Sonobe, Y. Ikeda, S. Sun, and E. E. Fullerton, J. Phys. D: Appl. Phys. 35, R157 (2002). 3S. N. Piramanayagam, J. Appl. Phys. 102, 011301 (2007). 4J. C. Slonczewski and J. Magn, Magn. 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1.4991663.pdf	Indirect excitation of self-oscillation in perpendicular ferromagnet by spin Hall effect Tomohiro Taniguchi Citation: Appl. Phys. Lett. 111, 022410 (2017); doi: 10.1063/1.4991663 View online: http://dx.doi.org/10.1063/1.4991663 View Table of Contents: http://aip.scitation.org/toc/apl/111/2 Published by the American Institute of Physics Articles you may be interested in Investigation of the Dzyaloshinskii-Moriya interaction and room temperature skyrmions in W/CoFeB/MgO thin films and microwires Applied Physics Letters 111, 022409 (2017); 10.1063/1.4991360 Inversion of the domain wall propagation in synthetic ferrimagnets Applied Physics Letters 111, 022407 (2017); 10.1063/1.4993604 Spin pumping torque in antiferromagnets Applied Physics Letters 110, 192405 (2017); 10.1063/1.4983196 Electrical switching of the magnetic vortex circulation in artificial multiferroic structure of Co/Cu/PMN-PT(011) Applied Physics Letters 110, 262405 (2017); 10.1063/1.4990987 Field-free spin-orbit torque switching of composite perpendicular CoFeB/Gd/CoFeB layers utilized for three- terminal magnetic tunnel junctions Applied Physics Letters 111, 012402 (2017); 10.1063/1.4990994 Reduction in write error rate of voltage-driven dynamic magnetization switching by improving thermal stability factor Applied Physics Letters 111, 022408 (2017); 10.1063/1.4990680Indirect excitation of self-oscillation in perpendicular ferromagnet by spin Hall effect Tomohiro Taniguchi National Institute of Advanced Industrial Science and Technology (AIST), Spintronics Research Center, Tsukuba 305-8568, Japan (Received 25 May 2017; accepted 21 June 2017; published online 14 July 2017) A possibility to excite a stable self-oscillation in a perpendicularly magnetized ferromagnet by the spin Hall effect is investigated theoretically. It had been shown that such self-oscillation cannot be stabilized solely by the direct spin torque by the spin Hall effect. Here, we consider adding anotherferromagnet, referred to as pinned layer, on the free layer. The pinned layer provides another spin torque through the reﬂection of the spin current. The study shows that the stable self-oscillation is excited by the additional spin torque when the magnetization in the pinned layer is tilted from theﬁlm plane. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4991663 ] It has been experimentally demonstrated that the spin Hall effect (SHE) 1,2in nonmagnetic heavy metals generates pure spin current ﬂowing in the direction perpendicular to an external voltage and excites spin torque on a magnetization in an adjacent ferromagnet.3–12The spin torque induces the magnetization dynamics such as switching and self- oscillation. Substantial efforts have been made to develop practical devices based on the spin Hall effect, for example,magnetic random access memory, microwave generator, high sensitivity sensor, and new direction such as bio- inspired computing. 13,14 The spintronics devices based on the spin Hall effect, however, face a serious problem because of the geometricalrestriction of the spin torque direction. Let us assume that an electric current ﬂows in the nonmagnet along xdirection, while the ferromagnet is set in zdirection. Then, the direc- tion of the spin polarization in the spin current generated by the spin Hall effect is ﬁxed to ydirection. The device designs and performances are subject to limitation due to suchrestriction of spin polarization. For example, the magnetiza-tion switching of a perpendicular ferromagnet solely by the spin Hall effect is impossible because the spin torque does not break the symmetry with respect to the ﬁlm plane,whereas a perpendicular ferromagnet is suitable for a high density memory. Using external magnetic ﬁeld, 7tilted anisotropy,15,16or exchange bias17has been proposed to overcome this issue. It was also shown that an excitation of the self-oscillation in a perpendicular ferromagnet solely by the spin Hall effect is impossible due to the symmetry,18 although a large amplitude oscillation excited in a perpendic- ular ferromagnet is preferable for an enhancement of emis- sion power. Contrary to the case of the switching, thisproblem has not been solved yet. The purpose of this letter is to investigate the possibility to excite the self-oscillation in a perpendicular ferromagnet by the spin Hall effect. The work is motivated by recent theoreti- cal studies on the spin-orbit torque in the presence of anadditional ferromagnet to the free layer. 19–21These theories predict the existence of additional torques and/or enhance- ment of the spin accumulation. Here, we consider addinganother ferromagnet, referred to as the pinned layer, on thetop of the free layer. The pinned layer provides an additional spin torque due to the reﬂection of the spin current at theinterface and the diffusion in bulk. This additional torque has a different angular dependence from the conventional spin-orbit torque and results in the excitation of the self-oscillation. In the following, we describe the system in thisstudy and show the spin torque formula applied in the geome- try. Then, we investigate the magnetization dynamics by solv- ing the Landau-Lifshitz-Gilbert (LLG) equation numerically.It is shown that the self-oscillation can be excited when themagnetization in the pinned layer is tilted from the ﬁlm plane. The system under consideration is schematically shown in Fig. 1(a). The bottom layer is a nonmagnetic heavy metal showing the spin Hall effect. Applying an external voltage along the in-plane ( x) direction, the electric current with the density J 0is converted to pure spin current ﬂowing in zdirection. We also consider placing a pinned layer onto the free layer. The spin current excites the magnetizationdynamics in the free layer through the spin-transfer effect.We denote the unit vectors pointing in the magnetization direction of the free and pinned layers as mandp, respec- tively. The magnetization dynamics in the free layer isdescribed by the LLG equation as dm dt¼/C0cm/C2HþTþam/C2dm dt; (1) where cand aare the gyromagnetic ratio and the Gilbert damping constant, respectively. The magnetic ﬁeld Hcon- sists of a perpendicular anisotropy ﬁeld and the stray ﬁeldfrom the pinned layer, and is given by H¼/C0H dpyeyþ2HdpzþðHK/C04pMÞmz ½/C138 ez; (2) where Hdcharacterizes the magnitude of the stray ﬁeld, whereas HKand 4 pMwith the saturation magnetization Mare the crystalline and shape (demagnetization) ﬁelds, respec-tively. The magnitude of the stray ﬁeld found in the experi-ments is typically on the order 100 Oe, 7which is consistent with a theoretical evaluation; see the supplementary material . Thus, in this paper, we use the value of Hd¼100 Oe in the following calculations. We assume that HK>4pM, and the 0003-6951/2017/111(2)/022410/5/$30.00 Published by AIP Publishing. 111, 022410-1APPLIED PHYSICS LETTERS 111, 022410 (2017) free layer consequently becomes perpendicularly magnetized in the absence of the pinned layer. The spin torque in Eq. (1) isT, which in the present geometry is given by T¼/C0/C22hg1J0 2eMdm/C2ey/C2m ðÞ /C0/C22hg2J0 2eMdmym/C1pðÞ m/C2p/C2m ðÞ ½/C138 1/C0k2m/C1pðÞ2; (3) where dis the thickness of the free layer, whereas eð>0Þis the elementary charge. The ﬁrst term on the right hand side of Eq. (3)is the conventional spin Hall torque directly excited by the spin current generated by the bottom nonmag-net. For convention, we call this torque the direct spin torque(DST) [see Fig. 1(a)]. On the other hand, the second term arises from the spin current transmitted through the free layer and reﬂected by the pinned layer, which is also schematically shown in Fig. 1(a). In a same manner, we call this torque the reﬂection spin torque (RST). Before solving the LLG equation, let us explain the physical meaning, as well as its derivation, of these spin tor- ques in this system. The spin torque has been calculated the-oretically by using several methods such as the ballistic spintransport theory with the interface scattering, 22,23the ﬁrst- principles calculations,24,25the Boltzmann approach,26,27 and the diffusive spin transport theory in bulk.28,29Although the parameters characterizing the spin torque depend on themodels, these theories basically deduce the same angulardependence of the spin torque. The derivation of Eq. (3)in the present geometry using the diffusive spin transport the-ory in bulk and the interface scattering theory is summarized in the supplementary material . The spin torque efﬁciency g 1 of the direct spin torque is proportional to the spin Hall angle in the bottom nonmagnet. It also depends on the interfaceand bulk properties. Note that the direct spin torque isexcited by an absorption of the transverse component (per- pendicular to m) of a spin current generated by the spin Hall effect in the bottom nonmagnet. The spin polarization of thespin current points to the ydirection, and thus, the direct spin torque moves the magnetization parallel or antiparallel to theyaxis, as schematically shown in Fig. 1(b). On the other hand, the reﬂection spin torque arises from the spin current passing through the free layer. Such spincurrent is again injected into the free layer from the top inter- face due to the reﬂection from the top interface and diffusivespin transport in the pinned layer. Note that the spin current generated in the bottom nonmagnet has the spin polarization in the ydirection. Because of the absorption of the transverse component of the spin current from the bottom nonmagnetmentioned earlier, the reﬂection spin torque includes the fac- torm y; i.e., when my¼0, the spin current generated in the bottom nonmagnet is completely absorbed to the free layerat the bottom interface, and therefore, the reﬂection spin tor- que becomes zero because the spin current passing through the free layer is unpolarized. Similarly, the spin polarizationparallel to psurvives during the transport through the pinned layer. As a result, the reﬂection spin torque also includes the factor m/C1pon the numerator in Eq. (3). Moreover, the direc- tion of the reﬂection spin torque is given by m/C2ðp/C2mÞ,a s schematically shown in Fig. 1(b), in comparison to that of the direct spin torque pointing to the direction of m/C2ðe y/C2mÞ. The spin torque efﬁciency g2and the parame- terkdetermining the angular dependence characterize the amount of the spin current reinjected from the top interfaceto the free layer. Their values depend not only on the spin Hall angle in the bottom nonmagnet but also on the interface and bulk properties of the pinned layer, such as spin diffu-sion length and mixing conductance. The details of the deri-vation of the reﬂection spin torque, as well as the relation to material parameters in the diffusive model, are summarized insupplementary material . We note that the present model is applicable to a metal- lic multilayer. When a spacer between the free and pinned layers is replaced by an oxide barrier, as in the case of amagnetic tunnel junction, the spin current cannot penetrateinto the pinned layer, and thus, the reﬂection spin torque becomes zero. When an electric voltage is applied along the perpendicular direction, as in the case of the experiment toobtain an electric signal through tunnel magnetoresistanceeffect, 7a spin current will be driven between the free and pinned layer, and a torque similar to the reﬂection spin tor- que will appear. The angular dependence of such a torque,however, might be different from the reﬂection spin torque. We investigate the magnetization dynamics in this geometry by solving Eq. (1)numerically. The values of the parameters are brought from typical experimental FIG. 1. (a) Schematic view of the system in this study. The spin Hall effect (SHE) in the bottom nonmagnet injects pure spin current into the free layer. T he spin current reﬂected by the pinned layer is again injected into the free layer. The direct and reﬂection spin torques are referred to as DST and RST, for simplic- ity. (b) Schematic view of the ﬂow of the spin current and the direction of spin torques. Passing through the free layer from bottom to top, the spin polar ization transverse to mis absorbed and excites direct spin torque. The spin polarization of the reﬂected spin current is parallel (or antiparallel) to p. The transverse component of the reﬂected spin current is absorbed to the free layer and excites the reﬂection spin torque.022410-2 Tomohiro Taniguchi Appl. Phys. Lett. 111, 022410 (2017)values in spin torque oscillator,30i.e., M¼1448.3 emu/c.c., HK¼18:6 kOe, d¼2n m , c¼1:764/C2107rad/(Oe s), and a¼0:005. The spin torque parameters are g1¼0:14;g2 ¼0:07, and k¼0:82, respectively; see the supplementary material for the evaluations of these parameters. The magne- tization in the pinned layer is p¼0 /C0sinhp coshp0 B@1 CA; (4) where hpis the tilted angle from zaxis. We note that efforts have been made to realize a tilted state ( hp6¼0/C14nor 90/C14) of a magnetization in a ferromagnet by making use of a higher-order anisotropy or an interlayer exchange couplingbetween a perpendicular and an in-plane magnetized ferro- magnets. 31–34The initial state is the energetically stable state given by mð0Þ¼ð 0;sinh0;cosh0Þ, where h0is the tilted angle of the magnetization from zaxis, which minimizes the energy density given by E¼/C0MÐdm/C1H¼/C0M½Hd sinhpmyþ2HdcoshpmzþðHK/C04pMÞm2 z=2/C138. We note that the magnetization in equilibrium is destabilized by the spin torques when the current density is larger than a critical value given by Jc¼2aeMd /C22hPHXþHY 2/C18/C19 ; (5) where an effective spin polarization Pis derived as P¼ g1sinh0þg2 1/C0k2p2 Zp2 Zsin2h0/C0n 2pX/C18/C19 : (6) Here, pZ¼/C0sinðhpþh0Þand pX¼cosðhpþh0Þ, whereas n¼KpZsinh0þpXsinh0þpZcosh0withK¼2k2pXpZ= ð1/C0k2p2 ZÞ. The ﬁelds HXandHYin Eq. (5)are expressed as HX¼Hdð2 cos hpcosh0þsinhpsinh0Þ þðHK/C04pMÞcos 2 h0; (7) HY¼Hdð2 cos hpcosh0þsinhpsinh0Þ þðHK/C04pMÞcos2h0: (8) We note that the ferromagnetic resonance (FMR) frequency is related to HXandHYas fFMR¼c 2pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HXHYp : (9) The derivation of Eq. (5)based on the linearized LLG equa- tion is summarized in the supplementary material . Equation (5)diverges when the magnetization in the pinned layer points to the perpendicular direction, hp¼0. This fact indi- cates that the linearized LLG equation is inapplicable to study the instability analysis of the magnetization dynamics. In this case, the critical current will be independent of the damping constant, as studied in Ref. 35, and does not show self-oscillation. Figures 2(a) and2(b) show examples of the magnetiza- tion dynamics obtained from Eq. (1),w h e r e hpis 10/C14in (a) and 60/C14in (b). The current density is 20 MA/cm2in thesecalculations, whereas the critical current density estimated from Eq. (5)is 9.5 MA/cm2forhp¼10/C14and 7.4 MA/cm2for hp¼60/C14. As shown, a stable oscillation is excited for hp¼10/C14; this is the main ﬁnding in this study. The magneti- zation precesses around an axis slightly tilted from zaxis. The oscillation frequency is 1.60 GHz, which is slightly smaller than the FMR frequency, 1.72 GHz. The relaxation time to theself-oscillation time is about 50 ns. We note here that theinverse of the relaxation time is proportional to the current, 36 and therefore, the relaxation time will be shortened by apply-ing a large current. On the other hand, when h p¼60/C14,t h e magnetization switches to the direction antiparallel to the y direction without showing a self-oscillation, as shown in Fig. 2(b), for a current larger than the critical current. This behav- ior is similar to that excited solely by the direct spin torquestudied in Ref. 18. The current dependences of the oscillation frequency for several values of h pare summarized in Fig. 3. Random tor- que,/C0cm/C2h, originated from thermal ﬂuctuation is added to the right hand side of Eq. (1)to evaluate the magnoise frequency below the threshold. The components of the random torque satisfy the ﬂuctuation-dissipation theorem,hh kðtÞh‘ðt0Þ i¼½ 2akBT=ðcMVÞ/C138dk‘dðt/C0t0Þ, where the tem- perature Tand the cross-section area S(V¼Sd) are assumed as 300 K and p/C2602nm2,30respectively. The oscillation frequency is estimated from the Fourier transformation ofm yðtÞ, where the spectra are averaged over 103realizations. When the current density is smaller than the critical current density, the magnetization oscillates around the equilibrium state, and thus, the magnoise appears around the FMRFIG. 2. Time evolutions of the magnetization ( mxin red, myin blue, and mz in black) for (a) hp¼10/C14and (b) 60/C14. The current density is 20 MA/cm2. The inset in (a) shows the oscillation of the magnetization in a steady state.022410-3 Tomohiro Taniguchi Appl. Phys. Lett. 111, 022410 (2017)frequency. When hp¼90/C14and the current is larger than the critical value, the magnetization switches its direction to the negative ydirection without showing a self-oscillation, simi- lar to that shown in Fig. 2(b). As a result, a discontinuous change of the oscillation frequency appears near the critical value, Jc¼10:1 MA/cm2. For hp¼30/C14;40/C14, and 50/C14, the magnetization shows the self-oscillation when the current islarger than the critical value. In the self-oscillation, the oscil- lation frequency decreases with increasing current magni- tude. Above certain values of the current, however, the magnetization switching occurs, and thus, the discontinuous drops of the oscillation frequency are observed. On the otherhand, self-oscillations are observed for the present range of the current ( J 0/C2050 MA/cm2) when hp¼10/C14and 20/C14 (a switching for hp¼10/C14occurs at a sufﬁciently large cur- rent J0>115 MA/cm2). These results indicate that the self- oscillation is stably excited when the magnetization in the pinned layer is tilted, particularly in close range, from theperpendicular ( z) axis. A possible reason why a current over which the stable self-oscillation terminates becomes smaller when h pbecomes larger is due to the characteristics of angular dependence of the reﬂection spin torque. As men- tioned above, the angular dependence of the reﬂection spin torque includes the term m/C1p. As can be seen in Fig. 2(a), the self-oscillation is excited closely around the zaxis. Consequently, the magnitude of the reﬂection spin torque becomes small for a large hp(p!ey), making the effect of the reﬂection spin torque on the oscillation small and region of the stable self-oscillation narrow. We note, however, that the self-oscillation cannot be excited when the magnetizationin the pinned layer completely points to the zdirection, as mentioned earlier. Finally, let us discuss the role of the reﬂection spin tor- que in the above-mentioned results. We emphasize that the reﬂection spin torque plays a key role in stabilizing the self- oscillation. To understand this argument, we revisit the theo-retical conditions to excite the self-oscillation studied in our previous work. 18First, the spin torque should supply a ﬁnite positive energy to the free layer during the oscillation to cancel the energy dissipation due to the damping torque. In the con- ventional geometry of the spin Hall devices consisting of asingle perpendicular ferromagnet and in the absence of an external ﬁeld, the energy supplied by the spin torque becomes totally zero due to the axial symmetry of the oscil-lation orbit. Therefore, a self-oscillation cannot be excited. 18 A way to solve this problem is to apply an external mag- netic ﬁeld. The ﬁeld breaks the symmetry of the oscillationorbit, and makes the supplied energy by the spin torque ﬁnite. In this work, the stray ﬁeld from the pinned layer plays the role of such external ﬁeld. This is, however, not sufﬁcientenough to stabilize the self-oscillation. The second condition necessary to stabilize the self-oscillation is that a current magnitude should be larger than the critical current destabi-lizing the equilibrium state. 18If this condition is unfulﬁlled, the free layer undergoes the magnetization switching above the critical current without showing a self-oscillation,18as in the case shown in Fig. 2(b). It was shown in Ref. 18that the direct spin torque is not sufﬁcient to stabilize the self-oscillation in the spin Hall geometry because the second condition is not satisﬁed evenin the presence of an external ﬁeld. On the other hand, in the present study, the self-oscillation is excited, as shown in Fig. 2(a). This fact indicates that the reﬂection spin torque fulﬁlls the second condition and stabilizes the self-oscillation. One might be interested in proving the stabilization of the self-oscillation by the reﬂection spin torque analytically,instead of the numerical approach done in the earlier calcula- tions. It is relatively easy to conﬁrm whether the ﬁrst condi- tion to stabilize the self-oscillation is satisﬁed by focusing onthe symmetries of the oscillation orbit and the angular depen-dence of the spin torque. On the other hand, as far as we know, it cannot be easily conﬁrmed to fulﬁll the second con- dition. It should individually be examined for each system.In principle, the second condition can be studied theoreti- cally by deriving an analytical formula of mcorresponding to an oscillation orbit and solving the energy balance equa-tion. 37Then, it becomes, for example, possible to derive ana- lytical conditions on the material parameters to stabilize the self-oscillation. These calculations are, however, generallycomplicated, in practice, except for a few cases. In the pre- sent system, the solution of the oscillation orbit can be described by the elliptic functions, in principle. It involves,however, complex mathematics, and thus, analytical calcula-tions to study the satisfaction of the second condition are beyond the scope of this paper. In conclusion, the magnetization dynamics in the spin Hall geometry in the presence of an additional ferromagnet was studied theoretically. In addition to the direct spin torque by the spin Hall effect, the additional ferromagnet providesanother spin torque through the reﬂection of the spin current. Solving the LLG equation with the direct and reﬂection spin torques numerically, it was found that a stable self-oscillation can be excited when the magnetization in the pinned layer is tilted from the ﬁlm-plane. Seesupplementary material for the derivations of Eqs. (2),(3), and (5). FIG. 3. Current dependences of the oscillation frequency of the magnetiza- tion at ﬁnite temperature for hp¼10/C14(red square), 20/C14(green square), 30/C14 (blue circle), 40/C14(magenta circle), 50/C14(turquoise triangle), and 90/C14(purple triangle). The critical current densities at zero temperature for these hpare 9.5, 6.3, 5.8, 6.1, 6.7, and 10.1 MA/cm2.022410-4 Tomohiro Taniguchi Appl. Phys. Lett. 111, 022410 (2017)The author is grateful to Takehiko Yorozu and Hitoshi Kubota for valuable discussion. The author is also thankful to Satoshi Iba, Aurelie Spiesser, Hiroki Maehara, and Ai Emura for their support and encouragement. This work wassupported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) 16K17486. 1M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971). 2J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 3T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007). 4K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. 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1.2839900.pdf	Measurement of vocal-tract inﬂuence during saxophone performance Gary P . Scavone,a/H20850Antoine Lefebvre, and Andrey R. da Silva Computational Acoustic Modeling Laboratory, Centre for Interdisciplinary Research in Music Media and Technology, Music Technology, Schulich School of Music, McGill University, Montreal, Québec, CanadaH3A 1E3 /H20849Received 5 July 2007; revised 11 January 2008; accepted 13 January 2008 /H20850 This paper presents experimental results that quantify the range of inﬂuence of vocal tract manipulations used in saxophone performance. The experiments utilized a measurement system thatprovides a relative comparison of the upstream windway and downstream air column impedancesunder normal playing conditions, allowing researchers and players to investigate the effect ofvocal-tract manipulations in real time. Playing experiments explored vocal-tract inﬂuence over thefull range of the saxophone, as well as when performing special effects such as pitch bending,multiphonics, and “bugling.” The results show that, under certain conditions, players can create anupstream windway resonance that is strong enough to override the downstream system incontrolling reed vibrations. This can occur when the downstream air column provides only weaksupport of a given note or effect, especially for notes with fundamental frequencies an octave belowthe air column cutoff frequency and higher. Vocal-tract inﬂuence is clearly demonstrated when pitchbending notes high in the traditional range of the alto saxophone and when playing in thesaxophone’s extended register. Subtle timbre variations via tongue position changes are possible formost notes in the saxophone’s traditional range and can affect spectral content from at least800–2000 Hz. © 2008 Acoustical Society of America. /H20851DOI: 10.1121/1.2839900 /H20852 PACS number /H20849s/H20850: 43.75.Ef, 43.75.Pq, 43.75.St, 43.75.Yy /H20851NFH /H20852 Pages: 2391–2400 I. INTRODUCTION Since the late 1970s, there has been signiﬁcant interest in understanding the role and inﬂuence of a player’s vocaltract in wind instrument performance. Acousticians generallyagree that, in order for such inﬂuence to exist in reed-valveinstruments, the player’s upstream windway must exhibit in-put impedance maxima of similar or greater magnitude thanthose of the downstream air column. 1Most musicians concur that they can inﬂuence sound via vocal-tract manipulations,though there is less consensus in terms of the extent of suchinﬂuence or the speciﬁcs of how this is done. 2A number of previous studies have been reported but attempts to demon-strate and/or quantify vocal-tract inﬂuence have not beenconclusive. These analyses have been complicated by thefact that measurement of the upstream windway conﬁgura-tion and impedance are difﬁcult under performance condi-tions. It is the aim of this study to provide experimental results that substantiate the discussion on the role of vocal tractmanipulations in wind instrument performance. The resultsare obtained using a measurement system that allows theanalysis of vocal tract inﬂuence in real time during perfor-mance. Most previous acoustical studies of vocal-tract inﬂuence have focused on the measurement of the input impedance ofplayers’ upstream windways while they simulate or mimicoral cavity shapes used in playing conditions. 1,3–6These measurements have then been compared with the input im-pedance of the downstream instrument air column to show instances where the vocal tract might be able to inﬂuence thereed vibrations. The majority of these investigations are inagreement with regard to the existence of an adjustable up-stream wind-way resonance in the range of 500–1500 Hz,which corresponds to the second vocal-tract resonance. 3–6 On the other hand, musicians do not appear to manipulate the ﬁrst vocal-tract resonance, which is typically below about300 Hz. 6In Ref. 6, players reported using a fairly stable vocal-tract shape for most normal playing conditions, withupstream manipulations taking place mainly in the altissimoregister and for special effects. A brief review of previousstudies and a discussion of their limitations is provided byFritz and Wolfe. 6 Numerical investigations have also been reported that couple an upstream windway model to a complete wind in-strument system. 7–10Simpliﬁed, single-resonance models of a player’s windway have been shown to reproduce bugling,pitchbend, multiphonics, and glissando characteristics inclarinet and saxophone models, effects generally associatedwith vocal-tract inﬂuence. 8,9 Many performers have used x-ray ﬂuoroscopic or endo- scopic approaches to analyze the vocal tract shapes and ma-nipulations used while playing their instruments. 2,11–15These studies have reported some general trends in vocal-tract con-ﬁgurations for different registers, produced mainly by varia-tions of the tongue position, but have not well addressed theunderlying acoustic principles involved. A fundamental limitation of previous acoustical investi- gations is related to the fact that the measurements were notconducted in real time. Subjects were required to mime and a/H20850Electronic mail: gary@music.mcgill.ca. J. Acoust. Soc. Am. 123 /H208494/H20850, April 2008 © 2008 Acoustical Society of America 2391 0001-4966/2008/123 /H208494/H20850/2391/10/$23.00hold a particular vocal-tract setting for up to 10 s, a task that is likely difﬁcult and uncomfortable for the player. Despitethe results of Fritz and Wolfe 6that indicate players’ have consistent “muscle” /H20849or procedural /H20850memory, it is not clear that subjects can accurately reproduce the exact vocal-tractconﬁgurations of interest in this study without playing theinstrument because auditory, and perhaps vibrotactile, feed-back is important in ﬁne-tuning and maintaining an oral cav-ity conﬁguration. Further, these methods only provide datafor the held setting, without indicating the time-varying char-acteristics of vocal-tract manipulations. This last issue re-mains for a recently reported approach that allows the up-stream impedance to be measured while the instrument isplayed. 16 This paper addresses the previously mentioned limita- tions by providing a running analysis of vocal-tract inﬂuenceover time /H20849not solely at discrete moments in time /H20850. This is achieved using a measurement system embedded in an E /flat alto saxophone that allows a relative comparison of the up-stream windway and downstream air column impedances un-der normal playing conditions. The same approach couldalso be applied to clarinets, though we chose to concentrateon saxophones given our own playing experience with them. This paper is organized as follows: Section II describes the measurement approach, including the system and proce-dures used to evaluate vocal-tract inﬂuence. Section III pre-sents the measured data for playing tasks involving tradi-tional and extended registers, pitch bend, bugling,multiphonics, and timbre variation. Finally, Sec. IV con-cludes with an analysis of the results in the context of saxo-phone performance practice. II. MEASUREMENT APPROACH Measurements made for this study were based on conti- nuity of volume ﬂow at the reed junction, Uu=Ud=Pu Zu=Pd Zd⇒Zu Zd=Pu Pd, /H208491/H20850 where uanddsubscripts represent upstream and downstream quantities, respectively. This expression is attributed ﬁrst toElliot and Bowsher 17in the context of brass instrument mod- eling. This implies that a relative measure of the upstream and downstream impedances can be obtained from the en-trance pressures in the player’s mouth and the instrumentmouthpiece, which is sufﬁcient to indicate when vocal-tractinﬂuence is being exerted. That is, it is assumed that theupstream system can be inﬂuential when values of Z uare close to or greater than those of Zdand this can be deter- mined from their ratio without knowing the distinct value ofeach. Because measurements of the mouth and mouthpiecepressures can be acquired while the instrument is being played, this approach allows the player to interact normallywith the instrument. Further, as modern computers can cal-culate and display the pressure spectra in real time, playersand researchers can instantly see how vocal tract changesdirectly affect the upstream impedance. In this context, the reed is assumed to be primarily con- trolled by the pressure difference across it and thus, a mea-surement of these two pressures can provide a good indica- tion of how the reed oscillations are inﬂuenced by the twosystems on either side of it. While it may not be possible touse this approach to reconstruct or calculate an upstreamimpedance characteristic from measurements of Z d,Pd, and Pu/H20849as attempted by Wilson5/H20850, it is valid to indicate instances of upstream inﬂuence. This argument ignores the possibleeffect of hydrodynamic forces on the behavior of the reeddue to unsteady ﬂow. 18–20 A. Measurement system The measurement approach used in this study requires the simultaneous acquisition of the “input” pressures on theupstream and downstream sides of the instrument reed undernormal playing conditions. Therefore, it was necessary toﬁnd a nonintrusive method for measuring pressures in theinstrument mouthpiece and the player’s mouth near the reedtip. Such measurements are complicated by the fact that thesound pressure levels /H20849SPL /H20850in a saxophone mouthpiece un- der playing conditions can reach 160 dB or more 21and that the dimensions of the alto saxophone mouthpiece requirerelatively small microphones to allow normal playing condi-tions and to avoid interference with the normal reed/mouthpiece interaction. After tests with a prototype system, 22a special saxo- phone mouthpiece was developed as shown in Fig. 1.A n Endevco 8510B-1 pressure transducer was threaded throughthe top of the mouthpiece 55 mm from the mouthpiece tip toobtain internal pressure values. For pressure measurementsinside the player’s mouth, a groove was carved in the top ofthe saxophone mouthpiece for an Endevco 8507C-1 minia-ture pressure transducer of 2.34 mm diameter such that thetransducer tip was positioned 3 mm behind the front edge of FIG. 1. Saxophone mouthpiece system. 2392 J. Acoust. Soc. Am., Vol. 123, No. 4, April 2008 Scavone et al. : Vocal-tract inﬂuence in saxophone performancethe mouthpiece. This position is well within the mouth given that the player’s teeth typically rest anywhere from about 13to 20 mm from the mouthpiece tip. Both Endevco transduc-ers are rated for maximum SPLs between 170 and 180 dBand were found to be unaffected by moisture. Ideally, both microphones would be located at the front edge of the mouthpiece /H20849inside and outside /H20850, which repre- sents the input to both the downstream and upstream aircolumns. Wilson 5experimented with two mouthpiece trans- ducer locations on a clarinet and selected the more distantposition /H2084939 mm /H20850because of noise due to unsteady ﬂow nearer the reed tip. The results from a digital waveguide simulation using a cylinder-cone model 23suggest that the signal recorded by the 8510B-1 transducer at a distance of55 mm from the tip differs from the downstream input im-pedance peak values by less than 2 dB for frequencies up to1500 Hz and less than 3.7 dB for frequencies up to 2000 Hz.Thus, the microphone positioning does not represent a limi-tation, considering that the frequency range analyzed in thisstudy does not exceed 2000 Hz and that we are mainly in- terested in the use of a vocal-tract resonance that runs fromabout 500 to 1500 Hz. The pressure transducers were connected to an Endevco 136 differential voltage ampliﬁer and the signals from therewere routed to a National Instruments /H20849NI/H20850PCI-4472 dy- namic signal acquisition board. The acquisition card samplerate was set to 12 000 Hz. An NI LABVIEW interface was designed to allow real-time display of the spectra of the twopressure signals. The Endevco transducers were calibratedrelative to one another prior to the experiment, as describedin the Appendix. To help distinguish between vocal tract and embouchure changes, a small circular /H2084912.7 mm diameter /H20850force sensing resistor /H20849FSR /H20850made by Interlink Electronics was placed un- der the cushion on the top of the saxophone mouthpiece toobtain a relative measure of the upper teeth force /H20849see Fig. 1/H20850. The time-varying sensor voltages were input to the signal acquisition board for storage and to provide a running dis-play of “embouchure movement” in the LABVIEW interface. Although this setup provided no data on movements of thelower lip, normal embouchure adjustments involve a simul- FIG. 2. /H20849Color online /H20850Spectrograms of the SPL ratio between the mouth- piece and the mouth pressures when playing a scale /H20849Subjects A and D, from top to bottom /H20850.5 10 15 20−40−30−20−1001020 Scale Note IndexFirst Partial SPL Ratios (dB)Subject A Subject B Subject C Subject D 0 5 10 15 20−40−30−20−1001020 Scale Note IndexFirst Partial SPL Ratios (dB) FIG. 3. Average SPL ratios for ﬁrst partials of scale: individual by subject /H20849top/H20850and for all subjects /H20849bottom /H20850. J. Acoust. Soc. Am., Vol. 123, No. 4, April 2008 Scavone et al. : Vocal-tract inﬂuence in saxophone performance 2393taneous variation of both the upper teeth and lower lip. The primary purpose of this sensor was to provide players withvisual feedback to help them focus on maintaining a ﬁxedembouchure setting, rather than acquiring an absolute mea-sure of lip or teeth force. Playing tests were conducted in an IAC /H20849Industrial Acoustics Company /H20850double-walled sound isolation booth to minimize external sound interference. In addition to themouthpiece described previously, a single Vandoren #3 /H20849me- dium hardness /H20850reed and Selmer Super Action Series II alto saxophone /H20849serial number 438024 /H20850were used for the entire experiment. B. Player tests Upstream inﬂuence was evaluated using the measure- ment system through a series of playing tests with a group offour professional saxophonists. Subject C was the ﬁrst authorof this paper. The subjects ﬁlled out a questionnaire abouttheir saxophone background and experience and were givenabout 5 min to become accustomed to the mouthpiece and saxophone setup. They were allowed to practice the re-quested tasks before the recording began. After the subjectswere comfortable with all the tasks, data storage was initi-ated and each task was performed in sequential order. Whensubjects had difﬁculty with a given task, they were allowedto repeat it. A large real-time display of the FSR reading wasprovided and the subjects were told to avoid making embou-chure changes while performing the tasks. III. RESULTS To examine variations of upstream inﬂuence over time, many of the results in the subsequent sections are displayedvia spectrograms of the ratio of the SPL in decibels /H20849or power spectral densities /H20850of the mouth and mouthpiece pres- sures. To reduce artifacts arising from the ratio calculation,spectral bins containing no signiﬁcant energy in both theupstream and downstream power spectra were masked. Themeasured upper teeth force is also plotted to indicate therelative steadiness of a player’s embouchure setting. A. Traditional and extended registers The traditional range of a saxophone is from written B/flat3–F/sharp6, which on an alto saxophone corresponds to the frequency range 138.6–880 Hz. Advanced players can play afurther octave or more using cross-ﬁngerings, a range re-ferred to as the “altissimo” or extended register. In order toevaluate general player trends over the full range of the in-strument, each subject was asked to play an ascending legatowritten B /flat/H20849D/flatconcert pitch /H20850scale from the lowest note on the instrument to the highest note that could comfortably beheld /H20849which was typically near a written C7 in the extended register /H20850, playing each note for about 0.5 s. Representative spectrograms of the SPL ratios of up- stream and downstream pressures for this task are shown inFig. 2, as played by Subjects A and D. Breaks in the mea- sured upper teeth force indicate instances where subjectsstopped to breath. SPL averages computed at the fundamen- FIG. 4. /H20849Color online /H20850Spectrograms of the SPL ratio between the mouth- piece and the mouth pressures when playing a written D6 without vibrato,with a slow lip vibrato, and bending via vocal tract manipulations /H20849Subjects B and D, from top to bottom /H20850.500 550 600 650 700 750 800 850 90060708090100110120130140150 Frequency (Hz)SPL dB (pref=2 0 µPa)Mouth Pressure Mouthpiece Pressure FIG. 5. Overlaid snapshots of the pitch bend spectra on an alto saxophone at the starting frequency /H20849700 Hz /H20850and at the lowest frequency trajectory /H20849580 Hz /H20850for Subject D. 2394 J. Acoust. Soc. Am., Vol. 123, No. 4, April 2008 Scavone et al. : Vocal-tract inﬂuence in saxophone performancetal frequencies for each scale note are shown in Fig. 3for each subject and averaged across all subjects. For most notesin the traditional range, the SPL ratios at the fundamentalfrequencies are below −20 dB. In other words, pressures inthe mouthpiece for these notes are typically 10 times greaterthan those in the players’ mouths. Based on the assumptionsoutlined in Sec. II, this indicates a similar ratio of inputimpedance peak levels at the fundamental playing frequen-cies on either side of the reed and thus minimal upstreaminﬂuence for notes in this range. These ratios display a localminima centered at the thirteenth note of the scale /H20849466 Hz /H20850, which may be related to the fact that notes in this range are relatively easy to play. The standard deviation of the SPLratios are shown by the error bars in the lower plot of Fig. 3. A fairly abrupt change in SPL ratios is evident when subjects prepare to enter the extended register, a result thatwas also reported by Fritz and Wolfe. 6Scale note index 19 in Fig.3is the highest note /H20849written F6 /H20850in the scale that falls within the traditional register. An alternate ﬁngering existsfor F6 on the saxophone that has a playing behavior morelike extended register notes /H20849it is based on the use of a third air column partial /H20850, though the ﬁngering used by subjects in this study was not speciﬁed or recorded. Averaged SPL ratiosacross all subjects for the extended register notes are be-tween 3 and 5 dB, though the standard deviation is signiﬁ-cant. In particular, Subjects A and B show large variations inSPL ratios from note to note in this range, whereas the re-sults for Subjects C and D are more consistent. It should benoted that the task called for a legato scale, or slurring fromnote to note. Future studies could investigate potential varia-tions of SPL ratios in the extended register when the notesare attacked individually, with or without breaks betweeneach. It has been suggested by Wilson 5that performers might tune upstream resonances with higher harmonics of a playednote. There are some instances in Fig. 2/H20849and the data for the other subjects /H20850where the ratios for upper partials of notes are near 0 dB, typically in the range 600–1600 Hz, though thisvaries signiﬁcantly among the subjects. However, no system-atic note-to-note tuning with an upper harmonic was found. It is unlikely that upstream tuning would help stabilize a noteunless the fundamental is weak and only one or two higherpartials exist below the cutoff frequency of the instrument.That said, variations of upper partial ratios could affect thetimbre of the instrument and this is investigated further inSec. III E. B. Pitch pending Saxophonists make frequent use of pitch bends in their playing, especially in jazz contexts. There are several wayssuch frequency modiﬁcations can be achieved, including lippressure and tonehole key height changes, as well as vocaltract manipulations. Lip pressure variations, which are usedto produce vibrato, affect the average reed tip opening andcan yield maximum frequency modulations of about half asemitone. The use of vocal tract manipulations for pitch bendcan achieve downward frequency shifts of a musical third ormore. Signiﬁcant upward frequency shifts using lip pressure0 500 1000 1500 200001234567x1 07 Frequency (Hz)Impedance Magnitude (Pa s m−3)Register Key Closed Register Key Open FIG. 6. Input impedance of the alto saxophone for the D6 ﬁngering. FIG. 7. /H20849Color online /H20850Spectrograms of the SPL ratio between the mouth and mouth piece pressures when performing the bugling task /H20849Subjects B and C, from top to bottom /H20850. J. Acoust. Soc. Am., Vol. 123, No. 4, April 2008 Scavone et al. : Vocal-tract inﬂuence in saxophone performance 2395or the vocal tract are not possible. On an alto saxophone, bends produced via vocal-tract manipulations are easiestwhen starting on notes above concert E /flat5. Bends that start above a concert C5 tend to have a lower limit around the C5frequency /H20849523 Hz /H20850. In other words, the ﬁngered note con- trols the starting /H20849and highest /H20850frequency of the bend range but the minimum frequency is generally always between 500and 600 Hz. Pitch bending has previously been investigated with re- spect to vocal tract inﬂuence. 5,6In the present study, subjects were asked to ﬁnger a written high D6 /H20849698 Hz /H20850with the ﬁrst palm key and to play the note normally, without vibrato, for about 3 s. They were then asked to play the note with a slowvibrato /H20849about 1 Hz /H20850, modulating the pitch up and down as much as possible using lip pressure only . Finally, subjects were instructed to perform, without varying their lip pres- sure, a slow downward pitch bend to the lowest note they could comfortably maintain, hold that note for about 1 s, andthen bend the note back to the starting D pitch. Figure 4shows representative spectrograms of the SPL ratios of upstream and downstream pressures for Subjects Band D when performing the requested tasks. The averageSPL ratio at the fundamental frequency, across all subjects, when played normally without vibrato was −25 dB, with astandard deviation of 9.3. Subjects B, C, and D demonstrateda consistent vibrato frequency range, using lip pressurevariations only, of about 677–700 Hz /H20849or 54 cents down and about 5 cents up /H20850. Subject A’s range for the same task was 655–690 Hz, but his SPL fundamental ratios were as high as3.4 dB at the vibrato frequency dips, indicating that he usedsome vocal-tract manipulation as well. For the pitch bendperformed without lip pressure variation, the lower fre-quency limits of the four subjects were 583, 597, 558, and570 Hz, respectively, with corresponding fundamental fre-quency SPL ratios of 29.1, 22.7, 30.7, and 25.1 dB. The sub-jects averaged a frequency drop of 330 cents with a SPL ratioof 26.9 and a standard deviation of 3.7. Snapshots of theupstream and downstream spectra at the starting frequencyand lowest frequency trajectory of the pitch bend task forSubject D are overlaid in Fig. 5. At the starting frequency of about 700 Hz, the upstream and downstream SPLs differ byabout −35 dB, while at the bottom of the bend, they differ byabout 25 dB. The input impedance of the alto saxophone used for this study with a D6 ﬁngering is shown in Fig. 6, from which it is clear that the instrument has no strong resonance between400 and 650 Hz. This measurement was obtained using atwo-microphone transfer function technique. 24,25Given the ﬁxed ﬁngering and relatively constant lip pressure used bythe subjects, as well as the fact that lip pressure variationsalone can only produce bends of about 54 cents, the pitchbends of 300 cents and more can only be the result of vocal-tract manipulation. The high SPL ratio during the bend indi-cates that a signiﬁcantly stronger resonance exists in theplayers’ mouths during the pitch bend than in the down-stream air column. The implication is that performers cancreate a resonance in the upstream windway in the range700–550 Hz /H20849for this task /H20850that is strong enough to override the downstream air column and assume control of the reedvibrations. The upstream resonance frequency is mainly controlled via tongue position variations. The pitch bend is only pos-sible for notes higher in the traditional range, where the aircolumn resonance structure is relatively weak. We speculatethat the lower frequency limit on the pitch bend is related tothe vocal-tract physiology and players’ control of the secondupstream resonance within an approximate range of about520–1500 Hz, which is close to that reported by Benade. 3 This is corroborated by the fact that a similar lower fre-quency limit is found when pitch bending on both sopranoand tenor saxophones. Informal tests on a B /flatclarinet indi-0 2 4 6 8−30−20−100102030 Overtone Series Note IndexSPL Ratios (dB)Subject A Subject B Subject C Subject D 0 1 2 3 4 5 6 7 8 9−30−20−100102030 Overtone Series Note IndexSPL Ratios (dB) FIG. 8. Average SPL ratios at each overtone series frequency: individual by subject /H20849top/H20850and for all subjects /H20849bottom /H20850. FIG. 9. /H20849Color online /H20850Fingerings and approximate sounding notes /H20849written and frequencies /H20850for four requested multiphonics. 2396 J. Acoust. Soc. Am., Vol. 123, No. 4, April 2008 Scavone et al. : Vocal-tract inﬂuence in saxophone performancecate a greater variance in the lower pitch bend range with different ﬁngerings, though the overall range still falls withinthe limits mentioned earlier. C. “Bugling” Bugling involves the articulation of the notes of an over- tone series while maintaining a ﬁxed low note ﬁngering. Be-ginning students can normally produce the ﬁrst and secondovertones, sometimes inadvertently while trying to play thefundamental. It typically takes many years of practice to de-velop the ﬂexibility that allows one to cleanly attack thehigher overtones. In this task, subjects were asked to ﬁnger awritten B /flat3/H20849all holes closed /H20850and to play an overtone se- ries, attacking each note individually. Representative spectrograms of the SPL ratios for the bugling exercise are shown in Fig. 7, as played by Subjects B and C. Individual subject averages, as well as averagesacross all subjects, computed at the fundamental frequenciesof each overtone are shown in Fig. 8. Subject D did not play overtones 8–9 and Subject A was unable to play overtones2–4 and 9. Vocal tract inﬂuence is not clearly evident untilthe third overtone, where downstream instrument resonancesbegin to weaken. In general, it is difﬁcult to play the thirdovertone without some vocal tract manipulation and it ap-pears that a ratio of about −7 dB is sufﬁcient to allow this tohappen. It also may be possible that, when playing the thirdovertone, players reinforce the 1130 Hz component with anupstream resonance rather than the component at 563 Hz,which falls at the lower end of the adjustable upstream reso-nance range. Averaging across Subjects B, C, and D, the SPLratio of the 1130 Hz component when playing the fundamen-tal of the overtone series is −8.7 dB. The ratio at this samefrequency when playing the third overtone is 4.8 dB. D. Multiphonics Multiphonics involve oscillations of the reed based on two inharmonic downstream air column resonance frequen-cies and their intermodulation components. 26Two relatively easy and two more difﬁcult multiphonics were chosen foranalysis with respect to potential vocal-tract inﬂuence. Thefour multiphonic ﬁngerings and their approximate soundingnotes are illustrated in Fig. 9. Figure 10shows the SPL ratios for four different multi- phonics, as played by Subjects B and D. The sounds pro-duced by the two subjects differed signiﬁcantly. From theplots, there are more spectral components evident in thesounds of Subject D. We expect that a player can inﬂuencethe sound quality of a multiphonic by aligning an upstreamresonance with one or more components. For example, bothsubjects appear to reinforce a group of partials in the range875–1075 Hz for the last multiphonic. SPL ratio levels forSubject D were, across all tasks, generally higher than thoseof Subject B. This suggests that Subject D makes use ofstronger upstream resonances, which likely explains both thehigher ratios for this task, as well as the greater spectraldensity of the multiphonic sounds.E. Timbre variations In a questionnaire, all subjects indicated that they thought they could inﬂuence the sound of the saxophone viavocal-tract variations. They reported using upstream inﬂu-ence to control and adjust tone color /H20849timbre /H20850, pitch, and extended register notes. To investigate timbre variations viavocal-tract manipulation, subjects were asked to play asteady written F4 /H20849207.7 Hz /H20850and to move their tongue peri- odically toward and away from the reed while maintaining a constant embouchure setting. Figure 11shows spectrograms of the SPL ratios result- ing from this task, as played by Subjects A and C. Partialsfour and higher show clear variations in SPL ratios withtongue motion. Snapshots of the mouth and mouthpiecespectra for Subject C are shown in Fig. 12at times of 4 and 5 seconds. Although both the upstream and downstreamspectra varied with tongue position, the upstream changeswere most signiﬁcant. For example, the downstream SPL ofthe fourth partial only changes by +3 dB between the two FIG. 10. /H20849Color online /H20850Spectrograms of the SPL ratio between the mouth and mouthpiece pressures when playing four multiphonics /H20849Subjects B and D, from top to bottom /H20850. J. Acoust. Soc. Am., Vol. 123, No. 4, April 2008 Scavone et al. : Vocal-tract inﬂuence in saxophone performance 2397times, whereas the upstream SPL increases by +13 dB. For this partial and subject, the SPL ratio goes from an averageof −0.7 to 9.9 dB, or a difference of 10.6 dB between thetwo tongue positions. The change of downstream and up-stream SPL for two tongue position extremes is collated forpartials 1–9 and Subjects A-C in Fig. 13. Subject D’s results showed no clear variation of SPL ratios over time and it islikely this person did not properly understand the requestedtask. In general, tongue movements toward the reed tendedto boost upstream frequency components from about 800 Hzto at least 2000 Hz. Note that these spectral changes occursimultaneously over a wide frequency range and thus arelikely the result of a more wide bandwidth upstream reso-nance. Below 800 Hz, the effect of the tongue positionmovement varied signiﬁcantly among the subjects. It is notpossible to say whether these variations were the result ofdifferences in physiology or tongue positions. IV. DISCUSSION Results of the tongue movement task indicate that vocal- tract manipulations can be used throughout the playing rangeof the saxophone to produce subtle timbre variations involv- ing frequency components from at least 800–2000 Hz.These timbre variations simultaneously affect partials over awide frequency range, which implies the use of a relativelywide bandwidth upstream resonance. The pitch bend, ex-tended register, and bugling tasks, however, indicate thatvocal-tract inﬂuence signiﬁcant enough to override down-stream air column control of reed vibrations is only possiblewhen the downstream system provides weak support of agiven note. This is normally the case for notes with funda-mental frequencies an octave below the downstream air col-umn cutoff frequency /H20849around 1500 Hz for the alto saxophone 27/H20850and higher. Thus, signiﬁcant vocal-tract inﬂu- ence can be exerted for notes near the top of the alto saxo-phone’s conventional range and on into the extended /H20849or al- tissimo /H20850register. This type of inﬂuence makes use of a narrow upstream resonance that the player manipulates tocontrol the fundamental vibrating frequency of the reed. Various special ﬁngerings are used when playing notes in the saxophone’s extended register but these provide onlyweak downstream support of a note. When students unaccus-tomed with altissimo register playing try these ﬁngerings,they produce a weak tone lower in the traditional range of FIG. 11. /H20849Color online /H20850Spectrograms of the SPL ratio between the mouth and mouthypiece pressures when varying tongue position while holding anF3/H20849Subjects A and C, from top to bottom /H20850.0 500 1000 1500 200060708090100110120130140150 Frequency (Hz)SPL dB (pref=2 0 µPa)Time = 4 seconds Time = 5 seconds 0 500 1000 1500 200060708090100110120130140150 Frequency (Hz)SPL dB (pref=2 0 µPa)Time = 4 seconds Time = 5 seconds FIG. 12. Snapshots of the mouthpiece /H20849top/H20850and mouth /H20849bottom /H20850spectra for the tongue variation task at times of 4 and 5 s as played by Subject C. 2398 J. Acoust. Soc. Am., Vol. 123, No. 4, April 2008 Scavone et al. : Vocal-tract inﬂuence in saxophone performancethe instrument. From the discussion above, this implies the need for vocal-tract inﬂuence when playing in the extendedregister. There were a few instances when the SPL ratios forSubjects A and B fell below 0 dB while playing in this range/H20849see Figs. 3and8/H20850. This might happen because a given ﬁn- gering produced a relatively strong downstream resonance atthe fundamental playing frequency or it is possible thesesubjects made embouchure adjustments that were not de-tected by the FSR on the top of the mouthpiece. The subjectswere instructed to avoid making embouchure changes for alltasks in this study. However, it is common for performers tomove their lower lip forward, away from the reed tip, inpreparation for playing in the saxophone’s altissimo register.The authors assume this reduces the reed damping, with acorresponding increase in the reed resonance frequency, andthat this may help to maintain normal operation of the reed in its “stiffness controlled” region. As well, extended registerplaying is generally easier with stiffer reeds. Notes near thetop of the extended register, with fundamentals in the range1400–2000 Hz, may be beyond the range of a high- Qsec- ond vocal-tract resonance. It is not clear whether playersstabilize these extreme high notes, which are difﬁcult to pro-duce consistently even for professionals, with a vocal-tract resonance and/or by manipulating the reed resonance fre-quency. One of the clearest examples of the use of vocal-tract inﬂuence involves pitch bending, with downward frequencymodulations 5–6 times greater than that possible via lip pres-sure variations alone. Pitch bends are not possible for noteslower than about a concert C5 /H20849523 Hz /H20850, at the lower ex- treme of the main adjustable upstream resonance and where several downstream air column resonances can help stabilizea note. The results of this study are less conclusive with respect to the production of multiphonics. Performers who are un-able to play in the extended register of the saxophone nor-mally also have trouble producing the more difﬁcult multi-phonics. This would imply the use of an upstream resonanceto support one or more intermodulation components. A futurestudy should compare successful and unsuccessful multi-phonic attempts for a ﬁxed ﬁngerings in an effort to verifythis behavior. There were considerable differences in thesound and quality of the multiphonics produced by the sub-jects of this study and this is likely related to variations in theupstream system. Finally, there is a common misconception among some scientists and players that vocal-tract inﬂuence is exerted ona nearly continuous basis while playing a single-reed instru-ment in its traditional range . 5,13For example, an early study by Clinch et al.13concluded “that vocal tract resonant fre- quencies must match the frequency of the required notes inclarinet and saxophone performance.” Likewise, Wilson 5 claims that for most tones in analyzed melodic phrases, “theperformer’s airways were tuned to the ﬁrst harmonic or tothe second harmonic, or there was a resonance aligned withboth the ﬁrst and second harmonics.” A related conclusion ismade by Thompson 28with respect to variations of the reed resonance. These suggestions have no basis in performancepractice. Although it is true that effects such as pitch slidescommon to jazz playing may involve some level of vocal-tract inﬂuence, most professional musicians use and advo-cate a relatively ﬁxed vocal-tract shape during normal play-ing. This behavior is corroborated by Fritz and Wolfe. 6It seems likely that players vary their embouchure and perhapstheir vocal-tract shape gradually when moving from low tohigh register notes, but this should be understood to varywith register and not on a note-by-note basis /H20849until one gets to the altissimo register /H20850. The results of this study indicate that vocal-tract inﬂuence for “normal” playing within thetraditional range of the alto saxophone is primarily limited totimbre modiﬁcation because most of these notes are wellsupported by the downstream air column. A website and video demonstrating the measurement system described in this paper is available online. 29 ACKNOWLEDGMENTS The authors thank the anonymous reviewers for their insightful comments and suggestions. They also acknowl-edge the support of the Natural Sciences and EngineeringResearch Council of Canada, the Canadian Foundation for1 2 3 4 5 6 7 8 9−20−15−10−5051015202530 Partial NumberSPL Differences (dB)Subject A Subject B Subject C 1 2 3 4 5 6 7 8 9−20−15−10−5051015202530 Partial NumberSPL Differences (dB) Subject A Subject B Subject C FIG. 13. SPL changes in mouthpiece /H20849top/H20850and mouth /H20849bottom /H20850between two tongue position extremes for partials 1–9 of held F3 /H20849Subjects A, B, and C /H20850. J. Acoust. Soc. Am., Vol. 123, No. 4, April 2008 Scavone et al. : Vocal-tract inﬂuence in saxophone performance 2399Innovation, and the Centre for Interdisciplinary Research in Music Media and Technology at McGill University. The doc-toral research of the second and third authors is supported bythe Fonds Québécois de la Recherche sur la Nature et lesTechnologies and CAPES /H20849Funding Council of the Brazilian Ministry of Education /H20850, respectively. Finally, we would like to thank Bertrand Scherrer for his help in developing the ﬁrstprototype of the National Instruments LABVIEW measurement system. APPENDIX: TRANSDUCER CALIBRATION A JBL 2426H compression driver was connected to a steel cylindrical pipe of 150 mm length and 25.4 mm diam-eter. The calibration is valid up to the cutoff frequency of theﬁrst higher order mode, which is f c=1.84 c//H208492/H9266a/H20850/H110158 kHz for a radius a=12.7 mm and a speed of sound c=347 m /s.30 At the far end of this pipe, the two transducers were ﬁt through a cap such that they extended a few millimeters be-yond the cap surface. A 60-s noise sequence was playedthrough the system and power spectral densities were deter-mined using a modiﬁed averaged periodogram of 1024 datapoints and 50% overlap /H20849Hanning windows /H20850at a sampling rate of 48 kHz. In order to match fast Fourier transform binsused in the LABVIEW interface and the subsequent data analy- sis, the calibration data was downsampled to 12 kHz. Thepower spectral densities were computed with the pwelch andcpsd functions in MATLAB . The transfer function relat- ing the gain and phase differences between the microphoneswas obtained as: 31 Hˆ12=Sˆ22−Sˆ11+/H20881/H20849Sˆ11−Sˆ22/H208502+4/H20841Sˆ12/H208412 2Sˆ21, where Sˆ11=PWELCH /H20849P1,HANNING /H20849N/H20850,N/2,N,FS/H20850, Sˆ22=PWELCH /H20849P2,HANNING /H20849N/H20850,N/2,N,FS/H20850, Sˆ12=CPSD /H20849P1,P2,HANNING /H20849N/H20850,N/2,N,FS/H20850, and Sˆ21=CPSD /H20849P2,P1,HANNING /H20849N/H20850,N/2,N,FS/H20850. A similar calibration technique was previously reported by Seybert and Ross.24 1J. Backus, “The effect of the player’s vocal tract on woodwind instrument tone,” J. Acoust. Soc. Am. 78, 17–20 /H208491985 /H20850. 2M. Watkins, “The saxophonist’s vocal tract. 1,” Saxophone Symp. 27, 51–78 /H208492002 /H20850. 3A. H. Benade, “Air column, reed, and player’s windway interaction in musical instruments,” in Vocal Fold Physiology, Biomechanics, Acoustics, and Phonatory Control , edited by I. R. Titze and R. C. Scherer /H20849Denver Center for the Performing Arts, Denver, CO, 1985 /H20850, Chap. 35, pp. 425– 452. 4P. L. Hoekje, “Intercomponent energy exchange and upstream/downstreamsymmetry in nonlinear self-sustained oscillations of reed instruments,”Ph.D. thesis, Case Western Reserve University, Cleveland, OH, 1986. 5T. D. Wilson, “The measured upstream impedance for clarinet perfor-mance and its role in sound production,” Ph.D. thesis, University of Wash-ington, Seattle, WA, 1996. 6C. Fritz and J. Wolfe, “How do clarinet players adjust the resonances of their vocal tracts for different playing effects?,” J. Acoust. Soc. Am. 118, 3306–3315 /H208492005 /H20850. 7S. D. Sommerfeldt and W. J. Strong, “Simulation of a player-clarinet system,” J. Acoust. Soc. Am. 83, 1908–1918 /H208491988 /H20850. 8R. Johnston, P. G. Clinch, and G. J. Troup, “The role of the vocal tract resonance in clarinet playing,” Acoust. Aust. 14, 67–69 /H208491986 /H20850. 9G. P. Scavone, “Modeling vocal-tract inﬂuence in reed wind instruments,” inProceedings of the 2003 Stockholm Musical Acoustics Conference , Stockholm, Sweden, pp. 291–294. 10P. Guillemain, “Some roles of the vocal tract in clarinet breath attacks:Natural sounds analysis and model-based synthesis,” J. Acoust. Soc. Am.121, 2396–2406 /H208492007 /H20850. 11R. L. Wheeler, “Tongue registration and articulation for single and double reed instruments,” Natl. Assoc. College Wind Percussion Instruct. J. 22, 3–12 /H208491973 /H20850. 12W. E. J. Carr, “A videoﬂuorographic investigation of tongue and throat positions in playing ﬂute, oboe, clarinet, bassoon, and saxophone,” Ph.D.thesis, University of Southern California, Los Angeles, CA, 1978. 13P. G. Clinch, G. J. Troup, and L. Harris, “The importance of vocal tractresonance in clarinet and saxophone performance: A preliminary account,”Acustica 50, 280–284 /H208491982 /H20850. 14J. T. Peters, “An exploratory study of laryngeal movements during perfor- mance on alto saxophone,” Master’s thesis, North Texas State University,Denton, TX, 1984. 15M. Patnode, “A ﬁber-optic study comparing perceived and actual tonguepositions of saxophonists successfully producing tones in the altissimoregister,” Ph.D. thesis, Arizona State University, 1999. 16J.-M. Chen, J. Smith, and J. Wolfe, “Vocal tract interactions in saxophoneperformance,” in Proceedings of the International Symposium on Musical Acoustics , Barcelona, Spain, 2007. 17S. Elliott and J. Bowsher, “Regeneration in brass wind instruments,” J. Sound Vib. 83, 181–217 /H208491982 /H20850. 18A. R. da Silva, G. P. Scavone, and M. van Walstijn, “Numerical simula- tions of ﬂuid-structure interactions in single-reed mouthpieces,” J. Acoust.Soc. Am. 122, 1798–1809 /H208492007 /H20850. 19J. van Zon, A. Hirschberg, J. Gilbert, and A. Wijnands, “Flow through the reed channel of a single reed instrument,” J. Phys. /H20849Paris /H20850, Colloq. 54,C 2 821–824 /H208491990 /H20850. 20A. Hirschberg, R. W. A. van de Laar, J. P. Marrou-Maurières, A. P. J. Wijnands, H. J. Dane, S. G. Kruijswijk, and A. J. M. Houtsma, “A quasi-stationary model of air ﬂow in the reed channel of single-reed woodwindinstruments,” Acustica 70, 146–154 /H208491990 /H20850. 21X. Boutillon and V. Gibiat, “Evaluation of the acoustical stiffness of saxo- phone reeds under playing conditions by using the reactive power ap-proach,” J. Acoust. Soc. Am. 100, 1178–1889 /H208491996 /H20850. 22G. P. Scavone, “Real-time measurement/viewing of vocal-tract inﬂuence during wind instrument performance /H20849A/H20850,” J. Acoust. Soc. Am. 119, 3382 /H208492006 /H20850. 23G. P. Scavone, “Time-domain synthesis of conical bore instrument sounds,” in Proceedings of the 2002 International Computer Music Con- ference , Göteborg, Sweden, pp. 9–15. 24A. Seybert and D. Ross, “Experimental determination of acoustic proper- ties using a two-microphone random excitation technique,” J. Acoust. Soc.Am. 61, 1362–1370 /H208491977 /H20850. 25A. Lefebvre, G. P. Scavone, J. Abel, and A. Buckiewicz-Smith, “A com- parison of impedance measurements using one and two microphones,” inProceedings of the International Symposium on Musical Acoustics , Barce- lona, Spain, 2007. 26J. Backus, “Multiphonic tones in the woodwind instrument,” J. Acoust.Soc. Am. 63, 591–599 /H208491978 /H20850. 27A. H. Benade and S. J. Lutgen, “The saxophone spectrum,” J. Acoust. Soc. Am. 83, 1900–1907 /H208491988 /H20850. 28S. C. Thompson, “The effect of the reed resonance on woodwind tone production,” J. Acoust. Soc. Am. 66, 1299–1307 /H208491979 /H20850. 29G. P. Scavone, http://www.music.mcgill.ca/~gary/vti/, 2007 /H20849last viewed 11 January 2008 /H20850. 30N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments /H20849Springer, New York, 1991 /H20850. 31J. H. P. R. White and M. H. Tan, “Analysis of the maximum likelihood, total least squares and principal component approaches for frequency re-sponse function estimation,” J. Sound Vib. 290, 676–689 /H208492006 /H20850. 2400 J. Acoust. Soc. Am., Vol. 123, No. 4, April 2008 Scavone et al. : Vocal-tract inﬂuence in saxophone performance
1.4904960.pdf	Magnetoimpedance effect at the high frequency range for the thin film geometry: Numerical calculation and experiment M. A. Corrêa, F. Bohn, R. B. da Silva, and R. L. Sommer Citation: Journal of Applied Physics 116, 243904 (2014); doi: 10.1063/1.4904960 View online: http://dx.doi.org/10.1063/1.4904960 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Wide frequency range magnetoimpedance in tri-layered thin NiFe/Ag/NiFe films: Experiment and numerical calculation J. Appl. Phys. 110, 093914 (2011); 10.1063/1.3658262 Nanostructured magnetic Fe–Ni–Co/Teflon multilayers for high-frequency applications in the gigahertz range Appl. Phys. Lett. 89, 242501 (2006); 10.1063/1.2402877 High-frequency permeability of thin NiFe/IrMn layers J. Appl. Phys. 93, 6668 (2003); 10.1063/1.1556098 High frequency permeability of patterned spin valve type thin films J. Appl. 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Sommer3 1Departamento de F /C19ısica Te /C19orica e Experimental, Universidade Federal do Rio Grande do Norte, 59078-900 Natal, Rio Grande do Norte, Brazil 2Departamento de F /C19ısica, Universidade Federal de Santa Maria, 97105-900 Santa Maria, Rio Grande do Sul, Brazil 3Centro Brasileiro de Pesquisas F /C19ısicas, Rua Dr. Xavier Sigaud 150, Urca, 22290-180 Rio de Janeiro, Rio de Janeiro, Brazil (Received 12 November 2014; accepted 11 December 2014; published online 24 December 2014) The magnetoimpedance effect is a versatile tool to investigate ferromagnetic materials, revealing aspects on the fundamental physics associated to magnetization dynamics, broadband magneticproperties, important issues for current and emerging technological applications for magnetic sensors, as well as insights on ferromagnetic resonance effect at saturated and even unsaturated samples. Here, we perform a theoretical and experimental investigation of the magnetoimpedanceeffect for the thin ﬁlm geometry at the high frequency range. We calculate the longitudinal magnetoimpedance for single layered, multilayered, or exchange biased systems from an approach that considers a magnetic permeability model for planar geometry and the appropriate magneticfree energy density for each structure. From numerical calculations and experimental results found in literature, we analyze the magnetoimpedance behavior and discuss the main features and advantages of each structure. To test the robustness of the approach, we directly comparetheoretical results with experimental magnetoimpedance measurements obtained at the range of high frequencies for an exchange biased multilayered ﬁlm. Thus, we provide experimental evidence to conﬁrm the validity of the theoretical approach employed to describe themagnetoimpedance in ferromagnetic ﬁlms, revealed by the good agreement between numerical calculations and experimental results. VC2014 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4904960 ] I. INTRODUCTION The study of dynamical phenomena has provided central advances on magnetization dynamics during the past decades.Usually, the investigations are based on traditional ferromag-netic resonance (FMR) experiments, 1in which the sample is submitted to an intense external magnetic ﬁeld, saturating itmagnetically. From FMR measurements, information regard-ing magnetic anisotropies, damping parameter, and otherimportant parameters related to the magnetic dynamics can bereached. However, nowadays, similar information can beaccessibly obtained also through the study of the magnetoim-pedance effect. This effect is a versatile tool commonlyemployed to investigate ferromagnetic materials, revealingaspects on the fundamental physics associated to magnetiza-tion dynamics, broadband magnetic properties, 2,3as well as on important issues for current and emerging technologicalapplications for magnetic sensors. 4–7Besides, further insights on FMR effect at saturated and even unsaturated samples8can be easily gotten, making possible the study of local resonan-ces and their inﬂuence in the dynamics magnetization. The magnetoimpedance effect (MI) corresponds to the change of the real and imaginary components of electricalimpedance of a ferromagnetic sample caused by the actionof an external static magnetic ﬁeld. In a typical MIexperiment, the studied sample is also submitted to an alter- nate magnetic ﬁeld associated to the electric current I ac¼Io expði2pftÞ,fbeing the probe current frequency. Irrespective to the sample geometry, the overall effect of these magneticﬁelds is to induce strong modiﬁcations of the effective mag-netic permeability. Experimentally, studies on MI have been performed in ferromagnetic ﬁlms with several structures, such as singlelayered, 2,9multilayered,10–14and structured multilayered samples.15–17 The general theoretical approach to the MI problem focuses on its determination as a function of magnetic ﬁeldfor a range of frequencies. Traditionally, the changes of mag- netic permeability and impedance with magnetic ﬁelds at dif- ferent frequency ranges are caused by three distinctmechanisms: 18–20magnetoinductive effect, skin effect, and FMR effect. Thus, MI can generally be classiﬁed into thethree frequency regimes. 21Moreover, the MI behavior with magnetic ﬁeld and probe current frequency becomes more complex, since it also depends on magnetic properties, suchas magnetic anisotropies, as well as sample dimensions andgeometry. Given that distinct effects affect the magnetic per-meability behavior at different frequency ranges and differ-ent properties inﬂuences the MI, the description of the magnetoimpedance effect over a wide range of frequency becomes a difﬁcult task. For this reason, the comprehensionon the theoretical and experimental point of views of the a)Electronic address: marciocorrea@dfte.ufrn.br b)Electronic address: felipebohn@dfte.ufrn.br 0021-8979/2014/116(24)/243904/12/$30.00 VC2014 AIP Publishing LLC 116, 243904-1JOURNAL OF APPLIED PHYSICS 116, 243904 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Wed, 12 Aug 2015 01:24:37magnetoimpedance effect is fundamental for the develop- ment of new materials with optimized response. Since the system geometry has an important role on MI results, several studies have been performed to obtain further information on this dependence. Considerable atten-tion has been given to describe the MI effect in samples pre-senting cylindrical geometry with distinct anisotropyconﬁgurations. 22–25For this case, e.g., Makhnovskiy et al.23 have reported a very strict study on the surface impedance tensor, in which theoretical results for the cylindrical geome-try are directly compared to experimental measurementsacquired for ferromagnetic wires. In addition, Usov et al. 25 have presented theoretical and experimental results for ferro- magnetic wires with weak helical anisotropy. Regarding the MI effect for the case of planar systems, an important study has been performed in single layers byKraus, 26who performed the calculation of the MI effect in a single planar conductor and studied the inﬂuence of theGilbert damping constant, the angle between the anisotropydirection and the applied magnetic ﬁeld on the MI effect.Moreover, Panina et al. 27and Sukstanskii et al.28investi- gated the MI behavior in multilayers, analyzing the inﬂuenceof width, length, and relative conductivity in MI effect. Although the MI results obtained are consistent and seem to reproduce experimental data, they are restricted to alimited frequency range. Since experimental measurementsare usually taken over a wide range of frequencies, in whichdifferent mechanisms contribute to the permeability, a gen-eral theoretical approach to the transverse magnetic perme-ability, which enables the MI calculation, consideringfrequency dependent magnetic permeability, becomes veryimportant for MI interpretation. In this paper, we report a theoretical and experimental investigation of the magnetoimpedance effect for the thinﬁlm geometry at the high frequency range. First of all, weperform numerical calculations of the longitudinal magneto-impedance for single layered, multilayered, and exchangebias systems, from a classical electromagnetic impedance fora planar system. To this end, we consider a theoreticalapproach that takes into account a magnetic permeabilitymodel for planar geometry and the appropriate magnetic freeenergy density for each structure. We analyze the magneto-impedance behavior, and discuss the main features andadvantages of each structure, as well as we relate the numeri-cal calculations with experimental results found in literature.Finally, to test the robustness of the approach, we comparetheoretical results calculated for an exchange biased multi-layered system with experimental magnetoimpedance meas-urements obtained in the range of high frequencies for anexchange biased multilayered ﬁlm. Thus, we provide experi- mental evidence to conﬁrm the validity of the theoretical approach to describe the magnetoimpedance in ferromag-netic ﬁlms. II. EXPERIMENT H e r e ,w ei n v e s t i g a t ea[ N i 20Fe80(40 nm)/Ir 20Mn80(20 nm)/ Ta(1 nm)] /C220 ferromagnetic exchange biased multilayered ﬁlm. The ﬁlm is deposited by magnetron sputtering onto a glasssubstrate, covered with a 2 nm-thick Ta buffer layer. The depo- sition process is performed with the following parameters: basevacuum of 8.0 /C210 /C08Torr, deposition pressure of 5.0 mTorr with a 99.99% pure Ar at 50 sccm constant ﬂow, and DC source with current of 150 mA for the deposition of the Ta and IrMn layers, as well as 65 W set in the RF power supplyfor the deposition of the NiFe layers. With these conditions,the obtained deposition rates are 0.08 nm/s, 0.67 nm/s, and0.23 nm/s for NiFe, IrMn, and Ta, respectively. During the dep-osition, the substrate with dimensions of 5 /C22m m 2is submit- ted to a constant magnetic ﬁeld of 2 kOe, applied along themain axis of the substrate in order to deﬁne an easy magnetiza-tion axis and induce a magnetic anisotropy and an exchangebias ﬁeld ~H EBin the interface between the NiFe and IrMn layers. Quasi-static magnetization curves are obtained with a vibrating sample magnetometer, measured along and perpen-dicular to the main axis of the ﬁlms, in order to verify themagnetic behavior. The magnetoimpedance effect is measured using a RF-impedance analyzer Agilent model E4991, with E4991 A test head connected to a microstrip in which the sample isthe central conductor, which is separated from the groundplane by the substrate. The electric contacts between the sample and the sample holder are made with 24 h cured low resistance silver paint. To avoid propagative effects andacquire just the sample contribution to MI, the RF imped-ance analyzer is calibrated at the end of the connection cableby performing open, short, and load (50 X) measurements using reference standards. The probe current is fed directlyto one side of the sample, while the other side is in short cir-cuit with the ground plane. The accurrent and external mag- netic ﬁeld are applied along the length of the sample. MImeasurement is taken over a wide frequency range, between0.5 GHz and 3.0 GHz, with maximum applied magneticﬁelds of 6350 Oe. While the external magnetic ﬁeld is swept, a 0 dBm (1 mW) constant power is applied to the sample characterizing a linear regime of driving signal. Thus, at a given ﬁeld value, the frequency sweep is madeand the real Rand imaginary Xparts of the impedance are simultaneously acquired. For further information on thewhole procedure, we suggest Ref. 17. The curves are known to exhibit hysteretic behavior, associated with the coerciveﬁeld. However, in order to clarify the general behavior, onlycurves depicting the ﬁeld going from negative to positivevalues are presented. III. THEORETICAL APPROACH A. Thin film planar geometry To investigate the MI effect, we perform numerical cal- culations of quasi-static magnetization curves, magneticpermeability, and magnetoimpedance for the thin ﬁlm geom-etry. To this end, from the appropriate magnetic free energydensity for the investigated structure, in a ﬁrst moment, weconsider a general magnetic susceptibility model, whichtakes into account its dependence with both frequency and magnetic ﬁeld. 29It is therefore possible to obtain the trans- verse magnetic permeability for planar geometry from243904-2 Corr ^eaet al. J. Appl. Phys. 116, 243904 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Wed, 12 Aug 2015 01:24:37susceptibility and in turn describe the MI behavior by using different models, according to system structure, for a widerange of frequencies and external magnetic ﬁelds. We focus on the study of ferromagnetic thin ﬁlms, which can be modeled as a planar system. Here, in particu-lar, we calculate the longitudinal magnetoimpedance effectfor single layered, multilayered, or exchange biased sys-tems. Figure 1(a) presents the theoretical system and the deﬁnitions of the relevant vectors considered to perform thenumerical calculations. In order to investigate the magneto-impedance effect in ﬁlms, we consider the single layered,multilayered, and exchange biased systems, as, respectively,shown in Figs. 1(b)–1(d) . Thus, from the appropriate magnetic free energy density nfor each structure, a routine for energy minimization determines the values the equilibrium angles h ManduMof magnetization for a given external magnetic ﬁeld ~H, and we obtain the magnetization curve, permeability tensor l, and longitudinal magnetoimpedance Zfor the respective struc- ture in a wide range of frequencies. B. Permeability Tensor Generally, the magnetization dynamics is governed by the Landau-Lifshitz-Gilbert equation, given by d~M dt¼/C0c~M/C2~Hef f/C16/C17 /C0ca M~M/C2 ~M/C2~Hef f/C16/C17hi ;(1) where ~Mis the magnetization vector, ~Hef fis the effective magnetic ﬁeld, and c¼jcGj=ð1þa2Þ, in which cGis the gyromagnetic ratio and ais the phenomenological Gilbert damping constant. In a MI experiment, the effective mag-netic ﬁeld presents two contributions and can be written as~H ef f¼ð~Hþ~HnÞþ~hac. The ﬁrst term, ð~Hþ~HnÞ, corre- sponds to the static component of the ﬁeld. It contains the external magnetic ﬁeld ~Hand the internal magnetic ﬁeld ~Hn¼/C0@n @~M,30due to different contributions to the magnetic free energy density n, such as magnetic anisotropies and induced internal magnetic ﬁelds. On the other hand, the sec- ond term corresponds to the alternate magnetic ﬁeld ~hacgen- erated by the Iacapplied to the sample, which in turn induces deviations of the magnetization vector from the static equi-librium position. Equation (1)is a general expression that can be applied to express the magnetization dynamics of anysystem, with any geometry. As previously cited, it is possible to understand the MI effect from the knowledge of the transverse magnetic perme-ability of a given material. This goal is achieved by consider-ing how magnetic dynamics transition takes place from onestate of equilibrium to another under both dcandacﬁelds. With this spirit, a very interesting approach to study the mag-netization dynamics was successfully undertaken by Spinu et al. 29This theory allows us to investigate the magnetic sus- ceptibility tensor and its dependence on both frequency andmagnetic ﬁeld, using knowledge of appropriate magneticfree energy density. From the approach, 29the magnetic susceptibility tensor, in spherical coordinates, for a general system with a givenmagnetic free energy density n, is written as vr;h;uðÞ ¼gc 21þa2 ðÞ00 0 0nuu sin2hM/C0nhu sinhM 0/C0nhu sinhMnhh0 BBBBBB@1 CCCCCCA þg00 0 0iM scxa iMscx 0/C0iMscx iMscxa0 BB@1 CCA; (2) where gis g¼1 x2 r/C0x2þixDx: (3) The quantities xrandDxin Eq. (3)are known, respectively, as the resonance frequency and width of the resonance absorption line, given by29 xr¼c MsinhMﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þa2p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ nhhnuu/C0n2 huq ; (4) and Dx¼ac Mnhhþnuu sin2hM/C18/C19 : (5) Here, nhh;nuu;nuh, and nhuare the second derivatives of the magnetic free energy density at an equilibrium position,deﬁned by the magnetization vector with h ManduM, as pre- viously shown in Fig. 1(a). FIG. 1. Ferromagnetic thin ﬁlms modeled as a planar system. (a) Schematic diagram of the theoretical ferromagnetic system and deﬁnitions of magnetiza- tion and magnetic ﬁeld vectors considered for the numerical calculation ofmagnetization, magnetic permeability, and magnetoimpedance curves. (b) Single layered (SL) system, composed by a 500 nm-thick ferromagnetic (FM) layer. (c) Multilayered (ML) system, composed by 250 nm-thick ferromag- netic layers and metallic non-magnetic (NM) layers with variable thicknesses. (d) Exchange-biased (EB) system, composed by a single 500 nm-thick ferro- magnetic layer and a single antiferromagnetic (AF) layer.243904-3 Corr ^eaet al. J. Appl. Phys. 116, 243904 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Wed, 12 Aug 2015 01:24:37Considering the matrix of the linear transformation of the unit vectors from spherical to Cartesian coordinates, the suscep- tibility tensor in the laboratory reference can be obtained. For instance, the real and imaginary components of the term vxxcan be, respectively, written as29 <vxx½/C138¼ja2þ1 ðÞ cx2 rnuucot2hMcos2uM/C02nhucothMsinuMcosuMþnhhsin2uM/C16/C17 /C0x2/C02a2þ1 ðÞ cnhucothMsinuMcosuM þa2þ1 ðÞ cnuu sin2hM/C0aMs/H17005x/C18/C19 cos2hMcos2uMþa2þ1 ðÞ cnhh/C0aMs/H17005x/C0/C1 sin2uM2 643 752 6666643 777775; (6) =vxx½/C138¼/C0jx/C02nhua2þ1 ðÞ c/H17005xcothMsinuMcosuMþ a2þ1 ðÞ c/H17005xnuu sin2hMþaMsx2/C18/C19 cos2hMcos2uM þa2þ1 ðÞ c/H17005xnhhþaMsx2/C0/C1 sin2uM/C0aMsx2 rcos2hMcos2uMþsin2uM/C0/C12 643 75;(7) where j¼c x2 r/C0x2/C0/C12þx2Dx2: In particular, the diagonal component of the suscepti- bility tensor presented in Eqs. (6)and(7),a sw e l la st h e vyy andvzzcomponents (not presented here for sake of simplic- ity), exhibit form similar to that presented in Ref. 29 when x!0, as expected. From the cited equations, it can be noticed a clear dependence of the magnetic susceptibil-ity with the equilibrium angles of the magnetization, aswell as with the derivatives of the magnetic free energy density. Thus, this general description to the susceptibility and, consequently, to the dynamic magnetic behavior cor-responds to a powerful tool, once it can be employed forany magnetic structure, using an appropriate energyconﬁguration. In ferromagnetic thin ﬁlms, which can be modeled as planar systems, the magnetization is frequently observed tobe in the plane of the ﬁlm. Thus, by considering h M¼90/C14 (see Fig. 1(a)), the expressions for the terms of the perme- ability tensor l¼1þ4pvcan be considerably simpliﬁed. The diagonal terms represented by lxx,lyy,andlzzcan be written as lxx¼1þ4pjsin2uM /C2ðx2 r/C0x2Þð1þa2ÞcnhhþaMsx2/H17005x þi½/C0ð1þa2Þcx/H17005xnhhþaMsxðx2 r/C0x2Þ/C138"# ; (8) lyy¼1þ4pjcos2uM /C2ðx2 r/C0x2Þð1þa2ÞcnhhþaMsx2Dx þi½/C0ð1þa2Þcx/H17005xnhhþaMsxðx2 r/C0x2Þ/C138"# ;(9) lzz¼1þ4pj /C2ðx2/C0x2 rÞð1þa2ÞcnuuþaMsx2/H17005x þi½/C0ð1þa2Þcx/H17005xnuuþaMsxðx2 r/C0x2Þ/C138"# : (10)Moreover, the off-diagonal terms are lxy¼lyx¼1þ2pjsinð2uMÞ /C2/C0ðx2 r/C0x2Þð1þa2Þcnhh/C0aMsx2/H17005x þi½ð1þa2Þcx/H17005xnhh/C0aMsxðx2 r/C0x2Þ/C138"# ; (11) lxz¼1þ4pjsinuM /C2/C0ðx2 r/C0x2Þð1þa2Þcnhu/C0Msx2/H17005x þi½ð1þa2Þcx/H17005xnhuþMsxðx2 r/C0x2Þ/C138"# ;(12) lyz¼1þ4pjcosuM /C2ðx2 r/C0x2Þð1þa2ÞcnhuþMsx2/H17005x þi½/C0ð1þa2Þcx/H17005xnhu/C0Msxðx2 r/C0x2Þ/C138"# ; (13) lzx¼1þ4pjsinuM /C2/C0ðx2 r/C0x2Þð1þa2ÞcnhuþMsx2/H17005x þi½ð1þa2Þcx/H17005xnhuþMsxðx2 r/C0x2Þ/C138"# ;(14) lzy¼1þ4pjcosuM /C2ðx2 r/C0x2Þð1þa2Þcnhu/C0Msx2/H17005x þi½/C0ð1þa2Þcx/H17005xnhu/C0Msxðx2 r/C0x2Þ/C138"# : (15) For all the numerical calculations, we consider that the magnetic ﬁeld ~His applied, as well as the electrical current is ﬂowing, along the y-direction (see Fig. 1(a)), hH¼uH¼90/C14. Then, the lxxterm can be understood as the transverse magnetic permeability lt. In Secs. IV A –IV C ,w e present the MI calculations for single layered, multilayered, and exchange biased systems. To this end, we consider MImodels, such as the classical MI expression for a slab con-ductor 26,31or the model proposed by Panina for multi- layers,32previously explored in a limited frequency range. In particular, this limitation is due to the employed permeabilitycalculation. Here, we consider a general approach to the per-meability and, consequently, we are able to explore the MI243904-4 Corr ^eaet al. J. Appl. Phys. 116, 243904 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Wed, 12 Aug 2015 01:24:37behavior in several planar structures in a wide frequency range. IV. RESULTS AND DISCUSSION A. Single layered system First of all, we perform numerical calculation for the longitudinal MI effect for a single layered system, as pre-sented in Fig. 1(b). We consider a Stoner-Wohlfarth modiﬁed model to describe the magnetic free energy density. In this case, it canbe written as n¼/C0 ~M/C1~H/C0 Hk 2Ms~M/C1^uk/C0/C12þ4pM2 s^M/C1^nðÞ ; (16) where the ﬁrst term is the Zeeman interaction, the second term describes the uniaxial anisotropy, and the third one cor-responds to the demagnetizing energy density for a thin pla-nar system, such as a thin ﬁlm. In this case, in addition to thevectors ~H;~M;^u k, and ^nalready discussed in Fig. 1(a), Hk¼2Ku/Msis the known anisotropy ﬁeld, Kuis the uniaxial anisotropy constant, and Msis the saturation magnetization of the ferromagnetic material. The longitudinal impedance is strongly dependent of the sample geometry. Here, to describe the magnetoimpedancein a single layered system, we consider the approachreported by Kraus 26for an inﬁnite slab magnetic conductor. Thus, for a single layered system, the impedance can be writ-ten as 26 Z Rdc¼kt 2coth kt 2/C18/C19 ; (17) where Rdcis the electrical dc resistance, tis the thickness of the system, and k¼(1/C0i)/d, where dis the classic skin depth, given by d¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2q=xlp ; (18) in which qis the electric resistivity, xis the angular fre- quency, and lis the magnetic permeability. In our case, we consider l¼lxx¼lt. Thus, from the magnetic free energy density, given by Eq.(16), and the calculation of the transverse magnetic per- meability, Eq. (8), the longitudinal magnetoimpedance for a single layered system, Eq. (17), can be obtained. The other terms of the permeability tensor previously presented can beused to calculate the Zbehavior, since a speciﬁc calculation of the Ztensor is done. For a single layered system, to perform the numerical calculation, we consider the following parameters: M s ¼780 emu/cm3,Hk¼5 Oe, hk¼90/C14,uk¼2/C14,a¼0.018, cG/2p¼2.9 MHz/Oe,33t¼500 nm, hH¼90/C14, and uH¼90/C14. We intentionally chose uk6¼0/C14since small deviations in the sample position or of the magnetic ﬁeld in an experiment arereasonable. Figure 2shows the numerical calculations for the real Rand imaginary Xcomponents of the longitudinal impedance as a function of the external magnetic ﬁeld forselected frequency values.It is important to point out that, experimentally, the MI measurements present a frequency dependent shift of the realand imaginary components, a feature related to the electrical/metallic contributions of the sam ple and of the microwave cav- ity or microstrip employed in the experiment. In order todirectly compare experimental data with numerical calculation,this dependence can be removed of the experimental MIresults, according to Ref. 34, or can be inserted in the MI nu- merical calculation by ﬁtting the measured RandXcurves as a function of the frequency for th e highest magnetic ﬁeld value, where the sample is magnetically saturated and it is not in aresonant regime. 21In this case of a single layered system, we consider a ﬁtting obtained from the data reported in Ref. 17. Thus, from Fig. 2, the well-known symmetric magneto- impedance behavior around H¼0 for anisotropic systems is FIG. 2. (a) Real Rand (b) imaginary Xcomponents of the longitudinal im- pedance as a function of the external magnetic ﬁeld for selected frequency values. The numerical calculations are obtained, using the described approach, for a single layered system with Ms¼780 emu/cm3,Hk¼5 Oe, hk¼90/C14,uk¼2/C14,a¼0.018, cG/2p¼2.9 MHz/Oe,33t¼500 nm, hH¼90/C14, anduH¼90/C14.243904-5 Corr ^eaet al. J. Appl. Phys. 116, 243904 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Wed, 12 Aug 2015 01:24:37veriﬁed, including the dependence with the magnetic ﬁeld amplitude, frequency, and the orientation between theapplied magnetic ﬁeld and accurrent with respect to the magnetic anisotropies. A double peak behavior is present for the whole frequency range, a feature of the FMR relation dis- persion, 7,21,26in a signature of the parallel alignment of the external magnetic ﬁeld and accurrent along the hard mag- netization axis. At low and intermediate frequencies (not shown), below 0.5 GHz, the position of the peaks remains unchanged andthey are close to H k. This feature reﬂects the fact that, at this frequency range, the skin effect is the main responsible forthe magnetization dynamics and MI variations. Beyond 0.5 GHz, besides the skin effect, the FMR effect also becomes an important mechanism responsible for variationsin MI effect, a fact evidenced by the displacement of thepeak position in the double peak structure toward higherﬁelds as the frequency is increased following the behaviorpredicted for the FMR effect. 7,21,26The contribution of the FMR effect to Zis also veriﬁed using the method described by Barandiar /C19anet al.,35and previously employed by our group in Ref. 17. In particular, the classical FMR signature is observed in the numerical calculation of the longitudinalMI response at the high frequency range strictly due to the fact that we employ a magnetic permeability model derived from the FMR theory. 29 These numerical calculation results are in qualitative agreement with several experimental results for single lay-ered thin ﬁlms 2,9with uniaxial magnetic behavior, when the magnetic ﬁeld and current are transverse to the easy magnet-ization axis during the experiment. B. Multilayered system Here, we perform the numerical calculation of the longi- tudinal MI effect for a multilayered system, as presented inFig.1(c). The multilayered system consists of two ferromagnetic layers separated by a metallic non-magnetic layer. To modelit, we consider a Stoner-Wohlfarth modiﬁed model, similarto that discussed in Subsection IV A , and the magnetic free energy density can be written as n¼X 2 i¼1/C0~Mi/C1~H/C0Hki 2Msi~Mi/C1^uki/C0/C12þ4pM2 si^Mi/C1^n/C0/C1/C20/C21 ;(19) where ~MsiandMsiare the magnetization vector and satura- tion magnetization for each ferromagnetic layer, respec- tively, Hki¼2Kui/Msiis the anisotropy ﬁeld for each layer, and Kuiis the uniaxial anisotropy constant, directed along ^uki, for each layer. In a traditional multilayered system, it is reasonable to consider Ms1¼Ms2¼Ms,Ku1¼Ku2¼Ku, ^uk1¼^uk2¼^uk, since the two layers are made of similar ferromagnets. To describe the magnetoimpedance behavior in a multi- layered system, we consider the approach to study the mag- netoimpedance effect in a trilayered system reported byPanina et al. 32and investigated by our group.21In this model, the trilayered system has ﬁnite width 2 band length lfor all layers, thicknesses t1andt2, and conductivity values r1andr2for the metallic non-magnetic and ferromagnetic layers, respectively, and variable ﬂux leaks across the inner non-magnetic conductor. When bis sufﬁciently large and the edge effect is neglected, impedance is dependent on the ﬁlmthickness t. Therefore, for a tri-layered system, impedance can be written as Z Rdc¼gmgfðÞcothgmr2 lr1/C18/C19 coth gfðÞþ2gm k1t1 cothgmr2 lr1/C18/C19 þ2gm k1t1coth gfðÞ2 66643 7775; (20) where lis the magnetic permeability for the ferromagnetic layers, in our case, we consider l¼lxx¼lt, and gm¼k1t1 2lr1 r2/C18/C19 ;gf¼k2t2; k1¼1/C0iðÞ d1;k2¼1/C0iðÞ d2; d1¼2pr1x ðÞ/C01=2;d2¼2pr2xl ðÞ/C01=2: To perform the numerical calculation for a multilay- ered system, we consider the parameters similar to thosepreviously employed: M s1¼Ms2¼780 em/cm3,Hk1¼Hk2 ¼5O e , hk1¼hk2¼90/C14,uk1¼uk2¼2/C14;a¼0:018;cG=2p ¼2:9M H z =Oe, t1¼100 nm, t2¼250 nm, hH¼90/C14,a n d uH¼90/C14. In particular, the thickness of the metallic non- magnetic layer is thick enough to neglect the bilinear and biquadratic coupling between the ferromagnetic layers. Moreover, we employ r1¼6/C2107(Xm)/C01andr2¼r1/4. Thus, from Eqs. (19),(8), and (20), Fig. 3shows the nu- merical calculations for the real Rand imaginary Xcompo- nents of the longitudinal impedance as a function of the external magnetic ﬁeld for selected frequency values. In par- ticular, for a multilayered system, the MI curves present allthe typical features described for an anisotropic single layered system, including the double peak MI structure due to the ori- entation between ~H,I acsense and ^uk, as well as the R,X,and Zbehavior with frequency. In order to consider the frequency dependent shift of RandXfor a multilayered system, we con- sider a ﬁtting obtained from the data reported in Ref. 17. To contrast the MI behavior veriﬁed for the studied sys- tems, Fig. 4presents a comparison between the numerical calculations of the real Rand imaginary Xcomponents of the longitudinal impedance for single layered and multilayered systems. The positions in ﬁeld of the MI peaks are similar, irrespective of the frequency. This feature is expected in thiscase, once similar parameter values are employed for both numerical calculations, and, consequently, both systems have the same quasi-static magnetic properties. The primary difference between the results for single layered or multilayered systems resides basically in the am- plitude of the MI curves. This fact is veriﬁed in Fig. 4and evidenced in Fig. 5. In this case, the MI variations are ampli- ﬁed for the multilayered system, a fact directly associated to the insertion of a metallic non-magnetic layer with high elec-tric conductivity r 1(Ref. 17) and thickness t1.243904-6 Corr ^eaet al. J. Appl. Phys. 116, 243904 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Wed, 12 Aug 2015 01:24:37Regarding the electric properties and the conductivity of the system, it is well-known that multilayered systems pres-ent a clear dependence of the MI variations with the r 1/r2ra- tio. This behavior has been veriﬁed and detailed discussed in Ref. 32, as well as also previously calculated by our group for a trilayered system.21 Concerning the size of the system, the MI variations are strongly dependent on the thickness t1of the metallic non- magnetic layer, as shown in Fig. 5. The higher MI variation values are veriﬁed for the thicker systems, with large t1val- ues. This fact is due to the reduction of the electric resistanceof the whole system with the increase of t 1, which is affected for both the increase of the system cross section and higherconductivity of the system. On the other hand, theoretically,in the limit of t 1!0, Eq. (20) for the impedance is reducedto Eq. (17), as expected, since the multilayered system becomes a single layered system for t1¼0. In this line, these numerical calculation results obtained for multilayered systems with different r1/r2ratio or t1val- ues are in qualitative concordance with several experimental results found in literature for multilayered ﬁlms.13,14 The damping parameter ais also an important element for the determination of magnetoimpedance because of its relationship with the magnetization dynamics at high fre- quencies. Experimentally, the avalue is inﬂuenced by the kind of the employed ferromagnetic material,36structural character,36and structure of the sample (single layered, mul- tilayered, sandwiched samples). From the numerical calcula-tions, we carry out an analysis similar to that presented byKraus, 26although here we consider a higher frequency FIG. 4. Comparison of the (a) real Rand (b) imaginary Xcomponents of the longitudinal impedance, as a function of the external magnetic ﬁeld for selected frequency values, calculated for single layered (dashed lines) and multilayered (solid lines) systems. The numerical calculations are performedusing parameters similar to those previously employed for single layered and multilayered systems. FIG. 3. (a) Real Rand (b) imaginary Xcomponents of the longitudinal im- pedance as a function of the external magnetic ﬁeld for selected frequency values. The numerical calculations are obtained for a multilayered system with Ms1¼Ms2¼780 em/cm3,Hk1¼Hk2¼5 Oe, hk1¼hk2¼90/C14, uk1¼uk2¼2/C14,a¼0.018, cG/2p¼2.9 MHz/Oe, t1¼100 nm, t2¼250 nm, r1¼6/C2107(Xm)/C01,r2¼r1/4,hH¼90/C14, and uH¼90/C14.243904-7 Corr ^eaet al. J. Appl. Phys. 116, 243904 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Wed, 12 Aug 2015 01:24:37range, where FMR signatures can be veriﬁed in the MI results. Figure 6presents the numerical calculations of the real Rand imaginary Xcomponents of the longitudinal imped- ance, as well as the impedance Z, for multilayered systems with different values of the damping parameter a. Here, it can be clearly noticed that the amplitude of R,X,a n d Z increases as the avalue decreases. Moreover, a displace- ment of the peak position in ﬁeld is observed when differ- entavalues are considered. This displacement leads to changes in the FMR frequency for a given external mag- netic ﬁeld and, therefore, it modiﬁes the frequency limit between the regimes where distinct mechanisms are re- sponsible for the MI effect vari ations. The features veriﬁed in these numerical calculations are present in experimentalresults measured in ﬁlms with low damping parameter a values.36 C. Exchange biased system Finally, we perform the numerical calculation for the longitudinal MI effect for an exchange biased system, as pre-sented in Fig. 1(d). The exchange biased system is composed by a ferro- magnetic layer directly coupled to an antiferromagneticlayer. The sample conﬁguration favors the appearance of theexchange interaction in the ferromagnetic/antiferromagnetic interface, described through a bias ﬁeld ~H EB.37Thus, the magnetic free energy density can be written as37 n¼/C0 ~M/C1~H/C0Hk 2Ms~M/C1^uk/C0/C12þ4pM2 s^M/C1^nðÞ /C0~M/C1~HEB:(21) For the numerical calculation for an exchange biased system, we consider the following parameters previouslyemployed: M s¼780 emu/cm3,Hk¼5 Oe, hk¼90/C14, variable uk;a¼0:018, cG/2p¼2.9 MHz/Oe, thickness of the ferro- magnetic layer t¼500 nm, hH¼90/C14, and uH¼90/C14. Beyond the traditional parameters, HEB¼50 Oe, oriented along ^uk. In particular, the thickness of the antiferromagnetic layer isnot considered for the numerical calculations. Figure 7shows the numerical calculations for the nor- malized magnetization curves and real Rand imaginary X components obtained as a function of the external magneticﬁeld at 2 GHz for two different orientations between ~H EB and^ukwith ~HandIac, together with the schematic represen- tations of the two conﬁgurations. In particular, in this case,the calculations are performed considering Eqs. (21),(8), and(17). Considering the magnetization curves (see Fig. 7(a)), the exchange bias can be clearly identiﬁed through the shiftof the curve, where the maximum exchange bias ﬁeld isobserved when ~Hk~H EB(Fig. 7(c)), as expected. As the angle between ~HEBand ^ukwith ~Hand Iacis increased, a reduction of the component of the exchange bias ﬁeld along ~His veriﬁed, evidenced by the decrease of the shift (Not shown here). For the limit case of ~H?~HEB(Fig. 7(d)), none FIG. 6. (a) Real Rcomponent, (b) imaginary Xcomponent, and (c) imped- ance Z, as a function of the external magnetic ﬁeld at 2 GHz, calculated for multilayered systems with different values of the damping parameter a. The numerical calculations are performed using the same parameters previously employed for multilayered systems. FIG. 5. (a) Real Rand (b) imaginary Xcomponents of the longitudinal im- pedance, as a function of the external magnetic ﬁeld at 2 GHz, calculated for multilayered systems with different values of the thickness t1of the metallic non-magnetic layer. For t1¼0, the multilayered system becomes a single layered system. The numerical calculations are performed using the same parameters previously employed for multilayered systems.243904-8 Corr ^eaet al. J. Appl. Phys. 116, 243904 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Wed, 12 Aug 2015 01:24:37shift of the curve is observed. At the same time, an evolution of the shape of the magnetization curve is noticed as theangle increases. These exchange bias features are reﬂected inthe behavior of the MI curves. In particular, the shift of thecurves of the real and imaginary components of the imped- ance follows the one of the respective magnetization curve (see Fig. 7(b)). Figure 8shows the numerical calculations for the real R and imaginary Xcomponents of the longitudinal impedance as a function of the external magnetic ﬁeld for selected fre-quency values, calculated for an exchange biased system forthe conﬁguration of ~H EBk~H. In order to consider the fre- quency dependent shift of Rand X, we consider a ﬁtting obtained from the data reported in Ref. 17. For exchange biased systems, the well-known symmet- ric magnetoimpedance behavior around H¼0 for anisotropic systems38is entirely shifted to H¼HEB.37Besides, the MI curves reﬂect all classical features of the magnetoimpedancein systems without the exchange bias, including the Zbehav- ior for distinct orientation between the anisotropy and exter- nal magnetic ﬁeld, 7as well as the R,X,andZbehavior with frequency,20,21,26together with the new features owed to exchange bias effect.37In this case, a single peak placed at H¼HEB6Hc, where Hcis the coercive ﬁeld, can be observed from 0.15 GHz (not presented here) up to 0.5 GHz,and it is due to changes in the transverse magnetic perme-ability. The single peak becomes more pronounced with theincrease of the frequency. At around 0.6 GHz, the singlepeak splits in a double peak structure symmetric at H¼H EB. In classical MI experiments, this evolution of the curves from a single peak to a double peak structure is veriﬁed when both the external magnetic ﬁeld and electrical currentare applied along the easy magnetization axis,38and is owed to the typical shape of FMR dispersion relation for thisgeometry. 7,26 These numerical calculation results obtained for exchange biased systems are in qualitative agreement with experimental results found in literature for ferromagnetic ﬁlms with exchange bias.11,12 D. Comparison with the experiment The previous tests performed with the theoretical approach have qualitatively described the main features ofsingle layered, multilayered, and exchange biased systems . To verify the validity of the theoretical approach, we investi-gate the quasi-static and dynamical magnetic properties of an exchange biased multilayered ﬁlm and compare the ex- perimental results with numerical calculations obtained an FIG. 7. (a) Normalized magnetization curves and (b) real Rand imaginary X components of the longitudinal impedance Zas a function of the external ﬁeld at 2 GHz, calculated for a exchange biased system when EA and ~HEBare par- allel ( uk¼88/C14) and perpendicular ( uk¼2/C14)t o ~HandIac. Schematic repre- sentation of an exchange biased system and two conﬁgurations of the external and alternate magnetic ﬁelds and current sense when the easy magnetizationaxis (EA) and ~H EBare (c) parallel and (d) perpendicular to ~HandIac. FIG. 8. (a) Real Rand (b) imaginary Xcomponents of the longitudinal im- pedance as a function of the external magnetic ﬁeld for selected frequency values calculated for an exchange biased system with the same parameter employed previously and uk¼88/C14.243904-9 Corr ^eaet al. J. Appl. Phys. 116, 243904 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Wed, 12 Aug 2015 01:24:37exchange biased multilayered system . The complexity of the considered system, including different features previously studied, and the quantitative agreement with experimentalresults do conﬁrm the robustness of our theoretical approach. We perform numerical calculation for the quasi-static and dynamical magnetic properties of an exchange biasedmultilayered system, as shown in Fig. 9(a). To model the exchange biased multilayered system, we consider a magnetic free energy density that can be writtenas n¼X 20 i¼1/C0~Mi/C1~H/C0Hki 2Msi~Mi/C1^uki/C0/C12 þ4pM2 si^Mi/C1^n/C0/C1 /C0~HEB/C1~Mi2 643 75: (22) With respect to numerical calculations, the following parameters must be deﬁned to describe the experimentalmagnetization and MI curves: magnetization and saturation magnetization of each ferromagnetic layer, Miand Msi, respectively, uniaxial anisotropy ﬁeld Hki, uniaxial anisot- ropy versor ^uki, exchange bias ﬁeld HEB, thicknesses, t1and t2, and conductivities, r1andr2, of the non-magnetic and ferromagnetic layers, respectively, damping parameter a, gyromagnetic factor cG, and external magnetic ﬁeld ~H. The thickness of the antiferromagnetic layer is not considered for the numerical calculations, however, experimentally, it isthick enough to neglect the bilinear and biquadratic coupling between the ferromagnetic layers. The calculation of the magnetization curves is carried out using the same minimization process developed for the MI calculation, without the ~h acﬁeld. This process consists in to determine the hManduMvalues that minimize the mag- netic free energy density for the studied system for each external magnetic ﬁeld value. Thus, since the calculated magnetization curve validates the experimental magnetiza-tion behavior, the aforementioned parameters are ﬁxed to perform the numerical calculations of MI behavior. As previ- ously cited, there is an offset increase in the real and imagi-nary parts of the experimental impedance as a function of frequency, a feature of the electrical/metallic contribution to MI that is not taken into account in theoretical models. Thus,it is inserted in the MI numerical calculation from the ﬁtting of the measured Rand Xcurves as a function of the fre- quency for the highest magnetic ﬁeld value. 21 Figure 9(b) shows the normalized magnetization curves of the produced exchange biased multilayered ﬁlm. Experimental magnetization curves are obtained along twodifferent directions, when ~His applied along and perpendic- ular to the main axis of the ﬁlms. It is important to point out that a constant magnetic ﬁeld is applied along the main axisduring the deposition process. As a matter of fact, by com- paring experimental curves, it is possible to observe that magnetic anisotropy is induced during the ﬁlm growth, con-ﬁrming an easy magnetization axis and an exchange bias ﬁeld oriented along the main axis of the ﬁlm. From the magnetization curve measured along the main axis of the ﬁlm, we ﬁnd the coercive ﬁeld /C242O e a n d H EB/C2430 Oe. Thus, to the numerical calculation, we con- sider the following parameters Msi¼780 emu/cm3, Hki¼2O e , hki¼90/C14, variable uki,HEB¼30.5 Oe with ~HEB oriented along ^uki,a¼0.018, cG/2p¼2.73 MHz/Oe,39 t1¼1n m , t2¼40 nm, r1¼6/C2107(Xm)/C01,a n d r2¼r1/0.5. Since hH¼90/C14anduH¼90/C14, we obtain uki¼85/C14and uki¼4/C14, respectively, for the two measurement directions. A small misalignment between anisotropy and ﬁeld can beassociated to stress stored in the ﬁlm as the sample thickness increases, as well as small deviations in the sample position in an experiment are reasonable. This is conﬁrmed throughthe numerical calculation of the magnetization curves. Notice the striking quantitative agreement between experi- ment and theory. As mentioned above, parameters are ﬁxed from the cal- culation of magnetization curves and used to describe the MI behavior. Thus, from Eqs. (19),(8), and (20), the real Rand imaginary Xcomponents of the longitudinal impedance as a function of ﬁeld and frequency for an exchange biased FIG. 9. (a) Schematic diagram of an exchange biased multilayered system. Experimentally, we produce a [Ni 20Fe80(40 nm)/Ir 20Mn 80(20 nm)/ Ta(1 nm)] /C220 ferromagnetic multilayered ﬁlm, in which the easy magnet- ization axis EA and the exchange bias ﬁeld ~HEBare oriented in the same direction. (b) Normalized magnetization curves obtained experimentally when ~His applied along (0/C14) and perpendicular (90/C14) to the main axis of the ﬁlm, together with numerical calculations performed for an exchange biased multilayered system.243904-10 Corr ^eaet al. J. Appl. Phys. 116, 243904 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Wed, 12 Aug 2015 01:24:37multilayered system can be calculated. Fig. 10shows experi- mental data and numerical calculation of RandXas a func- tion of Hfor selected frequency values for both considered directions. For all cases, it is evident the quantitative agreement between the experimental data and numerical calculation. In particular, the numerical calculations performed using theconsidered magnetic permeability and MI models, with pa- rameters ﬁxed by analyzing the magnetization curves, are able to describe all the main features of each impedancecomponent for the whole frequency range. Although it is well-known that the changes of magnetic permeability and impedance with magnetic ﬁelds at differentfrequency ranges are caused by three distinct mecha- nisms, 18–20the determination of the precise frequency limits between regimes is a hard task, since the overlap of contribu-tions to MI of distinct mechanisms, such as the skin and FMR effects, is very likely to occur. Thus, the use of distinct models for magnetic permeability and their use in calculatingMI become restricted, since it is not possible to determine when to leave one model and start using another one as the frequency is changing. Even there are distinct mechanisms controlling MI var- iations at different frequency ranges, all of our experimental ﬁndings are well described by the theoretical results calcu-lated using the aforementioned magnetic permeability and MI models. This is due to the fact that the distinctmechanism contributions at different frequency ranges are included naturally in the numerical calculation through mag-netic permeability. V. CONCLUSION As an alternative to the traditional FMR experiment, the magnetoimpedance effect corresponds as a promising tool toinvestigate ferromagnetic materials, revealing aspects on thefundamental physics associated to magnetization dynamics,broadband magnetic properties, important issues for current,and emerging technological applications for magnetic sen-sors, as well as insights on ferromagnetic resonance effect atsaturated and even unsaturated samples. In this sense, itsstudy in ferromagnetic samples with distinct featuresbecomes a very important task. In this paper, we perform a theoretical and experimental investigation of the magnetoimpedance effect for the thinﬁlm geometry at the high frequency range. In particular, we calculate the longitudinal magnetoim- pedance for single layered, multilayered, or exchange biasedsystems from an approach that considers a magnetic perme-ability model for planar geometry and the appropriate mag-netic free energy density for each structure. Usually,theoretical models that describe magnetization dynamical properties and the MI of a given system consider more than one approach to magnetic permeability. This is due to the FIG. 10. Experimental results and numerical calculation of real Rand imaginary Xcomponents of the longitudinal impedance as a function of the external magnetic ﬁeld for selected frequency values. (a)–(c) Experimental data obtained when ~His applied along (0/C14) to the main axis of the ﬁlm, together with numer- ical calculations performed for an exchange biased multilayered system with uki¼85/C14. (d)–(f) Similar plot of experimental data when the ﬁeld is perpendicu- lar (90/C14) to the main axis of the ﬁlm, with numerical calculations performed with uki¼4/C14.243904-11 Corr ^eaet al. J. Appl. Phys. 116, 243904 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 134.124.28.17 On: Wed, 12 Aug 2015 01:24:37fact that these permeability approaches reﬂect distinct mech- anisms responsible for MI changes, applicable only for a re- stricted range of frequencies, where the mechanism isobserved. Thus, the difﬁcult task of choosing the correct magnetic permeability model to use at a certain frequency range explains the reduced number of reports comparing MIexperimental results and theoretical predictions for a wide frequency range. Anyway, even there are distinct mecha- nisms controlling MI variations at different frequencyranges, with the magnetic permeability and MI models con- sidered here, the distinct mechanism contributions at differ- ent frequency ranges are included naturally in the numericalcalculation through magnetic permeability. For this reason, the numerical calculations for different systems succeed to describe the main features of the MI effect in each structure,in concordance with experimental results found in literature. At the same time, we perform experimental magnetiza- tion and MI measurements in a multilayered ﬁlm withexchange bias. To interpret them, numerical calculations are performed using the described magnetic permeability and MI models. With parameters ﬁxed by analyzing the magnetizationcurves, quantitative agreement between the experimental MI data and numerical calculation is veriﬁed, and we are able to describe all the main features of each impedance componentfor the whole frequency range. Thus, we provide experimental evidence to conﬁrm the validity of the theoretical approach to describe the magnetoimpedance in ferromagnetic ﬁlms. Although we perform here all the analysis just for an exchange biased multilayered ﬁlm, since a general model is used to describe magnetic permeability, it can be consideredin the study of samples with any planar geometry, such as ﬁlms, ribbons, and sheets, given that an appropriate magnetic free energy density and adequate MI model are considered.In this sense, the simplicity and robustness place this theoret- ical approach as a powerful tool to investigate the permeabil- ity and longitudinal magnetoimpedance for the thin ﬁlmgeometry in a wide frequency range. In particular, we focus on the l xxterm of the magnetic permeability tensor and on the longitudinal magnetoimpe-dance. This is due to the fact that our experimental setup pro- vides information related to the transverse magnetic permeability. On the other hand, the l yy,lzz, and off-diagonal terms of the magnetic permeability tensor can bring relevant information on the MI effect, since the correct impedance expression is obtained. At the same time, this information canbe measured by considering a distinct experimental system. These next steps are currently in progress. ACKNOWLEDGMENTS The research was partially supported by the Brazilian agencies CNPq (Grant Nos. 471302/2013-9, 310761/2011-5, 476429/2010-2, and 555620/2010-7), CAPES, FAPERJ, and FAPERN (Grant PPP No. 013/2009, and Pronem No. 03/2012). M.A.C. and F.B. acknowledge ﬁnancial support of the INCT of Space Studies.1S. Chikazumi, Physics of Magnetism (Wiley, New York, 1964). 2R. L. Sommer and C. L. Chien, Appl. Phys. Lett. 67, 3346 (1995). 3J. P. Sinnecker, P. Tiberto, G. V. Kurlyandskaia, E. H. C. P. Sinnecker, M. V/C19azquez, and A. Hernando, J. Appl. Phys. 84, 5814 (1998). 4G. Kurlyandskaya, J. Magn. Magn. Mater. 321, 659 (2009). 5E. Fern /C19andez, G. V. 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1.5129724.pdf	AIP Advances 9, 125332 (2019); https://doi.org/10.1063/1.5129724 9, 125332 © 2019 Author(s).Dual-structure microwave-assisted magnetic recording using only a spin torque oscillator Cite as: AIP Advances 9, 125332 (2019); https://doi.org/10.1063/1.5129724 Submitted: 01 October 2019 . Accepted: 01 November 2019 . Published Online: 27 December 2019 Simon John Greaves , and Waka Saito COLLECTIONS Paper published as part of the special topic on 64th Annual Conference on Magnetism and Magnetic Materials Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials. ARTICLES YOU MAY BE INTERESTED IN Demonstration of spin transfer torque (STT) magnetic recording Applied Physics Letters 114, 243101 (2019); https://doi.org/10.1063/1.5097546 Synchronization of spin-torque oscillators via spin pumping AIP Advances 9, 035310 (2019); https://doi.org/10.1063/1.5066560 Microwave-magnetic-field-induced magnetization excitation and assisted switching of antiferromagnetically coupled magnetic bilayer with perpendicular magnetization Journal of Applied Physics 125, 153901 (2019); https://doi.org/10.1063/1.5089799AIP Advances ARTICLE scitation.org/journal/adv Dual-structure microwave-assisted magnetic recording using only a spin torque oscillator Cite as: AIP Advances 9, 125332 (2019); doi: 10.1063/1.5129724 Presented: 8 November 2019 •Submitted: 1 October 2019 • Accepted: 1 November 2019 •Published Online: 27 December 2019 Simon John Greavesa) and Waka Saito AFFILIATIONS RIEC, Tohoku University, Katahira 2-1-1, Aoba ku, Sendai 980-8577, Japan Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials. a)simon@riec.tohoku.ac.jp ABSTRACT The selective switching of dual-structure magnetic dots under the influence of the stray field from a spin torque oscillator was investigated. A configuration was found which allowed selective switching of either structure when subject to ac magnetic fields oscillating at 9 GHz and 20 GHz. No other external magnetic fields were needed to switch the magnetisation of the structures. ©2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5129724 .,s I. INTRODUCTION Microwave-assisted magnetic recording (MAMR) makes use of high frequency (HF) magnetic fields which, when oscillating at, or near, the resonance frequency of a magnetic material, reduce the switching field of the magnetic material.1When the HF field is applied in conjunction with the field from a write head magnetic grains with much higher uniaxial anisotropy, Ku, can be switched than when using the head field alone. The HF field is usually generated by a spin torque oscillator (STO).2However, it is difficult to obtain stable STO oscillation when the STO is integrated into a write head due to the large fields acting on the STO.3In some cases it is possible to switch the magnetisation of a recording medium or magnetic dot using only the field from the STO and with no other external field sources.4–6This is true even if the coercivity or switching field of the magnetic dot is much higher than the strength of the HF field from the STO. The sense of rotation of the HF field, or chirality, determines the direction of magnetisa- tion switching, e.g., from up to down, or vice-versa,7and this can be controlled by the direction in which the current flows through the device.8 Another advantage of MAMR is the ability to record on media or dots with multiple recording structures.9–11If each structure has a different ferromagnetic resonance frequency the magnetisa- tion of each structure can be switched independently. For media or dots with two structures this theoretically allows the recordingdensity to be doubled. In this work we investigate the possibility of selective recording on dual structure magnetic dots using only the field from a STO. No write head or any other external applied fields were used. II. THE MODEL A simplified model of a spin torque oscillator consisting of just a field generating layer (FGL) was used in this work. The FGL was modelled as a uniformly-magnetised cuboid with a thickness ( z) of 10 nm and in-plane ( x,y) dimensions of 20 nm ×20 nm. The FGL had Msof 1591 emu/cm3and the magnetisation was assumed to rotate in the x-yplane at a constant angular velocity. The stray field arising from the FGL was calculated underneath the FGL and varied as a function of position and time. Two discrete, cylindrical magnetic data storage structures, RL1 and RL2, were centred underneath the STO, with RL1 being fur- thest from the STO along the zaxis. The saturation magnetisa- tion of the two structures was 750 emu/cm3and their diameter, D, was 20 nm. When D≤12 nm discretisation of the structures into 1 nm thick layers was sufficient to capture the magnetisa- tion dynamics, but for larger diameters the magnetisation reversal became non-uniform and in-plane discretisation was also necessary. As a result the structures were discretised into 1 nm cubes with exchange coupling of 1 ×10−6erg/cm acting between neighbouring cubes. In some cases exchange coupled composite (ECC) structures AIP Advances 9, 125332 (2019); doi: 10.1063/1.5129724 9, 125332-1 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv were used.12,13These comprised of magnetically hard and soft lay- ers exchange coupled together with a strength of 5 ×10−7erg/cm. The Landau-Lifshitz-Gilbert damping constant, α, was 0.02 for all structures. The switching probability of the structures was calculated at a temperature of 300 K. At the start of the simulations the magnetisa- tion of the structures was pointing up ( ⃗Malong the positive zaxis). After waiting for 0.25 ns in zero field the STO was turned on for 2 ns. Once the STO was turned off the simulation continued in zero field for a further 1 ns after which time the magnetisation of the struc- tures was evaluated. The main variables were the frequency of the HF field, the spacing between the structures and the STO ( dz), and the uniaxial anisotropy of the structures. The switching probabilities given here were evaluated after 100 trials. III. SINGLE STRUCTURE DOTS First the switching of single structures was investigated. The HF field from the STO decreased rapidly with distance from the STO surface. The structure nearest to the STO (RL2) experienced higher HF fields and should therefore have higher Kuthan the structure further away from the STO (RL1). If two recording structures are to be switched by the same STO both structures should be as thin as possible. However, a minimum structure thickness of 3 nm was imposed in order that thermally stable structures could be realised using realistic values of Ku. Fig. 1 shows the switching probabilities of single phase struc- tures with various values of Kuwhere the spacing between the STO and the structures was 1 nm. Switching probabilities of 1 were achieved for Kuvalues from 4.6 ×106erg/cm3to 8.1 ×106erg/cm3. The switching probability curves shifted to higher STO frequen- cies as Kuincreased as the resonance frequency of a single phase structure is proportional to the anisotropy field Hk. For selective switching of two structures to be possible the left edge of the switching probability curve for RL2 should reach zero at a sufficiently high STO frequency, e.g., 10 GHz. Fig. 1 shows that to achieve this condition Kushould be at least 7.1 ×106erg/cm3. A design for RL1 is then needed which switches at a STO frequency below 10 GHz. Fig. 2 shows the effect of the spacing between the STO and the recording structure on the switching probability when Kuwas 6.6×106erg/cm3. The switching probability dropped below 1 once FIG. 1 . Switching probability vs. STO frequency for 3 nm thick recording structures with various Ku. Spacing between recording structure and STO dz= 1 nm. FIG. 2 . Switching probability vs. STO frequency for 3 nm thick recording structures. dz= spacing between recording structure and STO. FIG. 3 . Switching probability of RL1 for a 3 nm hard layer and various soft layer thicknesses. dz= 6 nm, Kusoft = 1 ×106erg/cm3. dzexceeded 3 nm. Given the minimum structure thickness of 3 nm and a minimum spacing between structures of 1 nm, the smallest possible value of dzfor the structure furthest from the STO (RL1) is 5 nm. The switching probability could be increased by reducing Ku, e.g., the switching probability was 1 at 8 GHz when dz= 5 nm and Ku= 4.6 ×106erg/cm3, but a more flexible approach is to use an ECC structure for RL1. Fig. 3 shows switching probabilities of single phase and ECC structures for dz= 6 nm and hard layer Kuof 6.1 ×106erg/cm3. Adding a soft layer with Kuof 1×106erg/cm3and a thickness of 2 nm, or more, shifted the switching probability curves to lower STO frequencies whilst maintaining a maximum switching probability of 1. The right edges of the switching probability curves for soft layer thicknesses of 2 nm and 3 nm reached zero at a frequency of around 15 GHz. This can be below the peak in the switching probability curve of RL2 if RL2 has Kuof 7.6 ×106erg/cm3, or more. IV. DUAL-STRUCTURE DOTS Based on the results in section III the two structure design shown in Fig. 4 was adopted. RL1 had an ECC structure with a 3 nm thick hard layer and 2 nm soft layer. The spacing between the two structures was 2 nm. Fig. 5(a) shows hysteresis loops of a dual structure dot calcu- lated at 4.2 K and 300 K. The loops show that the magnetisation AIP Advances 9, 125332 (2019); doi: 10.1063/1.5129724 9, 125332-2 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv FIG. 4 . Schematic of dual-structure dot and STO. of the two recording structures reversed independently at distinctly different applied fields, both at low temperature and at room tem- perature. Fig. 5(b) shows the magnitude of the in-plane component of the HF field from the STO at various distances from the STO sur- face. The fields shown were those at the centre of the dot and at the edge of the dot (10 nm from the centre), and were much smaller than the switching fields indicated by the hysteresis loops in Fig. 5(a). The switching probabilities of RL1 and RL2 in a dual-structure dot as a function of the STO frequency are shown in Fig. 6. In this calculation the magnetisation of the two structures was initially parallel. The thin lines show the switching probabilities for single structure dots whilst the bold lines and points are the results for a dual-structure dot. Compared with the single structure dots the switching probability curves of the dual-structure dot were narrower and the maximum switching probability decreased from 1 to 0.97 for RL1 and to 0.95 for RL2. The reduction was a consequence of FIG. 5 . (a) Hysteresis loops of a dual structure dot at 4.2 K and 300 K. (b) In-plane component of HF field at centre and edge of dot vs. vertical distance from STO. FIG. 6 . Switching probabilities of RL1 and RL2 in a dual-structure dot. Thin lines: single structure dots, bold lines and points: dual-structure dot. the magnetostatic interaction between the two recording structures which favoured the initial, parallel magnetisation alignment. Fig. 6 shows that for a dual-structure dot the maximum switch- ing probabilities for RL1 and RL2 occurred at STO frequencies of 9 GHz and 20 GHz, respectively. Fig. 7 shows some typical mag- netisation dynamics for RL1 and RL2 under HF fields at these fre- quencies. When the HF field was turned on the magnetisation of the target structure decreased and began to oscillate around Mz= 0 after about 0.5 ns. Subsequently the magnetisation gradually became increasingly negative over the following 0.5 - 1 ns. The magneti- sation of the non-target structure was slightly disturbed by the HF field in both cases but no correlation between the oscillation of the magnetisation of the two structures was observed. As can be seen in Fig. 7, the oscillation frequencies of the two structures when subjected to the same HF field were quite different. Magnetisation reversal took place via domain wall nucleation and motion. In addition to lateral motion of the domain wall, which was responsible for the magnetisation oscillations shown in Fig. 7, the domain wall also rotated multiple times during the reversal pro- cess. The speed of rotation was not constant, but the average fre- quency of rotation was about 12 GHz for RL1 when the HF field frequency was 9 GHZ. For RL2 the domain wall rotation frequency was about 22 GHz for a 20 GHz HF field. FIG. 7 . Switching of RL1 and RL2 when subject to HF fields of 9 GHz (top) and 20 GHz (bottom). 2 ns HF field pulse. AIP Advances 9, 125332 (2019); doi: 10.1063/1.5129724 9, 125332-3 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv FIG. 8 . Effect of HF pulse duration on switching probabilities of RL1 and RL2 in a two structure stack. FIG. 9 . Switching probabilities for anti-parallel initial magnetisation state. Pulse duration = 2 ns. Thin lines = switching probabilities for parallel initial magnetisation state. The effect of the HF field pulse duration was examined, vary- ing the pulse length from 0.5 ns to 4 ns. Fig. 8 shows the results. For pulse durations of less than 2 ns the switching probabilities decreased rapidly. For pulses longer than 2 ns the maximum switch- ing probability slowly increased, reaching 1 for a pulse length of 4 ns. As the pulse length increased the probability of RL1 switching at higher STO frequencies also increased. For example, when the STO frequency was 15 GHz the switching probability of RL1 increased from zero to 0.03 when the pulse length was extended from 2 ns to 4 ns. In a storage system the switching error rate (SER) determines the ultimate capacity of the system.14The SER of RL1 and RL2 at 9 GHz and 20 GHz was calculated for 1000 trials using 4 ns pulses. The SER was 0.001 for RL1 and 0.01 for RL2. Thus, the capacity of a system of Ndots would be 0.99 Nfor RL1 and 0.92 Nfor RL2.The switching probabilities for the anti-parallel initial magneti- sation state are shown in Fig. 9 for 2 ns HF field pulses. In contrast to the parallel initial magnetisation state the magnetostatic inter- action assisted magnetisation reversal of the target structure and the switching probability reached 1 over a wide range of STO fre- quencies. It should be noted that although the switching probability curves for RL1 and RL2 appear to overlap in the frequency range of 9 - 13 GHz, only the magnetisation of the target structure was switched in any given simulation. This was because when the ini- tial magnetisation state was anti-parallel only one of the structures responded to the HF field. The magnetisation of the other struc- ture precessed in the opposite direction and could not absorb energy from the HF field. V. CONCLUSIONS A design for a two structure dot in which either structure could be selectively switched by the application of a HF field from a STO at an appropriate frequency was shown. Switching probabilities of 1 for the target structure and 0 for the non-target structure were obtained after 100 trials for HF field pulses of 4 ns duration. ACKNOWLEDGMENTS The authors would like to thank the Advanced Storage Research Consortium for their support of this work. REFERENCES 1J. G. Zhu, X. Zhu, and Y. Tang, IEEE Trans. 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